code: import matplotlib.pyplot as plt
  import numpy as np
  def 		snake_case__  (		lowercase       ,							lowercase       ,							lowercase				):
     lowerCAmelCase_:       Union[str, Any]         =  len(lowercase				)  # Number of training samples
     lowerCAmelCase_:       Dict         =  np.eye(lowercase				)  # Initialize weights as identity matrix
     for j in range(lowercase				):
        lowerCAmelCase_:       Optional[Any]         =  point - x_train[j]
        lowerCAmelCase_:       Dict         =  np.exp(diff @ diff.T / (-2.0 * tau**2)				)
  
     return weights
  def 		snake_case__  (		lowercase       ,							lowercase       ,							lowercase       ,							lowercase				):
     lowerCAmelCase_:       str         =  weight_matrix(lowercase       ,							lowercase       ,							lowercase				)
     lowerCAmelCase_:       str         =  np.linalg.inv(x_train.T @ weight_mat @ x_train				) @ (
         x_train.T @ weight_mat @ y_train.T
     )
  
     return weight
  def 		snake_case__  (		lowercase       ,							lowercase       ,							lowercase				):
     lowerCAmelCase_:       List[Any]         =  np.zeros(len(lowercase				)				)  # Initialize array of predictions
     for i, item in enumerate(lowercase				):
        lowerCAmelCase_:       Optional[Any]         =  item @ local_weight(lowercase       ,							lowercase       ,							lowercase       ,							lowercase				)
  
     return y_pred
  def 		snake_case__  (		lowercase       ,							lowercase       ,							lowercase				):
     import seaborn as sns
  
     lowerCAmelCase_:       Any         =  sns.load_dataset(lowercase				)
     lowerCAmelCase_:       int         =  np.array(data[x_name]				)
     lowerCAmelCase_:       str         =  np.array(data[y_name]				)
  
     lowerCAmelCase_:       str         =  np.ones(len(lowercase				)				)
  
     # pairing elements of one and x_data
     lowerCAmelCase_:       int         =  np.column_stack((one, x_data)				)
  
     return x_train, x_data, y_data
  def 		snake_case__  (		lowercase       ,							lowercase       ,							lowercase       ,							lowercase       ,							lowercase       ,							lowercase       ,							):
     lowerCAmelCase_:       Union[str, Any]         =  np.sort(lowercase       ,							axis=0				)
     plt.scatter(lowercase       ,							lowercase       ,							color="blue"				)
     plt.plot(
         x_train_sorted[:, 1]       ,							preds[x_train[:, 1].argsort(0				)]       ,							color="yellow"       ,							linewidth=5       ,							)
     plt.title("Local Weighted Regression"				)
     plt.xlabel(lowercase				)
     plt.ylabel(lowercase				)
     plt.show()
  
  
  if __name__ == "__main__":
        import doctest
  
        doctest.testmod()
  
        # Demo with a dataset from the seaborn module
        a     :			List[Any]					    =						load_data("""tips""", """total_bill""", """tip""")
        a     :			List[Any]					    =						local_weight_regression(training_data_x, tip, 5)
        plot_preds(training_data_x, predictions, total_bill, tip, """total_bill""", """tip""")
style_context: import random
  
  class 				_lowercase						:
  
  
  
  
  
  
      '''simple docstring'''
  
  
  
  
      def __init__(						self					,	lowerCamelCase__					,	lowerCamelCase__   ):
         lowerCAmelCase_:       int         =  x
         lowerCAmelCase_:       List[str]         =  y
  
  
  
  
      def     _a						(						self   ):
         return (self.x**2 + self.y**2) <= 1
  
  
  
  
      @classmethod
      def     _a						(						cls   ):
         return cls(x=random.random()					,	y=random.random()   )
  
  
  
  
  
  def 		snake_case__  (		lowercase				):
     if number_of_simulations < 1:
        raise ValueError("At least one simulation is necessary to estimate PI."				)
  
     lowerCAmelCase_:       List[Any]         =  0
     for _ in range(lowercase				):
        lowerCAmelCase_:       Tuple         =  Point.random_unit_square()
  
        if random_point.is_in_unit_circle():
           number_in_unit_circle += 1
  
     return 4 * number_in_unit_circle / number_of_simulations
  
  
  if __name__ == "__main__":
   # import doctest
  
   # doctest.testmod()
   from math import pi
  
   a     :			Union[str, Any]					    =						"""Please enter the desired number of Monte Carlo simulations: """
   a     :			int					    =						estimate_pi(int(input(prompt).strip()))
   print(f'''An estimate of PI is {my_pi} with an error of {abs(my_pi - pi)}''')