



from copy import deepcopy



class 						UpperCAmelCase_					:







							def __init__(     self,    __a = None,    __a = None):

														'''simple docstring'''

														if arr is None and size is not None:
																					_lowerCAmelCase  :							int				     =					size
																					_lowerCAmelCase  :							Any				     =					[0] * size
														elif arr is not None:
																					self.init(__a)
														else:
																					raise ValueError("Either arr or size must be specified")







							def      snake_case__   (     self,    __a):

														'''simple docstring'''

														_lowerCAmelCase  :							Dict				     =					len(__a)
														_lowerCAmelCase  :							List[str]				     =					deepcopy(__a)
														for i in range(1,    self.size):
																					_lowerCAmelCase  :							List[Any]				     =					self.next_(__a)
																					if j < self.size:
																												self.tree[j] += self.tree[i]







							def      snake_case__   (     self):

														'''simple docstring'''

														_lowerCAmelCase  :							Union[str, Any]				     =					self.tree[:]
														for i in range(self.size - 1,    0,    -1):
																					_lowerCAmelCase  :							Union[str, Any]				     =					self.next_(__a)
																					if j < self.size:
																												arr[j] -= arr[i]
														return arr







							@staticmethod
							def      snake_case__   (     __a):

														'''simple docstring'''

														return index + (index & (-index))







							@staticmethod
							def      snake_case__   (     __a):

														'''simple docstring'''

														return index - (index & (-index))







							def      snake_case__   (     self,    __a,    __a):

														'''simple docstring'''

														if index == 0:
																					self.tree[0] += value
																					return
														while index < self.size:
																					self.tree[index] += value
																					_lowerCAmelCase  :							List[str]				     =					self.next_(__a)







							def      snake_case__   (     self,    __a,    __a):

														'''simple docstring'''

														self.add(__a,    value - self.get(__a))







							def      snake_case__   (     self,    __a):

														'''simple docstring'''

														if right == 0:
																					return 0
														_lowerCAmelCase  :							int				     =					self.tree[0]
														right -= 1  # make right inclusive
														while right > 0:
																					result += self.tree[right]
																					_lowerCAmelCase  :							Any				     =					self.prev(__a)
														return result







							def      snake_case__   (     self,    __a,    __a):

														'''simple docstring'''

														return self.prefix(__a) - self.prefix(__a)







							def      snake_case__   (     self,    __a):

														'''simple docstring'''

														return self.query(__a,    index + 1)







							def      snake_case__   (     self,    __a):

														'''simple docstring'''

														value -= self.tree[0]
														if value < 0:
																					return -1

														_lowerCAmelCase  :							int				     =					1  # Largest power of 2 <= size
														while j * 2 < self.size:
																					j *= 2

														_lowerCAmelCase  :							Optional[Any]				     =					0

														while j > 0:
																					if i + j < self.size and self.tree[i + j] <= value:
																												value -= self.tree[i + j]
																												i += j
																					j //= 2
														return i


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       (      _lowerCamelCase						):





							'''simple docstring'''




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






def        A       (      _lowerCamelCase       ,				_lowerCamelCase       ,				_lowerCamelCase						):





							'''simple docstring'''




							_lowerCAmelCase  :							Any				     =					np.nan
							for i in range(_lowerCamelCase						):
														_lowerCAmelCase  :							Tuple				     =					features[:, labels == i]
														_lowerCAmelCase  :							Dict				     =					data.mean(1						)
														# Centralize the data of class i
														_lowerCAmelCase  :							Union[str, Any]				     =					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)
																					_lowerCAmelCase  :							int				     =					np.dot(_lowerCamelCase       ,				centered_data.T						)

							return covariance_sum / features.shape[1]






def        A       (      _lowerCamelCase       ,				_lowerCamelCase       ,				_lowerCamelCase						):





							'''simple docstring'''




							_lowerCAmelCase  :							Optional[Any]				     =					features.mean(1						)
							_lowerCAmelCase  :							List[str]				     =					np.nan
							for i in range(_lowerCamelCase						):
														_lowerCAmelCase  :							str				     =					features[:, labels == i]
														_lowerCAmelCase  :							Optional[Any]				     =					data.shape[1]
														_lowerCAmelCase  :							Optional[Any]				     =					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)
																					_lowerCAmelCase  :							Optional[Any]				     =					device_data * np.dot(
																					    column_reshape(_lowerCamelCase						) - column_reshape(_lowerCamelCase						)       ,				(column_reshape(_lowerCamelCase						) - column_reshape(_lowerCamelCase						)).T       ,				)

							return covariance_sum / features.shape[1]






def        A       (      _lowerCamelCase       ,				_lowerCamelCase						):





							'''simple docstring'''




							if features.any():
														_lowerCAmelCase  :							List[Any]				     =					features.mean(1						)
														# Center the dataset
														_lowerCAmelCase  :							List[Any]				     =					features - np.reshape(_lowerCamelCase       ,				(data_mean.size, 1)						)
														_lowerCAmelCase  :							Optional[Any]				     =					np.dot(_lowerCamelCase       ,				centered_data.T						) / features.shape[1]
														_lowerCAmelCase     ,					_lowerCAmelCase  :							List[Any]				     =					np.linalg.eigh(_lowerCamelCase						)
														# Take all the columns in the reverse order (-1), and then takes only the first
														_lowerCAmelCase  :							Union[str, Any]				     =					eigenvectors[:, ::-1][:, 0:dimensions]
														# Project the database on the new space
														_lowerCAmelCase  :							List[Any]				     =					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        A       (      _lowerCamelCase       ,				_lowerCamelCase       ,				_lowerCamelCase       ,				_lowerCamelCase						):





							'''simple docstring'''




							assert classes > dimensions

							# Check if features have been already loaded
							if features.any:
														_lowerCAmelCase     ,					_lowerCAmelCase  :							List[str]				     =					eigh(
														    covariance_between_classes(_lowerCamelCase       ,				_lowerCamelCase       ,				_lowerCamelCase						)       ,				covariance_within_classes(_lowerCamelCase       ,				_lowerCamelCase       ,				_lowerCamelCase						)       ,				)
														_lowerCAmelCase  :							List[str]				     =					eigenvectors[:, ::-1][:, :dimensions]
														_lowerCAmelCase     ,					_lowerCAmelCase     ,					_lowerCAmelCase  :							Any				     =					np.linalg.svd(_lowerCamelCase						)
														_lowerCAmelCase  :							Optional[Any]				     =					svd_matrix[:, 0:dimensions]
														_lowerCAmelCase  :							str				     =					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        A       (      ):





							'''simple docstring'''




							_lowerCAmelCase  :							Optional[int]				     =					np.array([[1, 2, 3, 4, 5], [2, 3, 4, 5, 6], [3, 4, 5, 6, 7]]						)
							_lowerCAmelCase  :							List[Any]				     =					np.array([0, 0, 0, 1, 1]						)
							_lowerCAmelCase  :							List[Any]				     =					2
							_lowerCAmelCase  :							Union[str, Any]				     =					2

							# Assert that the function raises an AssertionError if dimensions > classes
							with pytest.raises(_lowerCamelCase						) as error_info:
														_lowerCAmelCase  :							Union[str, Any]				     =					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        A       (      ):





							'''simple docstring'''




							_lowerCAmelCase  :							Optional[Any]				     =					np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]]						)
							_lowerCAmelCase  :							List[str]				     =					2
							_lowerCAmelCase  :							List[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(_lowerCamelCase						) as error_info:
														_lowerCAmelCase  :							Tuple				     =					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()
