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.
  
  For a usage example, see
  :ref:`sphx_glr_auto_examples_decomposition_plot_pca_iris.py`
  
  Read more in the :ref:`User Guide <PCA>`.
  
  Parameters
  ----------
  n_components : int, float or 'mle', default=None
      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, 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 : {'auto', 'full', 'arpack', 'randomized'}, default='auto'
      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, default=0.0
      Tolerance for singular values computed by svd_solver == 'arpack'.
      Must be of range [0.0, infinity).
  
      .. versionadded:: 0.18.0
  
  iterated_power : int or 'auto', default='auto'
      Number of iterations for the power method computed by
      svd_solver == 'randomized'.
      Must be of range [0, infinity).
  
      .. versionadded:: 0.18.0
  
  n_oversamples : int, default=10
      This parameter is only relevant when `svd_solver="randomized"`.
      It corresponds to the additional number of random vectors to sample the
      range of `X` so as to ensure proper conditioning. See
      :func:`~sklearn.utils.extmath.randomized_svd` for more details.
  
      .. versionadded:: 1.1
  
  power_iteration_normalizer : {'auto', 'QR', 'LU', 'none'}, default='auto'
      Power iteration normalizer for randomized SVD solver.
      Not used by ARPACK. See :func:`~sklearn.utils.extmath.randomized_svd`
      for more details.
  
      .. versionadded:: 1.1
  
  random_state : int, RandomState instance or None, default=None
      Used when the 'arpack' or 'randomized' solvers are used. Pass an int
      for reproducible results across multiple function calls.
      See :term:`Glossary <random_state>`.
  
      .. versionadded:: 0.18.0
  
  Attributes
  ----------
  components_ : ndarray of shape (n_components, n_features)
      Principal axes in feature space, representing the directions of
      maximum variance in the data. Equivalently, the right singular
      vectors of the centered input data, parallel to its eigenvectors.
      The components are sorted by decreasing ``explained_variance_``.
  
  explained_variance_ : ndarray of shape (n_components,)
      The amount of variance explained by each of the selected components.
      The variance estimation uses `n_samples - 1` degrees of freedom.
  
      Equal to n_components largest eigenvalues
      of the covariance matrix of X.
  
      .. versionadded:: 0.18
  
  explained_variance_ratio_ : ndarray of 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_ : ndarray of 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_ : ndarray of 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_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.
  
  n_features_in_ : int
      Number of features seen during :term:`fit`.
  
      .. versionadded:: 0.24
  
  feature_names_in_ : ndarray of shape (`n_features_in_`,)
      Names of features seen during :term:`fit`. Defined only when `X`
      has feature names that are all strings.
  
      .. versionadded:: 1.0
  
  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 from:
  `Minka, T. P.. "Automatic choice of dimensionality for PCA".
  In NIPS, pp. 598-604 <https://tminka.github.io/papers/pca/minka-pca.pdf>`_
  
  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.
  <http://www.miketipping.com/papers/met-mppca.pdf>`_
  via the score and score_samples methods.
  
  For svd_solver == 'arpack', refer to `scipy.sparse.linalg.svds`.
  
  For svd_solver == 'randomized', see:
  :doi:`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.
  <10.1137/090771806>`
  and also
  :doi:`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.
  <10.1016/j.acha.2010.02.003>`
  
  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...]
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docstr: Fit the model by computing truncated SVD (by ARPACK or randomized)
  on X.
docstr_tok: ["Fit","the","model","by","computing","truncated","SVD","(","by","ARPACK","or","randomized",")","on","X","."]
code: def _fit_truncated(self, X, n_components, svd_solver):
      """Fit the model by computing truncated SVD (by ARPACK or randomized)
      on X.
      """
      xp, _ = get_namespace(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 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
      total_var = None
      if issparse(X):
          self.mean_, var = mean_variance_axis(X, axis=0)
          total_var = var.sum() * n_samples / (n_samples - 1)  # ddof=1
          X = _implicit_column_offset(X, self.mean_)
      else:
          self.mean_ = xp.mean(X, axis=0)
          X -= self.mean_
  
      if svd_solver == "arpack":
          v0 = _init_arpack_v0(min(X.shape), random_state)
          U, S, Vt = 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, Vt = svd_flip(U[:, ::-1], Vt[::-1])
  
      elif svd_solver == "randomized":
          # sign flipping is done inside
          U, S, Vt = randomized_svd(
              X,
              n_components=n_components,
              n_oversamples=self.n_oversamples,
              n_iter=self.iterated_power,
              power_iteration_normalizer=self.power_iteration_normalizer,
              flip_sign=True,
              random_state=random_state,
          )
  
      self.n_samples_ = n_samples
      self.components_ = Vt
      self.n_components_ = n_components
  
      # Get variance explained by singular values
      self.explained_variance_ = (S**2) / (n_samples - 1)
  
      # Workaround in-place variance calculation since at the time numpy
      # did not have a way to calculate variance in-place.
      #
      # TODO: update this code to either:
      # * Use the array-api variance calculation, unless memory usage suffers
      # * Update sklearn.utils.extmath._incremental_mean_and_var to support array-api
      # See: https://github.com/scikit-learn/scikit-learn/pull/18689#discussion_r1335540991
      if total_var is None:
          N = X.shape[0] - 1
          X **= 2
          total_var = xp.sum(X) / N
  
      self.explained_variance_ratio_ = (
          self.explained_variance_ / total_var
      )
      self.singular_values_ = xp.asarray(
          S, copy=True
      )  # Store the singular values.
  
      if self.n_components_ < min(n_features, n_samples):
          self.noise_variance_ = total_var - xp.sum(
              self.explained_variance_
          )
          self.noise_variance_ /= (
              min(n_features, n_samples) - n_components
          )
      else:
          self.noise_variance_ = 0.0
  
      return U, S, Vt
  
imports: from math import log, sqrt
  from numbers import Integral, Real
  import numpy
  from scipy import linalg
  from scipy.sparse import issparse
  from scipy.sparse.linalg import svds
  from scipy.special import gammaln
  from ..base import _fit_context
  from ..utils import check_random_state
  from ..utils._arpack import _init_arpack_v0
  from ..utils._array_api import _convert_to_numpy, get_namespace
  from ..utils._param_validation import Interval, RealNotInt, StrOptions
  from ..utils.extmath import fast_logdet, randomized_svd, stable_cumsum, svd_flip
  from ..utils.sparsefuncs import _implicit_column_offset, mean_variance_axis
  from ..utils.validation import check_is_fitted
  from ._base import _BasePCA
label_desc: Use image node_id 6 for calling the PCA obj's underlying member method code with example usage: obj._fit_truncated(X, n_components, svd_solver) and returns: U,  S,  Vt