






from numpy import exp, pi, sqrt





def  _lowerCamelCase						(						__lowerCamelCase ,							__lowerCamelCase = 0.0 ,							__lowerCamelCase = 1.0       ) ->			int:

     '''simple docstring'''
     return 1 / sqrt(2 * pi * sigma**2       ) * exp(-((x - mu) ** 2) / (2 * sigma**2)       )


if __name__ == "__main__":
      import doctest

      doctest.testmod()









import logging

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

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



def 			_A	(      SCREAMING_SNAKE_CASE      ):
	return input_array.reshape((input_array.size, 1)      )



def 			_A	(      SCREAMING_SNAKE_CASE	,SCREAMING_SNAKE_CASE	,SCREAMING_SNAKE_CASE      ):
	UpperCAmelCase__:	int									= np.nan
	for i in range(SCREAMING_SNAKE_CASE      ):
		UpperCAmelCase__:	Tuple									= features[:, labels == i]
		UpperCAmelCase__:	int									= data.mean(1      )
		# Centralize the data of class i
		UpperCAmelCase__:	Optional[Any]									= 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)
			UpperCAmelCase__:	Dict									= np.dot(SCREAMING_SNAKE_CASE	,centered_data.T      )

	return covariance_sum / features.shape[1]



def 			_A	(      SCREAMING_SNAKE_CASE	,SCREAMING_SNAKE_CASE	,SCREAMING_SNAKE_CASE      ):
	UpperCAmelCase__:	int									= features.mean(1      )
	UpperCAmelCase__:	Optional[Any]									= np.nan
	for i in range(SCREAMING_SNAKE_CASE      ):
		UpperCAmelCase__:	List[Any]									= features[:, labels == i]
		UpperCAmelCase__:	Optional[Any]									= data.shape[1]
		UpperCAmelCase__:	List[Any]									= 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)
			UpperCAmelCase__:	Tuple									= 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 			_A	(      SCREAMING_SNAKE_CASE	,SCREAMING_SNAKE_CASE      ):

	# Check if the features have been loaded
	if features.any():
		UpperCAmelCase__:	Tuple									= features.mean(1      )
		# Center the dataset
		UpperCAmelCase__:	List[Any]									= features - np.reshape(SCREAMING_SNAKE_CASE	,(data_mean.size, 1)      )
		UpperCAmelCase__:	Optional[int]									= np.dot(SCREAMING_SNAKE_CASE	,centered_data.T      ) / features.shape[1]
		UpperCAmelCase__							,						UpperCAmelCase__:	Dict									= np.linalg.eigh(SCREAMING_SNAKE_CASE      )
		# Take all the columns in the reverse order (-1), and then takes only the first
		UpperCAmelCase__:	Optional[Any]									= eigenvectors[:, ::-1][:, 0:dimensions]
		# Project the database on the new space
		UpperCAmelCase__:	Tuple									= 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 			_A	(      SCREAMING_SNAKE_CASE	,SCREAMING_SNAKE_CASE	,SCREAMING_SNAKE_CASE	,SCREAMING_SNAKE_CASE      ):
	assert classes > dimensions

	# Check if features have been already loaded
	if features.any:
		UpperCAmelCase__							,						UpperCAmelCase__:	List[str]									= 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      )	,)
		UpperCAmelCase__:	int									= eigenvectors[:, ::-1][:, :dimensions]
		UpperCAmelCase__							,						UpperCAmelCase__							,						UpperCAmelCase__:	Optional[Any]									= np.linalg.svd(SCREAMING_SNAKE_CASE      )
		UpperCAmelCase__:	Optional[Any]									= svd_matrix[:, 0:dimensions]
		UpperCAmelCase__:	Optional[int]									= 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 			_A	(      ):
	# Create dummy dataset with 2 classes and 3 features
	UpperCAmelCase__:	List[str]									= np.array([[1, 2, 3, 4, 5], [2, 3, 4, 5, 6], [3, 4, 5, 6, 7]]      )
	UpperCAmelCase__:	Optional[int]									= np.array([0, 0, 0, 1, 1]      )
	UpperCAmelCase__:	List[Any]									= 2
	UpperCAmelCase__:	Union[str, Any]									= 2

	# Assert that the function raises an AssertionError if dimensions > classes
	with pytest.raises(SCREAMING_SNAKE_CASE      ) as error_info:
		UpperCAmelCase__:	List[Any]									= 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 			_A	(      ):
	UpperCAmelCase__:	Optional[Any]									= np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]]      )
	UpperCAmelCase__:	Optional[int]									= 2
	UpperCAmelCase__:	Union[str, Any]									= np.array([[6.92_82_03_23, 8.66_02_54_04, 10.39_23_04_85], [3.0, 3.0, 3.0]]      )

	with pytest.raises(SCREAMING_SNAKE_CASE      ) as error_info:
		UpperCAmelCase__:	Any									= 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()