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-# -*- coding: utf-8 -*-
-#
-#
-# 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
-#
-#
-# Code from sklearn.decomposition.fastica_.py adapted by The Virtual Brain team - August 2019
-#
-#
-# TheVirtualBrain-Scientific Package. This package holds all simulators, and
-# analysers necessary to run brain-simulations. You can use it stand alone or
-# in conjunction with TheVirtualBrain-Framework Package. See content of the
-# documentation-folder for more details. See also http://www.thevirtualbrain.org
-#
-# (c) 2012-2020, Baycrest Centre for Geriatric Care ("Baycrest") and others
-#
-# This program is free software: you can redistribute it and/or modify it under the
-# terms of the GNU General Public License as published by the Free Software Foundation,
-# either version 3 of the License, or (at your option) any later version.
-# This program is distributed in the hope that it will be useful, but WITHOUT ANY
-# WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A
-# PARTICULAR PURPOSE. See the GNU General Public License for more details.
-# You should have received a copy of the GNU General Public License along with this
-# program. If not, see .
-#
-#
-# CITATION:
-# When using The Virtual Brain for scientific publications, please cite it as follows:
-#
-# Paula Sanz Leon, Stuart A. Knock, M. Marmaduke Woodman, Lia Domide,
-# Jochen Mersmann, Anthony R. McIntosh, Viktor Jirsa (2013)
-# The Virtual Brain: a simulator of primate brain network dynamics.
-# Frontiers in Neuroinformatics (7:10. doi: 10.3389/fninf.2013.00010)
-#
-#
-
-"""
-
-.. moduleauthor:: Gabriel Florea
-
-"""
-
-import warnings
-import numpy
-from scipy import linalg
-from scipy._lib._util import check_random_state
-from six import moves
-from six import string_types
-
-def _sym_decorrelation(W):
- """ Symmetric decorrelation
- i.e. W <- (W * W.T) ^{-1/2} * W
- """
- s, u = linalg.eigh(numpy.dot(W, W.T))
- # u (resp. s) contains the eigenvectors (resp. square roots of
- # the eigenvalues) of W * W.T
- return numpy.dot(numpy.dot(u * (1. / numpy.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 = numpy.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 /= numpy.sqrt((w ** 2).sum())
- i = None
- for i in moves.xrange(max_iter):
- gwtx, g_wtx = g(numpy.dot(w.T, X), fun_args)
-
- w1 = (X * gwtx).mean(axis=1) - g_wtx.mean() * w
-
- w1 -= numpy.dot(numpy.dot(w1, W[:j].T), W[:j])
-
- w1 /= numpy.sqrt((w1 ** 2).sum())
-
- lim = numpy.abs(numpy.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])
- ii = None
- for ii in moves.xrange(max_iter):
- gwtx, g_wtx = g(numpy.dot(W, X), fun_args)
- W1 = _sym_decorrelation(numpy.dot(gwtx, X.T) / p_
- - g_wtx[:, numpy.newaxis] * W)
- del gwtx, g_wtx
- # builtin max, abs are faster than numpy counter parts.
- lim = max(abs(abs(numpy.diag(numpy.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.')
-
- return W, ii + 1
-
-# Some standard non-linear functions.
-def _logcosh(x, fun_args=None):
- alpha = fun_args.get('alpha', 1.0) # comment it out?
-
- x *= alpha
- gx = numpy.tanh(x, x) # apply the tanh inplace
- g_x = numpy.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 = numpy.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 `.
-
- 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`
-
- """
- K = None
- X_mean = None
-
- random_state = check_random_state(random_state)
- fun_args = {} if fun_args is None else fun_args
-
- 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, 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[:, numpy.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 = numpy.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 *= numpy.sqrt(p)
- else:
- X1 = X
-
- if w_init is None:
- w_init = numpy.asarray(random_state.normal(size=(n_components,
- n_components)), dtype=X1.dtype)
-
- else:
- w_init = numpy.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 = numpy.dot(numpy.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 = numpy.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(object):
- """FastICA: a fast algorithm for Independent Component Analysis.
-
- Read more in the :ref:`User Guide `.
-
- 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).mean(axis=-1)
-
- 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.
-
- 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_ = numpy.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(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