"""
Linear Discriminant Analysis and Quadratic Discriminant Analysis
"""

# Authors: Clemens Brunner
#          Martin Billinger
#          Matthieu Perrot
#          Mathieu Blondel

# License: BSD 3-Clause

import warnings
import numpy as np
from scipy import linalg
from scipy.special import expit

from .base import BaseEstimator, TransformerMixin, ClassifierMixin
from .linear_model._base import LinearClassifierMixin
from .covariance import ledoit_wolf, empirical_covariance, shrunk_covariance
from .utils.multiclass import unique_labels
from .utils import check_array
from .utils.validation import check_is_fitted
from .utils.multiclass import check_classification_targets
from .utils.extmath import softmax
from .preprocessing import StandardScaler
from .utils.validation import _deprecate_positional_args


__all__ = ['LinearDiscriminantAnalysis', 'QuadraticDiscriminantAnalysis']


def _cov(X, shrinkage=None):
    """Estimate covariance matrix (using optional shrinkage).

    Parameters
    ----------
    X : array-like of shape (n_samples, n_features)
        Input data.

    shrinkage : {'empirical', 'auto'} or float, default=None
        Shrinkage parameter, possible values:
          - None or 'empirical': no shrinkage (default).
          - 'auto': automatic shrinkage using the Ledoit-Wolf lemma.
          - float between 0 and 1: fixed shrinkage parameter.

    Returns
    -------
    s : ndarray of shape (n_features, n_features)
        Estimated covariance matrix.
    """
    shrinkage = "empirical" if shrinkage is None else shrinkage
    if isinstance(shrinkage, str):
        if shrinkage == 'auto':
            sc = StandardScaler()  # standardize features
            X = sc.fit_transform(X)
            s = ledoit_wolf(X)[0]
            # rescale
            s = sc.scale_[:, np.newaxis] * s * sc.scale_[np.newaxis, :]
        elif shrinkage == 'empirical':
            s = empirical_covariance(X)
        else:
            raise ValueError('unknown shrinkage parameter')
    elif isinstance(shrinkage, float) or isinstance(shrinkage, int):
        if shrinkage < 0 or shrinkage > 1:
            raise ValueError('shrinkage parameter must be between 0 and 1')
        s = shrunk_covariance(empirical_covariance(X), shrinkage)
    else:
        raise TypeError('shrinkage must be of string or int type')
    return s


def _class_means(X, y):
    """Compute class means.

    Parameters
    ----------
    X : array-like of shape (n_samples, n_features)
        Input data.

    y : array-like of shape (n_samples,) or (n_samples, n_targets)
        Target values.

    Returns
    -------
    means : array-like of shape (n_classes, n_features)
        Class means.
    """
    classes, y = np.unique(y, return_inverse=True)
    cnt = np.bincount(y)
    means = np.zeros(shape=(len(classes), X.shape[1]))
    np.add.at(means, y, X)
    means /= cnt[:, None]
    return means


def _class_cov(X, y, priors, shrinkage=None):
    """Compute weighted within-class covariance matrix.

    The per-class covariance are weighted by the class priors.

    Parameters
    ----------
    X : array-like of shape (n_samples, n_features)
        Input data.

    y : array-like of shape (n_samples,) or (n_samples, n_targets)
        Target values.

    priors : array-like of shape (n_classes,)
        Class priors.

    shrinkage : 'auto' or float, default=None
        Shrinkage parameter, possible values:
          - None: no shrinkage (default).
          - 'auto': automatic shrinkage using the Ledoit-Wolf lemma.
          - float between 0 and 1: fixed shrinkage parameter.

    Returns
    -------
    cov : array-like of shape (n_features, n_features)
        Weighted within-class covariance matrix
    """
    classes = np.unique(y)
    cov = np.zeros(shape=(X.shape[1], X.shape[1]))
    for idx, group in enumerate(classes):
        Xg = X[y == group, :]
        cov += priors[idx] * np.atleast_2d(_cov(Xg, shrinkage))
    return cov


class LinearDiscriminantAnalysis(BaseEstimator, LinearClassifierMixin,
                                 TransformerMixin):
    """Linear Discriminant Analysis

    A classifier with a linear decision boundary, generated by fitting class
    conditional densities to the data and using Bayes' rule.

    The model fits a Gaussian density to each class, assuming that all classes
    share the same covariance matrix.

    The fitted model can also be used to reduce the dimensionality of the input
    by projecting it to the most discriminative directions, using the
    `transform` method.

    .. versionadded:: 0.17
       *LinearDiscriminantAnalysis*.

    Read more in the :ref:`User Guide <lda_qda>`.

    Parameters
    ----------
    solver : {'svd', 'lsqr', 'eigen'}, default='svd'
        Solver to use, possible values:
          - 'svd': Singular value decomposition (default).
            Does not compute the covariance matrix, therefore this solver is
            recommended for data with a large number of features.
          - 'lsqr': Least squares solution, can be combined with shrinkage.
          - 'eigen': Eigenvalue decomposition, can be combined with shrinkage.

    shrinkage : 'auto' or float, default=None
        Shrinkage parameter, possible values:
          - None: no shrinkage (default).
          - 'auto': automatic shrinkage using the Ledoit-Wolf lemma.
          - float between 0 and 1: fixed shrinkage parameter.

        Note that shrinkage works only with 'lsqr' and 'eigen' solvers.

    priors : array-like of shape (n_classes,), default=None
        The class prior probabilities. By default, the class proportions are
        inferred from the training data.

    n_components : int, default=None
        Number of components (<= min(n_classes - 1, n_features)) for
        dimensionality reduction. If None, will be set to
        min(n_classes - 1, n_features). This parameter only affects the
        `transform` method.

    store_covariance : bool, default=False
        If True, explicitely compute the weighted within-class covariance
        matrix when solver is 'svd'. The matrix is always computed
        and stored for the other solvers.

        .. versionadded:: 0.17

    tol : float, default=1.0e-4
        Absolute threshold for a singular value of X to be considered
        significant, used to estimate the rank of X. Dimensions whose
        singular values are non-significant are discarded. Only used if
        solver is 'svd'.

        .. versionadded:: 0.17

    Attributes
    ----------
    coef_ : ndarray of shape (n_features,) or (n_classes, n_features)
        Weight vector(s).

    intercept_ : ndarray of shape (n_classes,)
        Intercept term.

    covariance_ : array-like of shape (n_features, n_features)
        Weighted within-class covariance matrix. It corresponds to
        `sum_k prior_k * C_k` where `C_k` is the covariance matrix of the
        samples in class `k`. The `C_k` are estimated using the (potentially
        shrunk) biased estimator of covariance. If solver is 'svd', only
        exists when `store_covariance` is True.

    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 explained variances is equal to 1.0. Only available when eigen
        or svd solver is used.

    means_ : array-like of shape (n_classes, n_features)
        Class-wise means.

    priors_ : array-like of shape (n_classes,)
        Class priors (sum to 1).

    scalings_ : array-like of shape (rank, n_classes - 1)
        Scaling of the features in the space spanned by the class centroids.
        Only available for 'svd' and 'eigen' solvers.

    xbar_ : array-like of shape (n_features,)
        Overall mean. Only present if solver is 'svd'.

    classes_ : array-like of shape (n_classes,)
        Unique class labels.

    See also
    --------
    sklearn.discriminant_analysis.QuadraticDiscriminantAnalysis: Quadratic
        Discriminant Analysis

    Examples
    --------
    >>> import numpy as np
    >>> from sklearn.discriminant_analysis import LinearDiscriminantAnalysis
    >>> X = np.array([[-1, -1], [-2, -1], [-3, -2], [1, 1], [2, 1], [3, 2]])
    >>> y = np.array([1, 1, 1, 2, 2, 2])
    >>> clf = LinearDiscriminantAnalysis()
    >>> clf.fit(X, y)
    LinearDiscriminantAnalysis()
    >>> print(clf.predict([[-0.8, -1]]))
    [1]
    """
    @_deprecate_positional_args
    def __init__(self, *, solver='svd', shrinkage=None, priors=None,
                 n_components=None, store_covariance=False, tol=1e-4):
        self.solver = solver
        self.shrinkage = shrinkage
        self.priors = priors
        self.n_components = n_components
        self.store_covariance = store_covariance  # used only in svd solver
        self.tol = tol  # used only in svd solver

    def _solve_lsqr(self, X, y, shrinkage):
        """Least squares solver.

        The least squares solver computes a straightforward solution of the
        optimal decision rule based directly on the discriminant functions. It
        can only be used for classification (with optional shrinkage), because
        estimation of eigenvectors is not performed. Therefore, dimensionality
        reduction with the transform is not supported.

        Parameters
        ----------
        X : array-like of shape (n_samples, n_features)
            Training data.

        y : array-like of shape (n_samples,) or (n_samples, n_classes)
            Target values.

        shrinkage : 'auto', float or None
            Shrinkage parameter, possible values:
              - None: no shrinkage.
              - 'auto': automatic shrinkage using the Ledoit-Wolf lemma.
              - float between 0 and 1: fixed shrinkage parameter.

        Notes
        -----
        This solver is based on [1]_, section 2.6.2, pp. 39-41.

        References
        ----------
        .. [1] R. O. Duda, P. E. Hart, D. G. Stork. Pattern Classification
           (Second Edition). John Wiley & Sons, Inc., New York, 2001. ISBN
           0-471-05669-3.
        """
        self.means_ = _class_means(X, y)
        self.covariance_ = _class_cov(X, y, self.priors_, shrinkage)
        self.coef_ = linalg.lstsq(self.covariance_, self.means_.T)[0].T
        self.intercept_ = (-0.5 * np.diag(np.dot(self.means_, self.coef_.T)) +
                           np.log(self.priors_))

    def _solve_eigen(self, X, y, shrinkage):
        """Eigenvalue solver.

        The eigenvalue solver computes the optimal solution of the Rayleigh
        coefficient (basically the ratio of between class scatter to within
        class scatter). This solver supports both classification and
        dimensionality reduction (with optional shrinkage).

        Parameters
        ----------
        X : array-like of shape (n_samples, n_features)
            Training data.

        y : array-like of shape (n_samples,) or (n_samples, n_targets)
            Target values.

        shrinkage : 'auto', float or None
            Shrinkage parameter, possible values:
              - None: no shrinkage.
              - 'auto': automatic shrinkage using the Ledoit-Wolf lemma.
              - float between 0 and 1: fixed shrinkage constant.

        Notes
        -----
        This solver is based on [1]_, section 3.8.3, pp. 121-124.

        References
        ----------
        .. [1] R. O. Duda, P. E. Hart, D. G. Stork. Pattern Classification
           (Second Edition). John Wiley & Sons, Inc., New York, 2001. ISBN
           0-471-05669-3.
        """
        self.means_ = _class_means(X, y)
        self.covariance_ = _class_cov(X, y, self.priors_, shrinkage)

        Sw = self.covariance_  # within scatter
        St = _cov(X, shrinkage)  # total scatter
        Sb = St - Sw  # between scatter

        evals, evecs = linalg.eigh(Sb, Sw)
        self.explained_variance_ratio_ = np.sort(evals / np.sum(evals)
                                                 )[::-1][:self._max_components]
        evecs = evecs[:, np.argsort(evals)[::-1]]  # sort eigenvectors

        self.scalings_ = evecs
        self.coef_ = np.dot(self.means_, evecs).dot(evecs.T)
        self.intercept_ = (-0.5 * np.diag(np.dot(self.means_, self.coef_.T)) +
                           np.log(self.priors_))

    def _solve_svd(self, X, y):