class_docstr: 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,
      use fit_transform(X) instead.
  
  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[1]
      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_``.
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docstr: 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.
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code: 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()
  
file_path: pyod/pyod/models/pca.py
imports: from __future__ import division
  from __future__ import print_function
  import numpy
  from scipy.spatial.distance import cdist
  from sklearn.decomposition import PCA
  from sklearn.utils.validation import check_array
  from sklearn.utils.validation import check_is_fitted
  from .base import BaseDetector
  from ..utils.utility import check_parameter
  from ..utils.utility import standardizer
label_desc: Use image node_id 3 for calling the PCA obj's underlying member method code with example usage: obj.decision_function(X) and returns: np