class_docstr: Principal component analysis (PCA)
  
  Linear dimensionality reduction using Singular Value Decomposition of the
  data to project it to a lower dimensional space.
  
  It uses the "tsqr" algorithm from Benson et. al. (2013). See the References
  for more.
  
  Read more in the :ref:`User Guide <PCA>`.
  
  Parameters
  ----------
  n_components : int, or None
      Number of components to keep.
      if n_components is not set all components are kept::
  
          n_components == min(n_samples, n_features)
  
      .. note::
  
         Unlike scikit-learn, ``n_components='mle'`` and ``n_components``
         between ``(0, 1)`` are not currently supported.
  
  copy : bool (default True)
      ignored
  
  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', 'tsqr', '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 and select the components by postprocessing
      randomized :
          run randomized SVD by using ``da.linalg.svd_compressed``.
  
  tol : float >= 0, optional (default .0)
      ignored
  
  iterated_power : int >= 0, default 0
      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 `da.random`. Used when ``svd_solver`` == 'randomized'.
  
  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 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.
  
  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.
  
  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.
  
  References
  ----------
  Direct QR factorizations for tall-and-skinny matrices in
  MapReduce architectures.
  A. Benson, D. Gleich, and J. Demmel.
  IEEE International Conference on Big Data, 2013.
  http://arxiv.org/abs/1301.1071
  
  Examples
  --------
  >>> import numpy as np
  >>> import dask.array as da
  >>> from dask_ml.decomposition import PCA
  >>> X = np.array([[-1, -1], [-2, -1], [-3, -2], [1, 1], [2, 1], [3, 2]])
  >>> dX = da.from_array(X, chunks=X.shape)
  >>> pca = PCA(n_components=2)
  >>> pca.fit(dX)
  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
  [ 0.99244...  0.00755...]
  >>> print(pca.singular_values_)  # doctest: +ELLIPSIS
  [ 6.30061...  0.54980...]
  
  >>> pca = PCA(n_components=2, svd_solver='full')
  >>> pca.fit(dX)                 # 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
  [ 0.99244...  0.00755...]
  >>> print(pca.singular_values_)  # doctest: +ELLIPSIS
  [ 6.30061...  0.54980...]
  
  Notes
  -----
  Differences from scikit-learn:
  
  * svd_solver : 'randomized' uses ``dask.linalg.svd_compressed``
    'full' uses ``dask.linalg.svd``, 'arpack' is not valid.
  * iterated_power : defaults to ``0``, the default for
    ``dask.linalg.svd_compressed``.
  * n_components : ``n_components='mle'`` is not allowed.
    Fractional ``n_components`` between 0 and 1 is not allowed.
class_docstr_tok: ["Principal","component","analysis","(","PCA",")","Linear","dimensionality","reduction","using","Singular","Value","Decomposition","of","the","data","to","project","it","to","a","lower","dimensional","space",".","It","uses","the","``","tsqr","''","algorithm","from","Benson","et",".","al",".","(","2013",")",".","See","the","References","for","more",".","Read","more","in","the",":","ref",":","`","User","Guide","<","PCA",">","`",".","Parameters","--","--","--","--","--","n_components",":","int",",","or","None","Number","of","components","to","keep",".","if","n_components","is","not","set","all","components","are","kept",":",":","n_components","==","min","(","n_samples",",","n_features",")","..","note",":",":","Unlike","scikit-learn",",","``","n_components='mle","'","``","and","``","n_components","``","between","``","(","0",",","1",")","``","are","not","currently","supported",".","copy",":","bool","(","default","True",")","ignored","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","'",",","'tsqr","'",",","'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","and","select","the","components","by","postprocessing","randomized",":","run","randomized","SVD","by","using","``","da.linalg.svd_compressed","``",".","tol",":","float",">","=","0",",","optional","(","default",".0",")","ignored","iterated_power",":","int",">","=","0",",","default","0","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","`","da.random","`",".","Used","when","``","svd_solver","``","==","'randomized","'",".","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","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",".","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",".","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.","References","--","--","--","--","--","Direct","QR","factorizations","for","tall-and-skinny","matrices","in","MapReduce","architectures",".","A.","Benson",",","D.","Gleich",",","and","J.","Demmel",".","IEEE","International","Conference","on","Big","Data",",","2013.","http",":","\/\/arxiv.org\/abs\/1301.1071","Examples","--","--","--","--",">",">",">","import","numpy","as","np",">",">",">","import","dask.array","as","da",">",">",">","from","dask_ml.decomposition","import","PCA",">",">",">","X","=","np.array","(","[","[","-1",",","-1","]",",","[","-2",",","-1","]",",","[","-3",",","-2","]",",","[","1",",","1","]",",","[","2",",","1","]",",","[","3",",","2","]","]",")",">",">",">","dX","=","da.from_array","(","X",",","chunks=X.shape",")",">",">",">","pca","=","PCA","(","n_components=2",")",">",">",">","pca.fit","(","dX",")","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","[","0.99244","...","0.00755","...","]",">",">",">","print","(","pca.singular_values_",")","#","doctest",":","+ELLIPSIS","[","6.30061","...","0.54980","...","]",">",">",">","pca","=","PCA","(","n_components=2",",","svd_solver='full","'",")",">",">",">","pca.fit","(","dX",")","#","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","[","0.99244","...","0.00755","...","]",">",">",">","print","(","pca.singular_values_",")","#","doctest",":","+ELLIPSIS","[","6.30061","...","0.54980","...","]","Notes","--","--","-","Differences","from","scikit-learn",":","*","svd_solver",":","'randomized","'","uses","``","dask.linalg.svd_compressed","``","'full","'","uses","``","dask.linalg.svd","``",",","'arpack","'","is","not","valid",".","*","iterated_power",":","defaults","to","``","0","``",",","the","default","for","``","dask.linalg.svd_compressed","``",".","*","n_components",":","``","n_components='mle","'","``","is","not","allowed",".","Fractional","``","n_components","``","between","0","and","1","is","not","allowed","."]
code: def _fit(self, X):
      if isinstance(X, dd.DataFrame):
          X = X.values
  
      # Handle n_components==None
      if self.n_components is None:
          # TODO: handle nan shapes
          n_components = min(X.shape)
      elif 0 < self.n_components < 1:
          raise NotImplementedError(
              "Fractional 'n_components' is not " "currently supported"
          )
      else:
          n_components = self.n_components
  
      solver = self._get_solver(X, n_components)
  
      self.mean_ = X.mean(0)
      X -= self.mean_
  
      if solver in {"full", "tsqr"}:
          U, S, V = da.linalg.svd(X)
      else:
          # randomized
          random_state = check_random_state(self.random_state)
          seed = draw_seed(random_state, np.iinfo("int32").max)
          n_power_iter = self.iterated_power
          U, S, V = da.linalg.svd_compressed(
              X, n_components, n_power_iter=n_power_iter, seed=seed
          )
      if not DASK_2_26_0:
          U, V = svd_flip(U, V)
      components, singular_values = V, S
  
      if solver == "randomized":
          total_variance = X.var(ddof=1, axis=0).sum()
      else:
          total_variance = np.nan
  
      try:
          (
              (self.n_samples_, self.n_features_in_),
              self.n_components_,
              self.components_,
              self.singular_values_,
              total_variance,
          ) = compute(
              delayed(lambda x: x.shape)(X),
              n_components,
              components,
              singular_values,
              total_variance,
          )
      except ValueError as e:
          if np.isnan(X.shape).any():
              msg = (
                  "Computation of the SVD raised an error. It is possible "
                  "n_components is too large. i.e., "
                  "`n_components > np.nanmin(X.shape) = "
                  "np.nanmin({})`\n\n"
                  "A possible resolution to this error is to ensure that "
                  "n_components <= min(n_samples, n_features)"
              )
              raise ValueError(msg.format(X.shape)) from e
          raise e
  
      self.explained_variance_ = self.singular_values_**2 / (
          self.n_samples_ - 1
      )
  
      if solver == "randomized":
          self.explained_variance_ratio_ = (
              self.explained_variance_ / total_variance
          )
      else:
          self.explained_variance_ratio_ = (
              self.explained_variance_ / self.explained_variance_.sum()
          )
  
      # Compute noise covariance using Probabilistic PCA model
      # The sigma2 maximum likelihood (cf. eq. 12.46)
      self.noise_variance_ = 0.0
      if n_components < min(self.n_features_in_, self.n_samples_):
          if solver == "randomized":
              self.noise_variance_ = (
                  total_variance - self.explained_variance_.sum()
              ) / (
                  min(self.n_features_in_, self.n_samples_)
                  - n_components
              )
          else:
              if n_components < len(self.explained_variance_):
                  self.noise_variance_ = self.explained_variance_[
                      n_components:
                  ].mean()
  
      self.components_ = self.components_[:n_components]
      self.explained_variance_ = self.explained_variance_[:n_components]
      self.explained_variance_ratio_ = self.explained_variance_ratio_[
          :n_components
      ]
      self.singular_values_ = self.singular_values_[:n_components]
  
      if len(self.singular_values_) < n_components:
          self.n_components_ = len(self.singular_values_)
          msg = (
              "n_components={n} is larger than the number of singular values"
              " ({s}) (note: PCA has attributes as if n_components == {s})"
          )
          raise ValueError(
              msg.format(n=n_components, s=len(self.singular_values_))
          )
  
      return U, S, V
  
imports: import numbers
  import dask
  import dask.array
  import dask.dataframe
  import numpy
  import sklearn.decomposition
  from dask import compute, delayed
  from sklearn.utils.extmath import fast_logdet
  from sklearn.utils.validation import check_is_fitted, check_random_state
  from .._compat import DASK_2_26_0
  from .._utils import draw_seed
  from ..utils import svd_flip
label_desc: Use image node_id 4 for calling the PCA obj's underlying member method code with example usage: obj._fit(X) and returns: U,  S,  V