""" 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 numpy as np
from scipy import linalg
from scipy.special import gammaln
from scipy.sparse import issparse
from scipy.sparse.linalg import svds

from ..externals import six

from .base import _BasePCA
from ..base import BaseEstimator, TransformerMixin
from ..utils import deprecated
from ..utils import check_random_state, as_float_array
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.

    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
        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
        n_components cannot be equal to n_features for svd_solver == 'arpack'.

    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 < X.shape[1]
        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.

        .. 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 explained variances 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=1)`.

    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 n_features 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.

    References
    ----------
    For n_components == 'mle', this class uses the method of `Thomas P. Minka:
    Automatic Choice of Dimensionality for PCA. NIPS 2000: 598-604`

    Implements the probabilistic PCA model from:
    M. Tipping and C. Bishop, Probabilistic Principal Component Analysis,
    Journal of the Royal Statistical Society, Series B, 61, Part 3, pp. 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:
    `Finding structure with randomness: Stochastic algorithms
    for constructing approximate matrix decompositions Halko, et al., 2009
    (arXiv:909)`
    `A randomized algorithm for the decomposition of matrices
    Per-Gunnar Martinsson, Vladimir Rokhlin and Mark Tygert`


    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(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(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
    [ 0.99244...  0.00755...]
    >>> print(pca.singular_values_)  # doctest: +ELLIPSIS
    [ 6.30061...  0.54980...]

    >>> pca = PCA(n_components=1, svd_solver='arpack')
    >>> pca.fit(X)
    PCA(copy=True, iterated_power='auto', n_components=1, random_state=None,
      svd_solver='arpack', tol=0.0, whiten=False)
    >>> print(pca.explained_variance_ratio_)  # doctest: +ELLIPSIS
    [ 0.99244...]
    >>> print(pca.singular_values_)  # doctest: +ELLIPSIS
    [ 6.30061...]

    Notes
    -----
    PCA uses the maximum likelihood estimate of the eigenvalues, which does not
    include the Bessel correction, though in practice this should rarely make a
    difference in a machine learning context.

    See also
    --------
    KernelPCA
    SparsePCA
    TruncatedSVD
    IncrementalPCA
    """

    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