text: """ Principal Component Analysis
  """
  
  # Author: Alexandre Gramfort <alexandre.gramfort@inria.fr>
  #         Olivier Grisel <olivier.grisel@ensta.org>
  #         Mathieu Blondel <mathieu@mblondel.org>
  #         Denis A. Engemann <denis-alexander.engemann@inria.fr>
  #         Michael Eickenberg <michael.eickenberg@inria.fr>
  #         Giorgio Patrini <giorgio.patrini@anu.edu.au>
  #
  # License: BSD 3 clause
  
  from math import log, sqrt
  import numbers
  
  import numpy as np
  from scipy import linalg
  from scipy.special import gammaln
  from scipy.sparse import issparse
  from scipy.sparse.linalg import svds
  
  from ._base import _BasePCA
  from ..utils import check_random_state
  from ..utils import check_array
  from ..utils.extmath import fast_logdet, randomized_svd, svd_flip
  from ..utils.extmath import stable_cumsum
  from ..utils.validation import check_is_fitted
  
  
  def _assess_dimension_(spectrum, rank, n_samples, n_features):
      """Compute the likelihood of a rank ``rank`` dataset.
  
      The dataset is assumed to be embedded in gaussian noise of shape(n,
      dimf) having spectrum ``spectrum``.
  
      Parameters
      ----------
      spectrum : array of shape (n)
          Data spectrum.
      rank : int
          Tested rank value.
      n_samples : int
          Number of samples.
      n_features : int
          Number of features.
  
      Returns
      -------
      ll : float,
          The log-likelihood
  
      Notes
      -----
      This implements the method of `Thomas P. Minka:
      Automatic Choice of Dimensionality for PCA. NIPS 2000: 598-604`
      """
      if rank > len(spectrum):
          raise ValueError("The tested rank cannot exceed the rank of the"
                           " dataset")
  
      pu = -rank * log(2.)
      for i in range(rank):
          pu += (gammaln((n_features - i) / 2.) -
                 log(np.pi) * (n_features - i) / 2.)
  
      pl = np.sum(np.log(spectrum[:rank]))
      pl = -pl * n_samples / 2.
  
      if rank == n_features:
          pv = 0
          v = 1
      else:
          v = np.sum(spectrum[rank:]) / (n_features - rank)
          pv = -np.log(v) * n_samples * (n_features - rank) / 2.
  
      m = n_features * rank - rank * (rank + 1.) / 2.
      pp = log(2. * np.pi) * (m + rank + 1.) / 2.
  
      pa = 0.
      spectrum_ = spectrum.copy()
      spectrum_[rank:n_features] = v
      for i in range(rank):
          for j in range(i + 1, len(spectrum)):
              pa += log((spectrum[i] - spectrum[j]) *
                        (1. / spectrum_[j] - 1. / spectrum_[i])) + log(n_samples)
  
      ll = pu + pl + pv + pp - pa / 2. - rank * log(n_samples) / 2.
  
      return ll
  
  
  def _infer_dimension_(spectrum, n_samples, n_features):
      """Infers the dimension of a dataset of shape (n_samples, n_features)
  
      The dataset is described by its spectrum `spectrum`.
      """
      n_spectrum = len(spectrum)
      ll = np.empty(n_spectrum)
      for rank in range(n_spectrum):
          ll[rank] = _assess_dimension_(spectrum, rank, n_samples, n_features)
      return ll.argmax()
  
  
  class PCA(_BasePCA):
      """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 str
          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 : str {'auto', 'full', 'arpack', 'randomized'}
          If 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.
          If full :
              run exact full SVD calling the standard LAPACK solver via
              `scipy.linalg.svd` and select the components by postprocessing
          If arpack :
              run SVD truncated to n_components calling ARPACK solver via
              `scipy.sparse.linalg.svds`. It requires strictly
              0 < n_components < min(X.shape)
          If randomized :
              run randomized SVD by the method of Halko et al.
  
          .. versionadded:: 0.18.0
  
      tol : float >= 0, optional (default .0)
          Tolerance for singular values computed by svd_solver == 'arpack'.
  
          .. versionadded:: 0.18.0
  
      iterated_power : int >= 0, or 'auto', (default 'auto')
          Number of iterations for the power method computed by
          svd_solver == 'randomized'.
  
          .. versionadded:: 0.18.0
  
      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'.
  
          .. versionadded:: 0.18.0
  
      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.
  
          .. versionadded:: 0.18
  
      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 the ratios 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.
  
          .. versionadded:: 0.19
  
      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 the lesser value of n_features and n_samples
          if n_components is None.
  
      n_features_ : int
          Number of features in the training data.
  
      n_samples_ : int
          Number of samples in the training data.
  
      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.
  
      See Also
      --------
      KernelPCA : Kernel Principal Component Analysis.
      SparsePCA : Sparse Principal Component Analysis.
      TruncatedSVD : Dimensionality reduction using truncated SVD.
      IncrementalPCA : Incremental Principal Component Analysis.
  
      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 numpy as np
      >>> from sklearn.decomposition import PCA
      >>> X = np.array([[-1, -1], [-2, -1], [-3, -2], [1, 1], [2, 1], [3, 2]])
      >>> pca = PCA(n_components=2)
      >>> pca.fit(X)
      PCA(n_components=2)
      >>> print(pca.explained_variance_ratio_)
      [0.9924... 0.0075...]
      >>> print(pca.singular_values_)
      [6.30061... 0.54980...]
  
      >>> pca = PCA(n_components=2, svd_solver='full')
      >>> pca.fit(X)
      PCA(n_components=2, svd_solver='full')
      >>> print(pca.explained_variance_ratio_)
      [0.9924... 0.00755...]
      >>> print(pca.singular_values_)
      [6.30061... 0.54980...]
  
      >>> pca = PCA(n_components=1, svd_solver='arpack')
      >>> pca.fit(X)
      PCA(n_components=1, svd_solver='arpack')
      >>> print(pca.explained_variance_ratio_)
      [0.99244...]
      >>> print(pca.singular_values_)
      [6.30061...]
      """
  
      def __init__(self, n_components=None, copy=True, whiten=False,
                   svd_solver='auto', tol=0.0, iterated_power='auto',
                   random_state=None):
          self.n_components = n_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
  
      def fit(self, X, y=None):
          """Fit the model with X.
  
          Parameters
          ----------
          X : array-like, shape (n_samples, n_features)
              Training data, where n_samples is the number of samples
              and n_features is the number of features.
  
          y : None
              Ignored variable.
  
          Returns
          -------
          self : object
              Returns the instance itself.
          """
          self._fit(X)
          return self
  
      def fit_transform(self, X, y=None):
          """Fit the model with X and apply the dimensionality reduction on X.
  
          Parameters
          ----------
          X : array-like, shape (n_samples, n_features)
              Training data, where n_samples is the number of samples
              and n_features is the number of features.
  
          y : None
              Ignored variable.
  
          Returns
          -------
          X_new : array-like, shape (n_samples, n_components)
              Transformed values.
  
          Notes
          -----
          This method returns a Fortran-ordered array. To convert it to a
          C-ordered array, use 'np.ascontiguousarray'.
          """
          U, S, V = self._fit(X)
          U = U[:, :self.n_components_]
  
          if self.whiten:
              # X_new = X * V / S * sqrt(n_samples) = U * sqrt(n_samples)
              U *= sqrt(X.shape[0] - 1)
          else:
              # X_new = X * V = U * S * V^T * V = U * S
              U *= S[:self.n_components_]
  
          return U
  
      def _fit(self, X):
          """Dispatch to the right submethod depending on the chosen solver."""
  
          # Raise an error for sparse input.
          # This is more informative than the generic one raised by check_array.
          if issparse(X):
              raise TypeError('PCA does not support sparse input. See '
                              'TruncatedSVD for a possible alternative.')
  
          X = check_array(X, dtype=[np.float64, np.float32], ensure_2d=True,
                          copy=self.copy)
  
          # Handle n_components==None
          if self.n_components is None:
              if self.svd_solver != 'arpack':
                  n_components = min(X.shape)
              else:
                  n_components = min(X.shape) - 1
          else:
              n_components = self.n_components
  
          # Handle svd_solver
          self._fit_svd_solver = self.svd_solver
          if self._fit_svd_solver == 'auto':
              # Small problem or n_components == 'mle', just call full PCA
              if max(X.shape) <= 500 or n_components == 'mle':
                  self._fit_svd_solver = 'full'
              elif n_components >= 1 and n_components < .8 * min(X.shape):
                  self._fit_svd_solver = 'randomized'
              # This is also the case of n_components in (0,1)
              else:
                  self._fit_svd_solver = 'full'
  
          # Call different fits for either full or truncated SVD
          if self._fit_svd_solver == 'full':
              return self._fit_full(X, n_components)
          elif self._fit_svd_solver in ['arpack', 'randomized']:
              return self._fit_truncated(X, n_components, self._fit_svd_solver)
          else:
              raise ValueError("Unrecognized svd_solver='{0}'"
                               "".format(self._fit_svd_solver))
  
      def _fit_full(self, X, n_components):
          """Fit the model by computing full SVD on X"""
          n_samples, n_features = X.shape
  
          if n_components == 'mle':
              if n_samples < n_features:
                  raise ValueError("n_components='mle' is only supported "
                                   "if n_samples >= n_features")
          elif not 0 <= n_components <= min(n_samples, n_features):
              raise ValueError("n_components=%r must be between 0 and "
                               "min(n_samples, n_features)=%r with "
                               "svd_solver='full'"
                               % (n_components, min(n_samples, n_features)))
          elif n_components >= 1:
              if not isinstance(n_components, numbers.Integral):
                  raise ValueError("n_components=%r must be of type int "
                                   "when greater than or equal to 1, "
                                   "was of type=%r"
                                   % (n_components, type(n_components)))
  
          # Center data
          self.mean_ = np.mean(X, axis=0)
          X -= self.mean_
  
          U, S, V = linalg.svd(X, full_matrices=False)
          # flip eigenvectors' sign to enforce deterministic output
          U, V = svd_flip(U, V)
  
          components_ = V
  
          # Get variance explained by singular values
          explained_variance_ = (S ** 2) / (n_samples - 1)
          total_var = explained_variance_.sum()
          explained_variance_ratio_ = explained_variance_ / total_var
          singular_values_ = S.copy()  # Store the singular values.
  
          # Postprocess the number of components required
          if n_components == 'mle':
              n_components = \
                  _infer_dimension_(explained_variance_, n_samples, n_features)
          elif 0 < n_components < 1.0:
              # number of components for which the cumulated explained
              # variance percentage is superior to the desired threshold
              ratio_cumsum = stable_cumsum(explained_variance_ratio_)
              n_components = np.searchsorted(ratio_cumsum, n_components) + 1
  
          # Compute noise covariance using Probabilistic PCA model
          # The sigma2 maximum likelihood (cf. eq. 12.46)
          if n_components < min(n_features, n_samples):
              self.noise_variance_ = explained_variance_[n_components:].mean()
          else:
              self.noise_variance_ = 0.
  
          self.n_samples_, self.n_features_ = n_samples, n_features
          self.components_ = components_[:n_components]
          self.n_components_ = n_components
          self.explained_variance_ = explained_variance_[:n_components]
          self.explained_variance_ratio_ = \
              explained_variance_ratio_[:n_components]
          self.singular_values_ = singular_values_[:n_components]
  
          return U, S, V
  
      def _fit_truncated(self, X, n_components, svd_solver):
          """Fit the model by computing truncated SVD (by ARPACK or randomized)
          on X
          """
          n_samples, n_features = X.shape
  
          if isinstance(n_components, str):
              raise ValueError("n_components=%r cannot be a string "
                               "with svd_solver='%s'"
                               % (n_components, svd_solver))
          elif not 1 <= n_components <= min(n_samples, n_features):
              raise ValueError("n_components=%r must be between 1 and "
                               "min(n_samples, n_features)=%r with "
                               "svd_solver='%s'"
                               % (n_components, min(n_samples, n_features),
                                  svd_solver))
          elif not isinstance(n_components, numbers.Integral):
              raise ValueError("n_components=%r must be of type int "
                               "when greater than or equal to 1, was of type=%r"
                               % (n_components, type(n_components)))
          elif svd_solver == 'arpack' and n_components == min(n_samples,
                                                              n_features):
              raise ValueError("n_components=%r must be strictly less than "
                               "min(n_samples, n_features)=%r with "
                               "svd_solver='%s'"
                               % (n_components, min(n_samples, n_features),
                                  svd_solver))
  
          random_state = check_random_state(self.random_state)
  
          # Center data
          self.mean_ = np.mean(X, axis=0)
          X -= self.mean_
  
          if svd_solver == 'arpack':
              # random init solution, as ARPACK does it internally
              v0 = random_state.uniform(-1, 1, size=min(X.shape))
              U, S, V = svds(X, k=n_components, tol=self.tol, v0=v0)
              # svds doesn't abide by scipy.linalg.svd/randomized_svd
              # conventions, so reverse its outputs.
              S = S[::-1]
              # flip eigenvectors' sign to enforce deterministic output
              U, V = svd_flip(U[:, ::-1], V[::-1])
  
          elif svd_solver == 'randomized':
              # sign flipping is done inside
              U, S, V = randomized_svd(X, n_components=n_components,
                                       n_iter=self.iterated_power,
                                       flip_sign=True,
                                       random_state=random_state)
  
          self.n_samples_, self.n_features_ = n_samples, n_features
          self.components_ = V
          self.n_components_ = n_components
  
          # Get variance explained by singular values
          self.explained_variance_ = (S ** 2) / (n_samples - 1)
          total_var = np.var(X, ddof=1, axis=0)
          self.explained_variance_ratio_ = \
              self.explained_variance_ / total_var.sum()
          self.singular_values_ = S.copy()  # Store the singular values.
  
          if self.n_components_ < min(n_features, n_samples):
              self.noise_variance_ = (total_var.sum() -
                                      self.explained_variance_.sum())
              self.noise_variance_ /= min(n_features, n_samples) - n_components
          else:
              self.noise_variance_ = 0.
  
          return U, S, V
  
      def score_samples(self, X):
          """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 : array, shape(n_samples, n_features)
              The data.
  
          Returns
          -------
          ll : array, shape (n_samples,)
              Log-likelihood of each sample under the current model.
          """
          check_is_fitted(self)
  
          X = check_array(X)
          Xr = X - self.mean_
          n_features = X.shape[1]
          precision = self.get_precision()
          log_like = -.5 * (Xr * (np.dot(Xr, precision))).sum(axis=1)
          log_like -= .5 * (n_features * log(2. * np.pi) -
                            fast_logdet(precision))
          return log_like
  
      def score(self, X, y=None):
          """Return the average log-likelihood of all samples.
  
          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 : array, shape(n_samples, n_features)
              The data.
  
          y : None
              Ignored variable.
  
          Returns
          -------
          ll : float
              Average log-likelihood of the samples under the current model.
          """
          return np.mean(self.score_samples(X))
  