csp.py

# -*- coding: utf-8 -*-
# Authors: Romain Trachel <trachelr@gmail.com>
#          Alexandre Gramfort <alexandre.gramfort@telecom-paristech.fr>
#          Alexandre Barachant <alexandre.barachant@gmail.com>
#          Clemens Brunner <clemens.brunner@gmail.com>
#          Jean-Remi King <jeanremi.king@gmail.com>
#
# License: BSD (3-clause)

import copy as cp

import numpy as np
from scipy import linalg

from .mixin import TransformerMixin
from .base import BaseEstimator
from ..cov import _regularized_covariance


class CSP(TransformerMixin, BaseEstimator):
    u"""M/EEG signal decomposition using the Common Spatial Patterns (CSP).

    This object can be used as a supervised decomposition to estimate
    spatial filters for feature extraction in a 2 class decoding problem.
    CSP in the context of EEG was first described in [1]; a comprehensive
    tutorial on CSP can be found in [2]. Multiclass solving is implemented
    from [3].

    Parameters
    ----------
    n_components : int, defaults to 4
        The number of components to decompose M/EEG signals.
        This number should be set by cross-validation.
    reg : float | str | None, defaults to None
        if not None, allow regularization for covariance estimation
        if float, shrinkage covariance is used (0 <= shrinkage <= 1).
        if str, optimal shrinkage using Ledoit-Wolf Shrinkage ('ledoit_wolf')
        or Oracle Approximating Shrinkage ('oas').
    log : None | bool, defaults to None
        If transform_into == 'average_power' and log is None or True, then
        applies a log transform to standardize the features, else the features
        are z-scored. If transform_into == 'csp_space', then log must be None.
    cov_est : 'concat' | 'epoch', defaults to 'concat'
        If 'concat', covariance matrices are estimated on concatenated epochs
        for each class.
        If 'epoch', covariance matrices are estimated on each epoch separately
        and then averaged over each class.
    transform_into : {'average_power', 'csp_space'}
        If 'average_power' then self.transform will return the average power of
        each spatial filter. If 'csp_space' self.transform will return the data
        in CSP space. Defaults to 'average_power'.

    Attributes
    ----------
    ``filters_`` : ndarray, shape (n_channels, n_channels)
        If fit, the CSP components used to decompose the data, else None.
    ``patterns_`` : ndarray, shape (n_channels, n_channels)
        If fit, the CSP patterns used to restore M/EEG signals, else None.
    ``mean_`` : ndarray, shape (n_components,)
        If fit, the mean squared power for each component.
    ``std_`` : ndarray, shape (n_components,)
        If fit, the std squared power for each component.

    References
    ----------
    [1] Zoltan J. Koles, Michael S. Lazar, Steven Z. Zhou. Spatial Patterns
        Underlying Population Differences in the Background EEG. Brain
        Topography 2(4), 275-284, 1990.
    [2] Benjamin Blankertz, Ryota Tomioka, Steven Lemm, Motoaki Kawanabe,
        Klaus-Robert Müller. Optimizing Spatial Filters for Robust EEG
        Single-Trial Analysis. IEEE Signal Processing Magazine 25(1), 41-56,
        2008.
    [3] Grosse-Wentrup, Moritz, and Martin Buss. Multiclass common spatial
        patterns and information theoretic feature extraction. IEEE
        Transactions on Biomedical Engineering, Vol 55, no. 8, 2008.
    """

    def __init__(self, n_components=4, reg=None, log=None, cov_est="concat",
                 transform_into='average_power'):
        """Init of CSP."""
        # Init default CSP
        if not isinstance(n_components, int):
            raise ValueError('n_components must be an integer.')
        self.n_components = n_components

        # Init default regularization
        if (
            (reg is not None) and
            (reg not in ['oas', 'ledoit_wolf']) and
            ((not isinstance(reg, (float, int))) or
             (not ((reg <= 1.) and (reg >= 0.))))
        ):
            raise ValueError('reg must be None, "oas", "ledoit_wolf" or a '
                             'float in between 0. and 1.')
        self.reg = reg

        # Init default cov_est
        if not (cov_est == "concat" or cov_est == "epoch"):
            raise ValueError("unknown covariance estimation method")
        self.cov_est = cov_est

        # Init default transform_into
        if transform_into not in ('average_power', 'csp_space'):
            raise ValueError('transform_into must be "average_power" or '
                             '"csp_space".')
        self.transform_into = transform_into

        # Init default log
        if transform_into == 'average_power':
            if log is not None and not isinstance(log, bool):
                raise ValueError('log must be a boolean if transform_into == '
                                 '"average_power".')
        else:
            if log is not None:
                raise ValueError('log must be a None if transform_into == '
                                 '"csp_space".')
        self.log = log

    def _check_Xy(self, X, y=None):
        """Aux. function to check input data."""
        if y is not None:
            if len(X) != len(y) or len(y) < 1:
                raise ValueError('X and y must have the same length.')
        if X.ndim < 3:
            raise ValueError('X must have at least 3 dimensions.')

    def fit(self, X, y):
        """Estimate the CSP decomposition on epochs.

        Parameters
        ----------
        X : ndarray, shape (n_epochs, n_channels, n_times)
            The data on which to estimate the CSP.
        y : array, shape (n_epochs,)
            The class for each epoch.

        Returns
        -------
        self : instance of CSP
            Returns the modified instance.
        """
        if not isinstance(X, np.ndarray):
            raise ValueError("X should be of type ndarray (got %s)."
                             % type(X))
        self._check_Xy(X, y)
        n_channels = X.shape[1]

        self._classes = np.unique(y)
        n_classes = len(self._classes)
        if n_classes < 2:
            raise ValueError("n_classes must be >= 2.")

        covs = np.zeros((n_classes, n_channels, n_channels))
        sample_weights = list()
        for class_idx, this_class in enumerate(self._classes):
            if self.cov_est == "concat":  # concatenate epochs
                class_ = np.transpose(X[y == this_class], [1, 0, 2])
                class_ = class_.reshape(n_channels, -1)
                cov = _regularized_covariance(class_, reg=self.reg)
                weight = sum(y == this_class)
            elif self.cov_est == "epoch":
                class_ = X[y == this_class]
                cov = np.zeros((n_channels, n_channels))
                for this_X in class_:
                    cov += _regularized_covariance(this_X, reg=self.reg)
                cov /= len(class_)
                weight = len(class_)

            # normalize by trace and stack
            covs[class_idx] = cov / np.trace(cov)
            sample_weights.append(weight)

        if n_classes == 2:
            eigen_values, eigen_vectors = linalg.eigh(covs[0], covs.sum(0))
            # sort eigenvectors
            ix = np.argsort(np.abs(eigen_values - 0.5))[::-1]
        else:
            # The multiclass case is adapted from
            # http://github.com/alexandrebarachant/pyRiemann
            eigen_vectors, D = _ajd_pham(covs)

            # Here we apply an euclidean mean. See pyRiemann for other metrics
            mean_cov = np.average(covs, axis=0, weights=sample_weights)
            eigen_vectors = eigen_vectors.T

            # normalize
            for ii in range(eigen_vectors.shape[1]):
                tmp = np.dot(np.dot(eigen_vectors[:, ii].T, mean_cov),
                             eigen_vectors[:, ii])
                eigen_vectors[:, ii] /= np.sqrt(tmp)

            # class probability
            class_probas = [np.mean(y == _class) for _class in self._classes]

            # mutual information
            mutual_info = []
            for jj in range(eigen_vectors.shape[1]):
                aa, bb = 0, 0
                for (cov, prob) in zip(covs, class_probas):
                    tmp = np.dot(np.dot(eigen_vectors[:, jj].T, cov),
                                 eigen_vectors[:, jj])
                    aa += prob * np.log(np.sqrt(tmp))
                    bb += prob * (tmp ** 2 - 1)
                mi = - (aa + (3.0 / 16) * (bb ** 2))
                mutual_info.append(mi)
            ix = np.argsort(mutual_info)[::-1]

        # sort eigenvectors
        eigen_vectors = eigen_vectors[:, ix]

        self.filters_ = eigen_vectors.T
        self.patterns_ = linalg.pinv(eigen_vectors)

        pick_filters = self.filters_[:self.n_components]
        X = np.asarray([np.dot(pick_filters, epoch) for epoch in X])

        # compute features (mean band power)
        X = (X ** 2).mean(axis=-1)

        # To standardize features
        self.mean_ = X.mean(axis=0)
        self.std_ = X.std(axis=0)

        return self

    def transform(self, X):
        """Estimate epochs sources given the CSP filters.

        Parameters
        ----------
        X : array, shape (n_epochs, n_channels, n_times)
            The data.

        Returns
        -------
        X : ndarray
            If self.transform_into == 'average_power' then returns the power of
            CSP features averaged over time and shape (n_epochs, n_sources)
            If self.transform_into == 'csp_space' then returns the data in CSP
            space and shape is (n_epochs, n_sources, n_times)
        """
        if not isinstance(X, np.ndarray):
            raise ValueError("X should be of type ndarray (got %s)." % type(X))
        if self.filters_ is None:
            raise RuntimeError('No filters available. Please first fit CSP '
                               'decomposition.')

        pick_filters = self.filters_[:self.n_components]
        X = np.asarray([np.dot(pick_filters, epoch) for epoch in X])

        # compute features (mean band power)
        if self.transform_into == 'average_power':
            X = (X ** 2).mean(axis=-1)
            log = True if self.log is None else self.log
            if log:
                X = np.log(X)
            else:
                X -= self.mean_
                X /= self.std_
        return X

    def plot_patterns(self, info, components=None, ch_type=None, layout=None,
                      vmin=None, vmax=None, cmap='RdBu_r', sensors=True,
                      colorbar=True, scale=None, scale_time=1, unit=None,
                      res=64, size=1, cbar_fmt='%3.1f',
                      name_format='CSP%01d', proj=False, show=True,
                      show_names=False, title=None, mask=None,
                      mask_params=None, outlines='head', contours=6,
                      image_interp='bilinear', average=None, head_pos=None):
        """Plot topographic patterns of CSP components.

        The CSP patterns explain how the measured data was generated
        from the neural sources (a.k.a. the forward model).

        Parameters
        ----------
        info : instance of Info
            Info dictionary of the epochs used to fit CSP.
            If not possible, consider using ``create_info``.
        components : float | array of floats | None.
           The CSP patterns to plot. If None, n_components will be shown.
        ch_type : 'mag' | 'grad' | 'planar1' | 'planar2' | 'eeg' | None
            The channel type to plot. For 'grad', the gradiometers are
            collected in pairs and the RMS for each pair is plotted.
            If None, then first available channel type from order given
            above is used. Defaults to None.
        layout : None | Layout
            Layout instance specifying sensor positions (does not need to be
            specified for Neuromag data). If possible, the correct layout file
            is inferred from the data; if no appropriate layout file was found
            the layout is automatically generated from the sensor locations.
        vmin : float | callable
            The value specfying the lower bound of the color range.
            If None, and vmax is None, -vmax is used. Else np.min(data).
            If callable, the output equals vmin(data).
        vmax : float | callable
            The value specfying the upper bound of the color range.
            If None, the maximum absolute value is used. If vmin is None,
            but vmax is not, defaults to np.min(data).
            If callable, the output equals vmax(data).
        cmap : matplotlib colormap | (colormap, bool) | 'interactive' | None
            Colormap to use. If tuple, the first value indicates the colormap
            to use and the second value is a boolean defining interactivity. In
            interactive mode the colors are adjustable by clicking and dragging
            the colorbar with left and right mouse button. Left mouse button
            moves the scale up and down and right mouse button adjusts the
            range. Hitting space bar resets the range. Up and down arrows can
            be used to change the colormap. If None, 'Reds' is used for all
            positive data, otherwise defaults to 'RdBu_r'. If 'interactive',
            translates to (None, True). Defaults to 'RdBu_r'.

            .. warning::  Interactive mode works smoothly only for a small
                amount of topomaps.

        sensors : bool | str
            Add markers for sensor locations to the plot. Accepts matplotlib
            plot format string (e.g., 'r+' for red plusses). If True,
            a circle will be used (via .add_artist). Defaults to True.
        colorbar : bool
            Plot a colorbar.
        scale : dict | float | None
            Scale the data for plotting. If None, defaults to 1e6 for eeg, 1e13
            for grad and 1e15 for mag.
        scale_time : float | None
            Scale the time labels. Defaults to 1.
        unit : dict | str | None
            The unit of the channel type used for colorbar label. If
            scale is None the unit is automatically determined.
        res : int
            The resolution of the topomap image (n pixels along each side).
        size : float
            Side length per topomap in inches.
        cbar_fmt : str
            String format for colorbar values.
        name_format : str
            String format for topomap values. Defaults to "CSP%01d"
        proj : bool | 'interactive'
            If true SSP projections are applied before display.
            If 'interactive', a check box for reversible selection
            of SSP projection vectors will be show.
        show : bool
            Show figure if True.
        show_names : bool | callable
            If True, show channel names on top of the map. If a callable is
            passed, channel names will be formatted using the callable; e.g.,
            to delete the prefix 'MEG ' from all channel names, pass the
            function lambda x: x.replace('MEG ', ''). If `mask` is not None,
            only significant sensors will be shown.
        title : str | None
            Title. If None (default), no title is displayed.
        mask : ndarray of bool, shape (n_channels, n_times) | None
            The channels to be marked as significant at a given time point.
            Indices set to `True` will be considered. Defaults to None.
        mask_params : dict | None
            Additional plotting parameters for plotting significant sensors.
            Default (None) equals::

                dict(marker='o', markerfacecolor='w', markeredgecolor='k',
                     linewidth=0, markersize=4)

        outlines : 'head' | 'skirt' | dict | None
            The outlines to be drawn. If 'head', the default head scheme will
            be drawn. If 'skirt' the head scheme will be drawn, but sensors are
            allowed to be plotted outside of the head circle. If dict, each key
            refers to a tuple of x and y positions, the values in 'mask_pos'
            will serve as image mask, and the 'autoshrink' (bool) field will
            trigger automated shrinking of the positions due to points outside
            the outline. Alternatively, a matplotlib patch object can be passed
            for advanced masking options, either directly or as a function that
            returns patches (required for multi-axis plots). If None, nothing
            will be drawn. Defaults to 'head'.
        contours : int | False | None
            The number of contour lines to draw.
            If 0, no contours will be drawn.
        image_interp : str
            The image interpolation to be used.
            All matplotlib options are accepted.
        average : float | None
            The time window around a given time to be used for averaging
            (seconds). For example, 0.01 would translate into window that
            starts 5 ms before and ends 5 ms after a given time point.
            Defaults to None, which means no averaging.
        head_pos : dict | None
            If None (default), the sensors are positioned such that they span
            the head circle. If dict, can have entries 'center' (tuple) and
            'scale' (tuple) for what the center and scale of the head
            should be relative to the electrode locations.

        Returns
        -------
        fig : instance of matplotlib.figure.Figure
           The figure.
        """
        from .. import EvokedArray
        if components is None:
            components = np.arange(self.n_components)

        # set sampling frequency to have 1 component per time point
        info = cp.deepcopy(info)
        info['sfreq'] = 1.
        # create an evoked
        patterns = EvokedArray(self.patterns_.T, info, tmin=0)
        # the call plot_topomap
        return patterns.plot_topomap(times=components, ch_type=ch_type,
                                     layout=layout, vmin=vmin, vmax=vmax,
                                     cmap=cmap, colorbar=colorbar, res=res,
                                     cbar_fmt=cbar_fmt, sensors=sensors,
                                     scale=1, scale_time=1, unit='a.u.',
                                     time_format=name_format, size=size,
                                     show_names=show_names,
                                     mask_params=mask_params,
                                     mask=mask, outlines=outlines,
                                     contours=contours,
                                     image_interp=image_interp, show=show,
                                     head_pos=head_pos)

    def plot_filters(self, info, components=None, ch_type=None, layout=None,
                     vmin=None, vmax=None, cmap='RdBu_r', sensors=True,
                     colorbar=True, scale=None, scale_time=1, unit=None,
                     res=64, size=1, cbar_fmt='%3.1f',
                     name_format='CSP%01d', proj=False, show=True,
                     show_names=False, title=None, mask=None,
                     mask_params=None, outlines='head', contours=6,
                     image_interp='bilinear', average=None, head_pos=None):
        """Plot topographic filters of CSP components.

        The CSP filters are used to extract discriminant neural sources from
        the measured data (a.k.a. the backward model).

        Parameters
        ----------
        info : instance of Info
            Info dictionary of the epochs used to fit CSP.
            If not possible, consider using ``create_info``.
        components : float | array of floats | None.
           The CSP patterns to plot. If None, n_components will be shown.
        ch_type : 'mag' | 'grad' | 'planar1' | 'planar2' | 'eeg' | None
            The channel type to plot. For 'grad', the gradiometers are
            collected in pairs and the RMS for each pair is plotted.
            If None, then first available channel type from order given
            above is used. Defaults to None.
        layout : None | Layout
            Layout instance specifying sensor positions (does not need to be
            specified for Neuromag data). If possible, the correct layout file
            is inferred from the data; if no appropriate layout file was found
            the layout is automatically generated from the sensor locations.
        vmin : float | callable
            The value specfying the lower bound of the color range.
            If None, and vmax is None, -vmax is used. Else np.min(data).
            If callable, the output equals vmin(data).
        vmax : float | callable
            The value specfying the upper bound of the color range.
            If None, the maximum absolute value is used. If vmin is None,
            but vmax is not, defaults to np.min(data).
            If callable, the output equals vmax(data).
        cmap : matplotlib colormap | (colormap, bool) | 'interactive' | None
            Colormap to use. If tuple, the first value indicates the colormap
            to use and the second value is a boolean defining interactivity. In
            interactive mode the colors are adjustable by clicking and dragging
            the colorbar with left and right mouse button. Left mouse button
            moves the scale up and down and right mouse button adjusts the
            range. Hitting space bar resets the range. Up and down arrows can
            be used to change the colormap. If None, 'Reds' is used for all
            positive data, otherwise defaults to 'RdBu_r'. If 'interactive',
            translates to (None, True). Defaults to 'RdBu_r'.

            .. warning::  Interactive mode works smoothly only for a small
                amount of topomaps.

        sensors : bool | str
            Add markers for sensor locations to the plot. Accepts matplotlib
            plot format string (e.g., 'r+' for red plusses). If True,
            a circle will be used (via .add_artist). Defaults to True.
        colorbar : bool
            Plot a colorbar.
        scale : dict | float | None
            Scale the data for plotting. If None, defaults to 1e6 for eeg, 1e13
            for grad and 1e15 for mag.
        scale_time : float | None
            Scale the time labels. Defaults to 1.
        unit : dict | str | None
            The unit of the channel type used for colorbar label. If
            scale is None the unit is automatically determined.
        res : int
            The resolution of the topomap image (n pixels along each side).
        size : float
            Side length per topomap in inches.
        cbar_fmt : str
            String format for colorbar values.
        name_format : str
            String format for topomap values. Defaults to "CSP%01d"
        proj : bool | 'interactive'
            If true SSP projections are applied before display.
            If 'interactive', a check box for reversible selection
            of SSP projection vectors will be show.
        show : bool
            Show figure if True.
        show_names : bool | callable
            If True, show channel names on top of the map. If a callable is
            passed, channel names will be formatted using the callable; e.g.,
            to delete the prefix 'MEG ' from all channel names, pass the
            function lambda x: x.replace('MEG ', ''). If `mask` is not None,
            only significant sensors will be shown.
        title : str | None
            Title. If None (default), no title is displayed.
        mask : ndarray of bool, shape (n_channels, n_times) | None
            The channels to be marked as significant at a given time point.
            Indices set to `True` will be considered. Defaults to None.
        mask_params : dict | None
            Additional plotting parameters for plotting significant sensors.
            Default (None) equals::

                dict(marker='o', markerfacecolor='w', markeredgecolor='k',
                     linewidth=0, markersize=4)

        outlines : 'head' | 'skirt' | dict | None
            The outlines to be drawn. If 'head', the default head scheme will
            be drawn. If 'skirt' the head scheme will be drawn, but sensors are
            allowed to be plotted outside of the head circle. If dict, each key
            refers to a tuple of x and y positions, the values in 'mask_pos'
            will serve as image mask, and the 'autoshrink' (bool) field will
            trigger automated shrinking of the positions due to points outside
            the outline. Alternatively, a matplotlib patch object can be passed
            for advanced masking options, either directly or as a function that
            returns patches (required for multi-axis plots). If None, nothing
            will be drawn. Defaults to 'head'.
        contours : int | False | None
            The number of contour lines to draw.
            If 0, no contours will be drawn.
        image_interp : str
            The image interpolation to be used.
            All matplotlib options are accepted.
        average : float | None
            The time window around a given time to be used for averaging
            (seconds). For example, 0.01 would translate into window that
            starts 5 ms before and ends 5 ms after a given time point.
            Defaults to None, which means no averaging.
        head_pos : dict | None
            If None (default), the sensors are positioned such that they span
            the head circle. If dict, can have entries 'center' (tuple) and
            'scale' (tuple) for what the center and scale of the head
            should be relative to the electrode locations.

        Returns
        -------
        fig : instance of matplotlib.figure.Figure
           The figure.
        """
        from .. import EvokedArray
        if components is None:
            components = np.arange(self.n_components)

        # set sampling frequency to have 1 component per time point
        info = cp.deepcopy(info)
        info['sfreq'] = 1.
        # create an evoked
        filters = EvokedArray(self.filters_, info, tmin=0)
        # the call plot_topomap
        return filters.plot_topomap(times=components, ch_type=ch_type,
                                    layout=layout, vmin=vmin, vmax=vmax,
                                    cmap=cmap, colorbar=colorbar, res=res,
                                    cbar_fmt=cbar_fmt, sensors=sensors,
                                    scale=1, scale_time=1, unit='a.u.',
                                    time_format=name_format, size=size,
                                    show_names=show_names,
                                    mask_params=mask_params,
                                    mask=mask, outlines=outlines,
                                    contours=contours,
                                    image_interp=image_interp, show=show,
                                    head_pos=head_pos)


def _ajd_pham(X, eps=1e-6, max_iter=15):
    """Approximate joint diagonalization based on Pham's algorithm.

    This is a direct implementation of the PHAM's AJD algorithm [1].

    Parameters
    ----------
    X : ndarray, shape (n_epochs, n_channels, n_channels)
        A set of covariance matrices to diagonalize.
    eps : float, defaults to 1e-6
        The tolerance for stoping criterion.
    max_iter : int, defaults to 1000
        The maximum number of iteration to reach convergence.

    Returns
    -------
    V : ndarray, shape (n_channels, n_channels)
        The diagonalizer.
    D : ndarray, shape (n_epochs, n_channels, n_channels)
        The set of quasi diagonal matrices.

    References
    ----------
    [1] Pham, Dinh Tuan. "Joint approximate diagonalization of positive
    definite Hermitian matrices." SIAM Journal on Matrix Analysis and
    Applications 22, no. 4 (2001): 1136-1152.

    """
    # Adapted from http://github.com/alexandrebarachant/pyRiemann
    n_epochs = X.shape[0]

    # Reshape input matrix
    A = np.concatenate(X, axis=0).T

    # Init variables
    n_times, n_m = A.shape
    V = np.eye(n_times)
    epsilon = n_times * (n_times - 1) * eps

    for it in range(max_iter):
        decr = 0
        for ii in range(1, n_times):
            for jj in range(ii):
                Ii = np.arange(ii, n_m, n_times)
                Ij = np.arange(jj, n_m, n_times)

                c1 = A[ii, Ii]
                c2 = A[jj, Ij]

                g12 = np.mean(A[ii, Ij] / c1)
                g21 = np.mean(A[ii, Ij] / c2)

                omega21 = np.mean(c1 / c2)
                omega12 = np.mean(c2 / c1)
                omega = np.sqrt(omega12 * omega21)

                tmp = np.sqrt(omega21 / omega12)
                tmp1 = (tmp * g12 + g21) / (omega + 1)
                tmp2 = (tmp * g12 - g21) / max(omega - 1, 1e-9)

                h12 = tmp1 + tmp2
                h21 = np.conj((tmp1 - tmp2) / tmp)

                decr += n_epochs * (g12 * np.conj(h12) + g21 * h21) / 2.0

                tmp = 1 + 1.j * 0.5 * np.imag(h12 * h21)
                tmp = np.real(tmp + np.sqrt(tmp ** 2 - h12 * h21))
                tau = np.array([[1, -h12 / tmp], [-h21 / tmp, 1]])

                A[[ii, jj], :] = np.dot(tau, A[[ii, jj], :])
                tmp = np.c_[A[:, Ii], A[:, Ij]]
                tmp = np.reshape(tmp, (n_times * n_epochs, 2), order='F')
                tmp = np.dot(tmp, tau.T)

                tmp = np.reshape(tmp, (n_times, n_epochs * 2), order='F')
                A[:, Ii] = tmp[:, :n_epochs]
                A[:, Ij] = tmp[:, n_epochs:]
                V[[ii, jj], :] = np.dot(tau, V[[ii, jj], :])
        if decr < epsilon:
            break
    D = np.reshape(A, (n_times, -1, n_times)).transpose(1, 0, 2)
    return V, D