class_docstr: Principal component analysis (PCA)
  
  Linear dimensionality reduction using Singular Value Decomposition of the
  data to project it to a lower dimensional space. The input data is centered
  but not scaled for each feature before applying the SVD.
  
  It uses the LAPACK implementation of the full SVD or a randomized truncated
  SVD by the method of Halko et al. 2009, depending on the shape of the input
  data and the number of components to extract.
  
  It can also use the scipy.sparse.linalg ARPACK implementation of the
  truncated SVD.
  
  Notice that this class does not support sparse input. See
  :class:`TruncatedSVD` for an alternative with sparse data.
  
  Read more in the :ref:`User Guide <PCA>`.
  
  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. Use of ``n_components == 'mle'``
      will interpret ``svd_solver == 'auto'`` as ``svd_solver == 'full'``.
  
      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.
  
      If ``svd_solver == 'arpack'``, the number of components must be
      strictly less than the minimum of n_features and n_samples.
  
      Hence, the None case results in::
  
          n_components == min(n_samples, n_features) - 1
  
  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 < min(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'.
  
  Attributes
  ----------
  components_ : tensor, 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_ : tensor, 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_ : tensor, 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 the ratios is equal to 1.0.
  
  singular_values_ : tensor, 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_ : tensor, 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 the lesser value of n_features and n_samples
      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
      compute 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.
  
  References
  ----------
  For n_components == 'mle', this class uses the method of *Minka, T. P.
  "Automatic choice of dimensionality for PCA". In NIPS, pp. 598-604*
  
  Implements the probabilistic PCA model from:
  Tipping, M. E., and Bishop, C. M. (1999). "Probabilistic principal
  component analysis". Journal of the Royal Statistical Society:
  Series B (Statistical Methodology), 61(3), 611-622.
  via the score and score_samples methods.
  See http://www.miketipping.com/papers/met-mppca.pdf
  
  For svd_solver == 'arpack', refer to `scipy.sparse.linalg.svds`.
  
  For svd_solver == 'randomized', see:
  *Halko, N., Martinsson, P. G., and Tropp, J. A. (2011).
  "Finding structure with randomness: Probabilistic algorithms for
  constructing approximate matrix decompositions".
  SIAM review, 53(2), 217-288.* and also
  *Martinsson, P. G., Rokhlin, V., and Tygert, M. (2011).
  "A randomized algorithm for the decomposition of matrices".
  Applied and Computational Harmonic Analysis, 30(1), 47-68.*
  
  
  Examples
  --------
  >>> import mars.tensor as mt
  >>> from mars.learn.decomposition import PCA
  >>> X = mt.array([[-1, -1], [-2, -1], [-3, -2], [1, 1], [2, 1], [3, 2]])
  >>> pca = PCA(n_components=2)
  >>> pca.fit(X)  # doctest: +NORMALIZE_WHITESPACE
  PCA(copy=True, iterated_power='auto', n_components=2, random_state=None,
    svd_solver='auto', tol=0.0, whiten=False)
  >>> print(pca.explained_variance_ratio_)  # doctest: +ELLIPSIS
  [0.9924... 0.0075...]
  >>> print(pca.singular_values_)  # doctest: +ELLIPSIS
  [6.30061... 0.54980...]
  
  >>> pca = PCA(n_components=2, svd_solver='full')
  >>> pca.fit(X)                 # doctest: +ELLIPSIS +NORMALIZE_WHITESPACE
  PCA(copy=True, iterated_power='auto', n_components=2, random_state=None,
    svd_solver='full', tol=0.0, whiten=False)
  >>> print(pca.explained_variance_ratio_)  # doctest: +ELLIPSIS
  [0.9924... 0.00755...]
  >>> print(pca.singular_values_)  # doctest: +ELLIPSIS
  [6.30061... 0.54980...]
  
  See also
  --------
  KernelPCA
  SparsePCA
  TruncatedSVD
  IncrementalPCA
docstr: Return the log-likelihood of each sample.
  
  See. "Pattern Recognition and Machine Learning"
  by C. Bishop, 12.2.1 p. 574
  or http://www.miketipping.com/papers/met-mppca.pdf
  
  Parameters
  ----------
  X : tensor, shape(n_samples, n_features)
      The data.
  
  Returns
  -------
  ll : tensor, shape (n_samples,)
      Log-likelihood of each sample under the current model
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code: def score_samples(self, X, session=None):
      """Return the log-likelihood of each sample.
  
      See. "Pattern Recognition and Machine Learning"
      by C. Bishop, 12.2.1 p. 574
      or http://www.miketipping.com/papers/met-mppca.pdf
  
      Parameters
      ----------
      X : tensor, shape(n_samples, n_features)
          The data.
  
      Returns
      -------
      ll : tensor, shape (n_samples,)
          Log-likelihood of each sample under the current model
      """
      log_like = self._score_samples(X, session=session)
      log_like.execute(session=session)
      return log_like
  
imports: from math import log, sqrt
  import numbers
  import numpy
  from scipy.special import gammaln
  from sklearn.utils.validation import check_is_fitted
  from sklearn.utils.extmath import fast_logdet
  from ...None import tensor
  from ...None import remote
  from ...tensor.array_utils import get_array_module
  from ...tensor.core import TENSOR_TYPE
  from ...tensor.utils import check_random_state
  from ...tensor.linalg import randomized_svd
  from ...tensor.linalg.randomized_svd import svd_flip
  from ...lib.sparse import issparse
  from ...core import ExecutableTuple
  from ..utils import check_array
  from ._base import _BasePCA
label_desc: Use image node_id 9 for calling the PCA obj's underlying member method code with example usage: obj.score_samples(X, session) and returns: log_like