"""
Python implementation of the fast ICA algorithms.

Reference: Tables 8.3 and 8.4 page 196 in the book:
Independent Component Analysis, by  Hyvarinen et al.
"""

# Authors: Pierre Lafaye de Micheaux, Stefan van der Walt, Gael Varoquaux,
#          Bertrand Thirion, Alexandre Gramfort, Denis A. Engemann
# License: BSD 3 clause

import warnings

import numpy as np
from scipy import linalg

from ..base import BaseEstimator, TransformerMixin
from ..exceptions import ConvergenceWarning
from ..externals import six
from ..externals.six import moves
from ..externals.six import string_types
from ..utils import check_array, as_float_array, check_random_state
from ..utils.validation import check_is_fitted
from ..utils.validation import FLOAT_DTYPES

__all__ = ['fastica', 'FastICA']


def _gs_decorrelation(w, W, j):
    """
    Orthonormalize w wrt the first j rows of W

    Parameters
    ----------
    w : ndarray of shape(n)
        Array to be orthogonalized

    W : ndarray of shape(p, n)
        Null space definition

    j : int < p
        The no of (from the first) rows of Null space W wrt which w is
        orthogonalized.

    Notes
    -----
    Assumes that W is orthogonal
    w changed in place
    """
    w -= np.dot(np.dot(w, W[:j].T), W[:j])
    return w


def _sym_decorrelation(W):
    """ Symmetric decorrelation
    i.e. W <- (W * W.T) ^{-1/2} * W
    """
    s, u = linalg.eigh(np.dot(W, W.T))
    # u (resp. s) contains the eigenvectors (resp. square roots of
    # the eigenvalues) of W * W.T
    return np.dot(np.dot(u * (1. / np.sqrt(s)), u.T), W)


def _ica_def(X, tol, g, fun_args, max_iter, w_init):
    """Deflationary FastICA using fun approx to neg-entropy function

    Used internally by FastICA.
    """

    n_components = w_init.shape[0]
    W = np.zeros((n_components, n_components), dtype=X.dtype)
    n_iter = []

    # j is the index of the extracted component
    for j in range(n_components):
        w = w_init[j, :].copy()
        w /= np.sqrt((w ** 2).sum())

        for i in moves.xrange(max_iter):
            gwtx, g_wtx = g(np.dot(w.T, X), fun_args)

            w1 = (X * gwtx).mean(axis=1) - g_wtx.mean() * w

            _gs_decorrelation(w1, W, j)

            w1 /= np.sqrt((w1 ** 2).sum())

            lim = np.abs(np.abs((w1 * w).sum()) - 1)
            w = w1
            if lim < tol:
                break

        n_iter.append(i + 1)
        W[j, :] = w

    return W, max(n_iter)


def _ica_par(X, tol, g, fun_args, max_iter, w_init):
    """Parallel FastICA.

    Used internally by FastICA --main loop

    """
    W = _sym_decorrelation(w_init)
    del w_init
    p_ = float(X.shape[1])
    for ii in moves.xrange(max_iter):
        gwtx, g_wtx = g(np.dot(W, X), fun_args)
        W1 = _sym_decorrelation(np.dot(gwtx, X.T) / p_
                                - g_wtx[:, np.newaxis] * W)
        del gwtx, g_wtx
        # builtin max, abs are faster than numpy counter parts.
        lim = max(abs(abs(np.diag(np.dot(W1, W.T))) - 1))
        W = W1
        if lim < tol:
            break
    else:
        warnings.warn('FastICA did not converge. Consider increasing '
                      'tolerance or the maximum number of iterations.',
                      ConvergenceWarning)

    return W, ii + 1


# Some standard non-linear functions.
# XXX: these should be optimized, as they can be a bottleneck.
def _logcosh(x, fun_args=None):
    alpha = fun_args.get('alpha', 1.0)  # comment it out?

    x *= alpha
    gx = np.tanh(x, x)  # apply the tanh inplace
    g_x = np.empty(x.shape[0])
    # XXX compute in chunks to avoid extra allocation
    for i, gx_i in enumerate(gx):  # please don't vectorize.
        g_x[i] = (alpha * (1 - gx_i ** 2)).mean()
    return gx, g_x


def _exp(x, fun_args):
    exp = np.exp(-(x ** 2) / 2)
    gx = x * exp
    g_x = (1 - x ** 2) * exp
    return gx, g_x.mean(axis=-1)


def _cube(x, fun_args):
    return x ** 3, (3 * x ** 2).mean(axis=-1)


def fastica(X, n_components=None, algorithm="parallel", whiten=True,
            fun="logcosh", fun_args=None, max_iter=200, tol=1e-04, w_init=None,
            random_state=None, return_X_mean=False, compute_sources=True,
            return_n_iter=False):
    """Perform Fast Independent Component Analysis.

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

    Parameters
    ----------
    X : array-like, shape (n_samples, n_features)
        Training vector, where n_samples is the number of samples and
        n_features is the number of features.

    n_components : int, optional
        Number of components to extract. If None no dimension reduction
        is performed.

    algorithm : {'parallel', 'deflation'}, optional
        Apply a parallel or deflational FASTICA algorithm.

    whiten : boolean, optional
        If True perform an initial whitening of the data.
        If False, the data is assumed to have already been
        preprocessed: it should be centered, normed and white.
        Otherwise you will get incorrect results.
        In this case the parameter n_components will be ignored.

    fun : string or function, optional. Default: 'logcosh'
        The functional form of the G function used in the
        approximation to neg-entropy. Could be either 'logcosh', 'exp',
        or 'cube'.
        You can also provide your own function. It should return a tuple
        containing the value of the function, and of its derivative, in the
        point. The derivative should be averaged along its last dimension.
        Example:

        def my_g(x):
            return x ** 3, np.mean(3 * x ** 2, axis=-1)

    fun_args : dictionary, optional
        Arguments to send to the functional form.
        If empty or None and if fun='logcosh', fun_args will take value
        {'alpha' : 1.0}

    max_iter : int, optional
        Maximum number of iterations to perform.

    tol : float, optional
        A positive scalar giving the tolerance at which the
        un-mixing matrix is considered to have converged.

    w_init : (n_components, n_components) array, optional
        Initial un-mixing array of dimension (n.comp,n.comp).
        If None (default) then an array of normal r.v.'s is used.

    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`.

    return_X_mean : bool, optional
        If True, X_mean is returned too.

    compute_sources : bool, optional
        If False, sources are not computed, but only the rotation matrix.
        This can save memory when working with big data. Defaults to True.

    return_n_iter : bool, optional
        Whether or not to return the number of iterations.

    Returns
    -------
    K : array, shape (n_components, n_features) | None.
        If whiten is 'True', K is the pre-whitening matrix that projects data
        onto the first n_components principal components. If whiten is 'False',
        K is 'None'.

    W : array, shape (n_components, n_components)
        Estimated un-mixing matrix.
        The mixing matrix can be obtained by::

            w = np.dot(W, K.T)
            A = w.T * (w * w.T).I

    S : array, shape (n_samples, n_components) | None
        Estimated source matrix

    X_mean : array, shape (n_features, )
        The mean over features. Returned only if return_X_mean is True.

    n_iter : int
        If the algorithm is "deflation", n_iter is the
        maximum number of iterations run across all components. Else
        they are just the number of iterations taken to converge. This is
        returned only when return_n_iter is set to `True`.

    Notes
    -----

    The data matrix X is considered to be a linear combination of
    non-Gaussian (independent) components i.e. X = AS where columns of S
    contain the independent components and A is a linear mixing
    matrix. In short ICA attempts to `un-mix' the data by estimating an
    un-mixing matrix W where ``S = W K X.``

    This implementation was originally made for data of shape
    [n_features, n_samples]. Now the input is transposed
    before the algorithm is applied. This makes it slightly
    faster for Fortran-ordered input.

    Implemented using FastICA:
    `A. Hyvarinen and E. Oja, Independent Component Analysis:
    Algorithms and Applications, Neural Networks, 13(4-5), 2000,
    pp. 411-430`

    """
    random_state = check_random_state(random_state)
    fun_args = {} if fun_args is None else fun_args
    # make interface compatible with other decompositions
    # a copy is required only for non whitened data
    X = check_array(X, copy=whiten, dtype=FLOAT_DTYPES,
                    ensure_min_samples=2).T

    alpha = fun_args.get('alpha', 1.0)
    if not 1 <= alpha <= 2:
        raise ValueError('alpha must be in [1,2]')

    if fun == 'logcosh':
        g = _logcosh
    elif fun == 'exp':
        g = _exp
    elif fun == 'cube':
        g = _cube
    elif callable(fun):
        def g(x, fun_args):
            return fun(x, **fun_args)
    else:
        exc = ValueError if isinstance(fun, six.string_types) else TypeError
        raise exc("Unknown function %r;"
                  " should be one of 'logcosh', 'exp', 'cube' or callable"
                  % fun)

    n, p = X.shape

    if not whiten and n_components is not None:
        n_components = None
        warnings.warn('Ignoring n_components with whiten=False.')

    if n_components is None:
        n_components = min(n, p)
    if (n_components > min(n, p)):
        n_components = min(n, p)
        warnings.warn('n_components is too large: it will be set to %s' % n_components)

    if whiten:
        # Centering the columns (ie the variables)
        X_mean = X.mean(axis=-1)
        X -= X_mean[:, np.newaxis]

        # Whitening and preprocessing by PCA
        u, d, _ = linalg.svd(X, full_matrices=False)

        del _
        K = (u / d).T[:n_components]  # see (6.33) p.140
        del u, d
        X1 = np.dot(K, X)
        # see (13.6) p.267 Here X1 is white and data
        # in X has been projected onto a subspace by PCA
        X1 *= np.sqrt(p)
    else:
        # X must be casted to floats to avoid typing issues with numpy
        # 2.0 and the line below
        X1 = as_float_array(X, copy=False)  # copy has been taken care of

    if w_init is None:
        w_init = np.asarray(random_state.normal(size=(n_components,
                            n_components)), dtype=X1.dtype)

    else:
        w_init = np.asarray(w_init)
        if w_init.shape != (n_components, n_components):
            raise ValueError('w_init has invalid shape -- should be %(shape)s'
                             % {'shape': (n_components, n_components)})

    kwargs = {'tol': tol,
              'g': g,
              'fun_args': fun_args,
              'max_iter': max_iter,
              'w_init': w_init}

    if algorithm == 'parallel':
        W, n_iter = _ica_par(X1, **kwargs)
    elif algorithm == 'deflation':
        W, n_iter = _ica_def(X1, **kwargs)
    else:
        raise ValueError('Invalid algorithm: must be either `parallel` or'
                         ' `deflation`.')
    del X1

    if whiten:
        if compute_sources:
            S = np.dot(np.dot(W, K), X).T
        else:
            S = None
        if return_X_mean:
            if return_n_iter:
                return K, W, S, X_mean, n_iter
            else:
                return K, W, S, X_mean
        else:
            if return_n_iter:
                return K, W, S, n_iter
            else:
                return K, W, S

    else:
        if compute_sources:
            S = np.dot(W, X).T
        else:
            S = None
        if return_X_mean:
            if return_n_iter:
                return None, W, S, None, n_iter
            else:
                return None, W, S, None
        else:
            if return_n_iter:
                return None, W, S, n_iter
            else:
                return None, W, S


class FastICA(BaseEstimator, TransformerMixin):
    """FastICA: a fast algorithm for Independent Component Analysis.

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

    Parameters
    ----------
    n_components : int, optional
        Number of components to use. If none is passed, all are used.

    algorithm : {'parallel', 'deflation'}
        Apply parallel or deflational algorithm for FastICA.

    whiten : boolean, optional
        If whiten is false, the data is already considered to be
        whitened, and no whitening is performed.

    fun : string or function, optional. Default: 'logcosh'
        The functional form of the G function used in the
        approximation to neg-entropy. Could be either 'logcosh', 'exp',
        or 'cube'.
        You can also provide your own function. It should return a tuple
        containing the value of the function, and of its derivative, in the
        point. Example:

        def my_g(x):
            return x ** 3, 3 * x ** 2

    fun_args : dictionary, optional
        Arguments to send to the functional form.
        If empty and if fun='logcosh', fun_args will take value
        {'alpha' : 1.0}.

    max_iter : int, optional
        Maximum number of iterations during fit.

    tol : float, optional
        Tolerance on update at each iteration.

    w_init : None of an (n_components, n_components) ndarray
        The mixing matrix to be used to initialize the algorithm.

    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`.

    Attributes
    ----------
    components_ : 2D array, shape (n_components, n_features)
        The unmixing matrix.

    mixing_ : array, shape (n_features, n_components)
        The mixing matrix.

    n_iter_ : int
        If the algorithm is "deflation", n_iter is the
        maximum number of iterations run across all components. Else
        they are just the number of iterations taken to converge.

    Examples
    --------
    >>> from sklearn.datasets import load_digits
    >>> from sklearn.decomposition import FastICA
    >>> X, _ = load_digits(return_X_y=True)
    >>> transformer = FastICA(n_components=7,
    ...         random_state=0)
    >>> X_transformed = transformer.fit_transform(X)
    >>> X_transformed.shape
    (1797, 7)

    Notes
    -----
    Implementation based on
    `A. Hyvarinen and E. Oja, Independent Component Analysis:
    Algorithms and Applications, Neural Networks, 13(4-5), 2000,
    pp. 411-430`

    """
    def __init__(self, n_components=None, algorithm='parallel', whiten=True,
                 fun='logcosh', fun_args=None, max_iter=200, tol=1e-4,
                 w_init=None, random_state=None):
        super(FastICA, self).__init__()
        if max_iter < 1:
            raise ValueError("max_iter should be greater than 1, got "
                             "(max_iter={})".format(max_iter))
        self.n_components = n_components
        self.algorithm = algorithm
        self.whiten = whiten
        self.fun = fun
        self.fun_args = fun_args
        self.max_iter = max_iter
        self.tol = tol
        self.w_init = w_init
        self.random_state = random_state

    def _fit(self, X, compute_sources=False):
        """Fit the model

        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.

        compute_sources : bool
            If False, sources are not computes but only the rotation matrix.
            This can save memory when working with big data. Defaults to False.

        Returns
        -------
            X_new : array-like, shape (n_samples, n_components)
        """
        fun_args = {} if self.fun_args is None else self.fun_args
        whitening, unmixing, sources, X_mean, self.n_iter_ = fastica(
            X=X, n_components=self.n_components, algorithm=self.algorithm,
            whiten=self.whiten, fun=self.fun, fun_args=fun_args,
            max_iter=self.max_iter, tol=self.tol, w_init=self.w_init,
            random_state=self.random_state, return_X_mean=True,
            compute_sources=compute_sources, return_n_iter=True)

        if self.whiten:
            self.components_ = np.dot(unmixing, whitening)
            self.mean_ = X_mean
            self.whitening_ = whitening
        else:
            self.components_ = unmixing

        self.mixing_ = linalg.pinv(self.components_)

        if compute_sources:
            self.__sources = sources

        return sources

    def fit_transform(self, X, y=None):
        """Fit the model and recover the sources from 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 : Ignored

        Returns
        -------
        X_new : array-like, shape (n_samples, n_components)
        """
        return self._fit(X, compute_sources=True)

    def fit(self, X, y=None):
        """Fit the model to 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 : Ignored

        Returns
        -------
        self
        """
        self._fit(X, compute_sources=False)
        return self

    def transform(self, X, y='deprecated', copy=True):
        """Recover the sources from X (apply the unmixing matrix).

        Parameters
        ----------
        X : array-like, shape (n_samples, n_features)
            Data to transform, where n_samples is the number of samples
            and n_features is the number of features.
        y : (ignored)
            .. deprecated:: 0.19
               This parameter will be removed in 0.21.
        copy : bool (optional)
            If False, data passed to fit are overwritten. Defaults to True.

        Returns
        -------
        X_new : array-like, shape (n_samples, n_components)
        """
        if not isinstance(y, string_types) or y != 'deprecated':
            warnings.warn("The parameter y on transform() is "
                          "deprecated since 0.19 and will be removed in 0.21",
                          DeprecationWarning)

        check_is_fitted(self, 'mixing_')

        X = check_array(X, copy=copy, dtype=FLOAT_DTYPES)
        if self.whiten:
            X -= self.mean_

        return np.dot(X, self.components_.T)

    def inverse_transform(self, X, copy=True):
        """Transform the sources back to the mixed data (apply mixing matrix).

        Parameters
        ----------
        X : array-like, shape (n_samples, n_components)
            Sources, where n_samples is the number of samples
            and n_components is the number of components.
        copy : bool (optional)
            If False, data passed to fit are overwritten. Defaults to True.

        Returns
        -------
        X_new : array-like, shape (n_samples, n_features)
        """
        check_is_fitted(self, 'mixing_')

        X = check_array(X, copy=(copy and self.whiten), dtype=FLOAT_DTYPES)
        X = np.dot(X, self.mixing_.T)
        if self.whiten:
            X += self.mean_

        return X