_pca.py

""" 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
from ..utils.validation import _deprecate_positional_args


def _assess_dimension(spectrum, rank, n_samples):
    """Compute the log-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_features)
        Data spectrum.
    rank : int
        Tested rank value. It should be strictly lower than n_features,
        otherwise the method isn't specified (division by zero in equation
        (31) from the paper).
    n_samples : int
        Number of samples.

    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`
    """

    n_features = spectrum.shape[0]
    if not 1 <= rank < n_features:
        raise ValueError("the tested rank should be in [1, n_features - 1]")

    eps = 1e-15

    if spectrum[rank - 1] < eps:
        # When the tested rank is associated with a small eigenvalue, there's
        # no point in computing the log-likelihood: it's going to be very
        # small and won't be the max anyway. Also, it can lead to numerical
        # issues below when computing pa, in particular in log((spectrum[i] -
        # spectrum[j]) because this will take the log of something very small.
        return -np.inf

    pu = -rank * log(2.)
    for i in range(1, rank + 1):
        pu += (gammaln((n_features - i + 1) / 2.) -
               log(np.pi) * (n_features - i + 1) / 2.)

    pl = np.sum(np.log(spectrum[:rank]))
    pl = -pl * n_samples / 2.

    v = max(eps, 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) / 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):
    """Infers the dimension of a dataset with a given spectrum.

    The returned value will be in [1, n_features - 1].
    """
    ll = np.empty_like(spectrum)
    ll[0] = -np.inf  # we don't want to return n_components = 0
    for rank in range(1, spectrum.shape[0]):
        ll[rank] = _assess_dimension(spectrum, rank, n_samples)
    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, 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_ : 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...]
    """
    @_deprecate_positional_args
    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, Vt = 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 * Vt * 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 = self._validate_data(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, Vt = linalg.svd(X, full_matrices=False)
        # flip eigenvectors' sign to enforce deterministic output
        U, Vt = svd_flip(U, Vt)

        components_ = Vt

        # 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)
        elif 0 < n_components < 1.0:
            # number of components for which the cumulated explained
            # variance percentage is superior to the desired threshold
            # side='right' ensures that number of features selected
            # their variance is always greater than n_components float
            # passed. More discussion in issue: #15669
            ratio_cumsum = stable_cumsum(explained_variance_ratio_)
            n_components = np.searchsorted(ratio_cumsum, n_components,
                                           side='right') + 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, Vt

    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, 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_iter=self.iterated_power,
                                      flip_sign=True,
                                      random_state=random_state)

        self.n_samples_, self.n_features_ = n_samples, n_features
        self.components_ = Vt
        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, Vt

    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))