# Source code for pyod.models.pca

```# -*- coding: utf-8 -*-
"""Principal Component Analysis (PCA) Outlier Detector
"""
# Author: Yue Zhao <zhaoy@cmu.edu>

from __future__ import division
from __future__ import print_function

import numpy as np
from scipy.spatial.distance import cdist
from sklearn.decomposition import PCA as sklearn_PCA
from sklearn.utils.validation import check_is_fitted
from sklearn.utils.validation import check_array

from .base import BaseDetector
from ..utils.utility import check_parameter
from ..utils.utility import standardizer

[docs]class PCA(BaseDetector):
"""Principal component analysis (PCA) can be used in detecting outliers.
PCA is a linear dimensionality reduction using Singular Value Decomposition
of the data to project it to a lower dimensional space.

In this procedure, covariance matrix of the data can be decomposed to
orthogonal vectors, called eigenvectors, associated with eigenvalues. The
eigenvectors with high eigenvalues capture most of the variance in the
data.

Therefore, a low dimensional hyperplane constructed by k eigenvectors can
capture most of the variance in the data. However, outliers are different
from normal data points, which is more obvious on the hyperplane
constructed by the eigenvectors with small eigenvalues.

Therefore, outlier scores can be obtained as the sum of the projected
distance of a sample on all eigenvectors.
See :cite:`shyu2003novel,aggarwal2015outlier` for details.

Score(X) = Sum of weighted euclidean distance between each sample to the
hyperplane constructed by the selected eigenvectors

Parameters
----------
n_components : int, float, None or string
Number of components to keep.
if n_components is not set all components are kept::

n_components == min(n_samples, n_features)

if n_components == 'mle' and svd_solver == 'full', Minka\'s MLE is used
to guess the dimension
if ``0 < n_components < 1`` and svd_solver == 'full', select the number
of components such that the amount of variance that needs to be
explained is greater than the percentage specified by n_components
n_components cannot be equal to n_features for svd_solver == 'arpack'.

n_selected_components : int, optional (default=None)
Number of selected principal components
for calculating the outlier scores. It is not necessarily equal to
the total number of the principal components. If not set, use
all principal components.

contamination : float in (0., 0.5), optional (default=0.1)
The amount of contamination of the data set, i.e.
the proportion of outliers in the data set. Used when fitting to
define the threshold on the decision function.

copy : bool (default True)
If False, data passed to fit are overwritten and running
fit(X).transform(X) will not yield the expected results,

whiten : bool, optional (default False)
When True (False by default) the `components_` vectors are multiplied
by the square root of n_samples and then divided by the singular values
to ensure uncorrelated outputs with unit component-wise variances.

Whitening will remove some information from the transformed signal
(the relative variance scales of the components) but can sometime
improve the predictive accuracy of the downstream estimators by
making their data respect some hard-wired assumptions.

svd_solver : string {'auto', 'full', 'arpack', 'randomized'}
auto :
the solver is selected by a default policy based on `X.shape` and
`n_components`: if the input data is larger than 500x500 and the
number of components to extract is lower than 80% of the smallest
dimension of the data, then the more efficient 'randomized'
method is enabled. Otherwise the exact full SVD is computed and
optionally truncated afterwards.
full :
run exact full SVD calling the standard LAPACK solver via
`scipy.linalg.svd` and select the components by postprocessing
arpack :
run SVD truncated to n_components calling ARPACK solver via
`scipy.sparse.linalg.svds`. It requires strictly
0 < n_components < X.shape
randomized :
run randomized SVD by the method of Halko et al.

tol : float >= 0, optional (default .0)
Tolerance for singular values computed by svd_solver == 'arpack'.

iterated_power : int >= 0, or 'auto', (default 'auto')
Number of iterations for the power method computed by
svd_solver == 'randomized'.

random_state : int, RandomState instance or None, optional (default None)
If int, random_state is the seed used by the random number generator;
If RandomState instance, random_state is the random number generator;
If None, the random number generator is the RandomState instance used
by `np.random`. Used when ``svd_solver`` == 'arpack' or 'randomized'.

weighted : bool, optional (default=True)
If True, the eigenvalues are used in score computation.
The eigenvectors with small eigenvalues comes with more importance
in outlier score calculation.

standardization : bool, optional (default=True)
If True, perform standardization first to convert
data to zero mean and unit variance.
See http://scikit-learn.org/stable/auto_examples/preprocessing/plot_scaling_importance.html

Attributes
----------
components_ : array, shape (n_components, n_features)
Principal axes in feature space, representing the directions of
maximum variance in the data. The components are sorted by
``explained_variance_``.

explained_variance_ : array, shape (n_components,)
The amount of variance explained by each of the selected components.

Equal to n_components largest eigenvalues
of the covariance matrix of X.

explained_variance_ratio_ : array, shape (n_components,)
Percentage of variance explained by each of the selected components.

If ``n_components`` is not set then all components are stored and the
sum of explained variances is equal to 1.0.

singular_values_ : array, shape (n_components,)
The singular values corresponding to each of the selected components.
The singular values are equal to the 2-norms of the ``n_components``
variables in the lower-dimensional space.

mean_ : array, shape (n_features,)
Per-feature empirical mean, estimated from the training set.

Equal to `X.mean(axis=0)`.

n_components_ : int
The estimated number of components. When n_components is set
to 'mle' or a number between 0 and 1 (with svd_solver == 'full') this
number is estimated from input data. Otherwise it equals the parameter
n_components, or n_features if n_components is None.

noise_variance_ : float
The estimated noise covariance following the Probabilistic PCA model
from Tipping and Bishop 1999. See "Pattern Recognition and
Machine Learning" by C. Bishop, 12.2.1 p. 574 or
http://www.miketipping.com/papers/met-mppca.pdf. It is required to
computed the estimated data covariance and score samples.

Equal to the average of (min(n_features, n_samples) - n_components)
smallest eigenvalues of the covariance matrix of X.

decision_scores_ : numpy array of shape (n_samples,)
The outlier scores of the training data.
The higher, the more abnormal. Outliers tend to have higher
scores. This value is available once the detector is fitted.

threshold_ : float
The threshold is based on ``contamination``. It is the
``n_samples * contamination`` most abnormal samples in
``decision_scores_``. The threshold is calculated for generating
binary outlier labels.

labels_ : int, either 0 or 1
The binary labels of the training data. 0 stands for inliers
and 1 for outliers/anomalies. It is generated by applying
``threshold_`` on ``decision_scores_``.
"""

def __init__(self, n_components=None, n_selected_components=None,
contamination=0.1, copy=True, whiten=False, svd_solver='auto',
tol=0.0, iterated_power='auto', random_state=None,
weighted=True, standardization=True):

super(PCA, self).__init__(contamination=contamination)
self.n_components = n_components
self.n_selected_components = n_selected_components
self.copy = copy
self.whiten = whiten
self.svd_solver = svd_solver
self.tol = tol
self.iterated_power = iterated_power
self.random_state = random_state
self.weighted = weighted
self.standardization = standardization

# noinspection PyIncorrectDocstring
[docs]    def fit(self, X, y=None):
"""Fit detector. y is ignored in unsupervised methods.

Parameters
----------
X : numpy array of shape (n_samples, n_features)
The input samples.

y : Ignored
Not used, present for API consistency by convention.

Returns
-------
self : object
Fitted estimator.
"""
# validate inputs X and y (optional)
X = check_array(X)
self._set_n_classes(y)

# PCA is recommended to use on the standardized data (zero mean and
# unit variance).
if self.standardization:
X, self.scaler_ = standardizer(X, keep_scalar=True)

self.detector_ = sklearn_PCA(n_components=self.n_components,
copy=self.copy,
whiten=self.whiten,
svd_solver=self.svd_solver,
tol=self.tol,
iterated_power=self.iterated_power,
random_state=self.random_state)
self.detector_.fit(X=X, y=y)

# copy the attributes from the sklearn PCA object
self.n_components_ = self.detector_.n_components_
self.components_ = self.detector_.components_

# validate the number of components to be used for outlier detection
if self.n_selected_components is None:
self.n_selected_components_ = self.n_components_
else:
self.n_selected_components_ = self.n_selected_components
check_parameter(self.n_selected_components_, 1, self.n_components_,
include_left=True, include_right=True,
param_name='n_selected_components_')

# use eigenvalues as the weights of eigenvectors
self.w_components_ = np.ones([self.n_components_, ])
if self.weighted:
self.w_components_ = self.detector_.explained_variance_ratio_

# outlier scores is the sum of the weighted distances between each
# sample to the eigenvectors. The eigenvectors with smaller
# eigenvalues have more influence
# Not all eigenvectors are used, only n_selected_components_ smallest
# are used since they better reflect the variance change

self.selected_components_ = self.components_[
-1 * self.n_selected_components_:, :]
self.selected_w_components_ = self.w_components_[
-1 * self.n_selected_components_:]

self.decision_scores_ = np.sum(
cdist(X, self.selected_components_) / self.selected_w_components_,
axis=1).ravel()

self._process_decision_scores()
return self

[docs]    def decision_function(self, X):
"""Predict raw anomaly score of X using the fitted detector.

The anomaly score of an input sample is computed based on different
detector algorithms. For consistency, outliers are assigned with
larger anomaly scores.

Parameters
----------
X : numpy array of shape (n_samples, n_features)
The training input samples. Sparse matrices are accepted only
if they are supported by the base estimator.

Returns
-------
anomaly_scores : numpy array of shape (n_samples,)
The anomaly score of the input samples.
"""
check_is_fitted(self, ['components_', 'w_components_'])

X = check_array(X)
if self.standardization:
X = self.scaler_.transform(X)

return np.sum(
cdist(X, self.selected_components_) / self.selected_w_components_,
axis=1).ravel()

@property
def explained_variance_(self):
"""The amount of variance explained by each of the selected components.

Equal to n_components largest eigenvalues
of the covariance matrix of X.

Decorator for scikit-learn PCA attributes.
"""
return self.detector_.explained_variance_

@property
def explained_variance_ratio_(self):
"""Percentage of variance explained by each of the selected components.

If ``n_components`` is not set then all components are stored and the
sum of explained variances is equal to 1.0.

Decorator for scikit-learn PCA attributes.
"""
return self.detector_.explained_variance_ratio_

@property
def singular_values_(self):
"""The singular values corresponding to each of the selected
components. The singular values are equal to the 2-norms of the
``n_components`` variables in the lower-dimensional space.

Decorator for scikit-learn PCA attributes.
"""
return self.detector_.singular_values_

@property
def mean_(self):
"""Per-feature empirical mean, estimated from the training set.

Decorator for scikit-learn PCA attributes.
"""
return self.detector_.mean_

@property
def noise_variance_(self):
"""The estimated noise covariance following the Probabilistic PCA model
from Tipping and Bishop 1999. See "Pattern Recognition and
Machine Learning" by C. Bishop, 12.2.1 p. 574 or
http://www.miketipping.com/papers/met-mppca.pdf. It is required to
computed the estimated data covariance and score samples.

Equal to the average of (min(n_features, n_samples) - n_components)
smallest eigenvalues of the covariance matrix of X.

Decorator for scikit-learn PCA attributes.
"""
return self.detector_.noise_variance_
```