



"""simple docstring"""
import logging

import numpy as np
import pytest
from scipy.linalg import eigh

logging.basicConfig(level=logging.INFO, format="%(message)s")


def 	_SCREAMING_SNAKE_CASE  (__lowerCAmelCase							)       ->  Dict:







  '''simple docstring'''


  return input_array.reshape((input_array.size, 1)							)


def 	_SCREAMING_SNAKE_CASE  (__lowerCAmelCase       ,     __lowerCAmelCase       ,     __lowerCAmelCase							)       ->  Tuple:







  '''simple docstring'''


  lowercase_	    =       np.nan
  for i in range(__lowerCAmelCase							):
    lowercase_	    =       features[:, labels == i]
    lowercase_	    =       data.mean(1							)
    # Centralize the data of class i
    lowercase_	    =       data - column_reshape(__lowerCAmelCase							)
    if i > 0:
      # If covariance_sum is not None
      covariance_sum += np.dot(__lowerCAmelCase       ,     centered_data.T							)
    else:
      # If covariance_sum is np.nan (i.e. first loop)
      lowercase_	    =       np.dot(__lowerCAmelCase       ,     centered_data.T							)

  return covariance_sum / features.shape[1]


def 	_SCREAMING_SNAKE_CASE  (__lowerCAmelCase       ,     __lowerCAmelCase       ,     __lowerCAmelCase							)       ->  Dict:







  '''simple docstring'''


  lowercase_	    =       features.mean(1							)
  lowercase_	    =       np.nan
  for i in range(__lowerCAmelCase							):
    lowercase_	    =       features[:, labels == i]
    lowercase_	    =       data.shape[1]
    lowercase_	    =       data.mean(1							)
    if i > 0:
      # If covariance_sum is not None
      covariance_sum += device_data * np.dot(
          column_reshape(__lowerCAmelCase							) - column_reshape(__lowerCAmelCase							)       ,     (column_reshape(__lowerCAmelCase							) - column_reshape(__lowerCAmelCase							)).T       ,     )
    else:
      # If covariance_sum is np.nan (i.e. first loop)
      lowercase_	    =       device_data * np.dot(
          column_reshape(__lowerCAmelCase							) - column_reshape(__lowerCAmelCase							)       ,     (column_reshape(__lowerCAmelCase							) - column_reshape(__lowerCAmelCase							)).T       ,     )

  return covariance_sum / features.shape[1]


def 	_SCREAMING_SNAKE_CASE  (__lowerCAmelCase       ,     __lowerCAmelCase							)       ->  int:







  '''simple docstring'''


  if features.any():
    lowercase_	    =       features.mean(1							)
    # Center the dataset
    lowercase_	    =       features - np.reshape(__lowerCAmelCase       ,     (data_mean.size, 1)							)
    lowercase_	    =       np.dot(__lowerCAmelCase       ,     centered_data.T							) / features.shape[1]
    lowercase_	    =       np.linalg.eigh(__lowerCAmelCase							)
    # Take all the columns in the reverse order (-1), and then takes only the first
    lowercase_	    =       eigenvectors[:, ::-1][:, 0:dimensions]
    # Project the database on the new space
    lowercase_	    =       np.dot(filtered_eigenvectors.T       ,     __lowerCAmelCase							)
    logging.info("""Principal Component Analysis computed"""							)

    return projected_data
  else:
    logging.basicConfig(level=logging.ERROR       ,     format="""%(message)s"""       ,     force=__lowerCAmelCase							)
    logging.error("""Dataset empty"""							)
    raise AssertionError


def 	_SCREAMING_SNAKE_CASE  (__lowerCAmelCase       ,     __lowerCAmelCase       ,     __lowerCAmelCase       ,     __lowerCAmelCase							)       ->  int:







  '''simple docstring'''


  assert classes > dimensions

  # Check if features have been already loaded
  if features.any:
    lowercase_	    =       eigh(
        covariance_between_classes(__lowerCAmelCase       ,     __lowerCAmelCase       ,     __lowerCAmelCase							)       ,     covariance_within_classes(__lowerCAmelCase       ,     __lowerCAmelCase       ,     __lowerCAmelCase							)       ,     )
    lowercase_	    =       eigenvectors[:, ::-1][:, :dimensions]
    lowercase_	    =       np.linalg.svd(__lowerCAmelCase							)
    lowercase_	    =       svd_matrix[:, 0:dimensions]
    lowercase_	    =       np.dot(filtered_svd_matrix.T       ,     __lowerCAmelCase							)
    logging.info("""Linear Discriminant Analysis computed"""							)

    return projected_data
  else:
    logging.basicConfig(level=logging.ERROR       ,     format="""%(message)s"""       ,     force=__lowerCAmelCase							)
    logging.error("""Dataset empty"""							)
    raise AssertionError


def 	_SCREAMING_SNAKE_CASE  ()       ->  Optional[Any]:







  '''simple docstring'''


  lowercase_	    =       np.array([[1, 2, 3, 4, 5], [2, 3, 4, 5, 6], [3, 4, 5, 6, 7]]							)
  lowercase_	    =       np.array([0, 0, 0, 1, 1]							)
  lowercase_	    =       2
  lowercase_	    =       2

  # Assert that the function raises an AssertionError if dimensions > classes
  with pytest.raises(__lowerCAmelCase							) as error_info:
    lowercase_	    =       linear_discriminant_analysis(
        __lowerCAmelCase       ,     __lowerCAmelCase       ,     __lowerCAmelCase       ,     __lowerCAmelCase							)
    if isinstance(__lowerCAmelCase       ,     np.ndarray							):
      raise AssertionError(
          """Did not raise AssertionError for dimensions > classes"""							)
    assert error_info.type is AssertionError


def 	_SCREAMING_SNAKE_CASE  ()       ->  List[str]:







  '''simple docstring'''


  lowercase_	    =       np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]]							)
  lowercase_	    =       2
  lowercase_	    =       np.array([[6.92_820_323, 8.66_025_404, 10.39_23_04_85], [3.0, 3.0, 3.0]]							)

  with pytest.raises(__lowerCAmelCase							) as error_info:
    lowercase_	    =       principal_component_analysis(__lowerCAmelCase       ,     __lowerCAmelCase							)
    if not np.allclose(__lowerCAmelCase       ,     __lowerCAmelCase							):
      raise AssertionError
    assert error_info.type is AssertionError


if __name__ == "__main__":
       import doctest

       doctest.testmod()


'''simple docstring'''

from math import pow
def 			__UpperCamelCase       (   UpperCAmelCase		,     UpperCAmelCase		,     UpperCAmelCase		,     UpperCAmelCase		,     UpperCAmelCase		,     ):
 if current_sum == needed_sum:
  # If the sum of the powers is equal to needed_sum, then we have a solution.
  solutions_count += 1
  return current_sum, solutions_count

 lowercase__				:    Optional[Any]  =					int(pow(UpperCAmelCase		,     UpperCAmelCase     )     )
 if current_sum + i_to_n <= needed_sum:
  # If the sum of the powers is less than needed_sum, then continue adding powers.
  current_sum += i_to_n
  lowercase__ ,							lowercase__				:    Dict  =					backtrack(
      UpperCAmelCase		,     UpperCAmelCase		,     current_number + 1		,     UpperCAmelCase		,     UpperCAmelCase     )
  current_sum -= i_to_n
 if i_to_n < needed_sum:
  # If the power of i is less than needed_sum, then try with the next power.
  lowercase__ ,							lowercase__				:    str  =					backtrack(
      UpperCAmelCase		,     UpperCAmelCase		,     current_number + 1		,     UpperCAmelCase		,     UpperCAmelCase     )
 return current_sum, solutions_count
def 			__UpperCamelCase       (   UpperCAmelCase		,     UpperCAmelCase     ):
 if not (1 <= needed_sum <= 1000 and 2 <= power <= 10):
  raise ValueError(
      '''Invalid input\n'''
      '''needed_sum must be between 1 and 1000, power between 2 and 10.'''     )

 return backtrack(UpperCAmelCase		,     UpperCAmelCase		,     1		,     0		,     0     )[1]  # Return the solutions_count


if __name__ == "__main__":
  import doctest

  doctest.testmod()

