





import logging

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

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




def     __UpperCamelCase						(_SCREAMING_SNAKE_CASE	)       ->     np.ndarray:
  return input_array.reshape((input_array.size, 1)	)




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

  return covariance_sum / features.shape[1]




def     __UpperCamelCase						(_SCREAMING_SNAKE_CASE	,		_SCREAMING_SNAKE_CASE	,		_SCREAMING_SNAKE_CASE	)       ->     np.ndarray:
  lowercase__			    =						features.mean(1	)
  lowercase__			    =						np.nan
  for i in range(_SCREAMING_SNAKE_CASE	):
    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(_SCREAMING_SNAKE_CASE	) - column_reshape(_SCREAMING_SNAKE_CASE	)	,		(column_reshape(_SCREAMING_SNAKE_CASE	) - column_reshape(_SCREAMING_SNAKE_CASE	)).T	,		)
    else:
      # If covariance_sum is np.nan (i.e. first loop)
      lowercase__			    =						device_data * np.dot(
          column_reshape(_SCREAMING_SNAKE_CASE	) - column_reshape(_SCREAMING_SNAKE_CASE	)	,		(column_reshape(_SCREAMING_SNAKE_CASE	) - column_reshape(_SCREAMING_SNAKE_CASE	)).T	,		)

  return covariance_sum / features.shape[1]




def     __UpperCamelCase						(_SCREAMING_SNAKE_CASE	,		_SCREAMING_SNAKE_CASE	)       ->     np.ndarray:

  # Check if the features have been loaded
  if features.any():
    lowercase__			    =						features.mean(1	)
    # Center the dataset
    lowercase__			    =						features - np.reshape(_SCREAMING_SNAKE_CASE	,		(data_mean.size, 1)	)
    lowercase__			    =						np.dot(_SCREAMING_SNAKE_CASE	,		centered_data.T	) / features.shape[1]
    lowercase__    ,					lowercase__			    =						np.linalg.eigh(_SCREAMING_SNAKE_CASE	)
    # 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	,		_SCREAMING_SNAKE_CASE	)
    logging.info('Principal Component Analysis computed'	)

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




def     __UpperCamelCase						(_SCREAMING_SNAKE_CASE	,		_SCREAMING_SNAKE_CASE	,		_SCREAMING_SNAKE_CASE	,		_SCREAMING_SNAKE_CASE	)       ->     np.ndarray:
  assert classes > dimensions

  # Check if features have been already loaded
  if features.any:
    lowercase__    ,					lowercase__			    =						eigh(
        covariance_between_classes(_SCREAMING_SNAKE_CASE	,		_SCREAMING_SNAKE_CASE	,		_SCREAMING_SNAKE_CASE	)	,		covariance_within_classes(_SCREAMING_SNAKE_CASE	,		_SCREAMING_SNAKE_CASE	,		_SCREAMING_SNAKE_CASE	)	,		)
    lowercase__			    =						eigenvectors[:, ::-1][:, :dimensions]
    lowercase__    ,					lowercase__    ,					lowercase__			    =						np.linalg.svd(_SCREAMING_SNAKE_CASE	)
    lowercase__			    =						svd_matrix[:, 0:dimensions]
    lowercase__			    =						np.dot(filtered_svd_matrix.T	,		_SCREAMING_SNAKE_CASE	)
    logging.info('Linear Discriminant Analysis computed'	)

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




def     __UpperCamelCase						()       ->     None:
  # Create dummy dataset with 2 classes and 3 features
  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(_SCREAMING_SNAKE_CASE	) as error_info:
    lowercase__			    =						linear_discriminant_analysis(
        _SCREAMING_SNAKE_CASE	,		_SCREAMING_SNAKE_CASE	,		_SCREAMING_SNAKE_CASE	,		_SCREAMING_SNAKE_CASE	)
    if isinstance(_SCREAMING_SNAKE_CASE	,		np.ndarray	):
      raise AssertionError(
          'Did not raise AssertionError for dimensions > classes'	)
    assert error_info.type is AssertionError




def     __UpperCamelCase						()       ->     None:
  lowercase__			    =						np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]]	)
  lowercase__			    =						2
  lowercase__			    =						np.array([[6.9_2_8_2_0_3_2_3, 8.6_6_0_2_5_4_0_4, 10.39230485], [3.0, 3.0, 3.0]]	)

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


if __name__ == "__main__":
      import doctest

      doctest.testmod()









def     __UpperCamelCase						(_SCREAMING_SNAKE_CASE = 50	)       ->     int:
  lowercase__			    =						[1] * (length + 1)

  for row_length in range(3	,		length + 1	):
    for block_length in range(3	,		row_length + 1	):
      for block_start in range(row_length - block_length	):
        ways_number[row_length] += ways_number[
            row_length - block_start - block_length - 1
        ]

      ways_number[row_length] += 1

  return ways_number[length]


if __name__ == "__main__":
      print(f'''{solution() = }''')


