626 lines
20 KiB
Python
626 lines
20 KiB
Python
"""
|
|
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 ..utils import check_array, as_float_array, check_random_state
|
|
from ..utils.validation import check_is_fitted
|
|
from ..utils.validation import FLOAT_DTYPES
|
|
from ..utils.validation import _deprecate_positional_args
|
|
|
|
__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 range(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 range(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)
|
|
|
|
|
|
@_deprecate_positional_args
|
|
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, default=None
|
|
Used to initialize ``w_init`` when not specified, with a
|
|
normal distribution. Pass an int, for reproducible results
|
|
across multiple function calls.
|
|
See :term:`Glossary <random_state>`.
|
|
|
|
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)
|
|
The square matrix that unmixes the data after whitening.
|
|
The mixing matrix is the pseudo-inverse of matrix ``W K``
|
|
if K is not None, else it is the inverse of W.
|
|
|
|
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.``
|
|
While FastICA was proposed to estimate as many sources
|
|
as features, it is possible to estimate less by setting
|
|
n_components < n_features. It this case K is not a square matrix
|
|
and the estimated A is the pseudo-inverse of ``W K``.
|
|
|
|
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*
|
|
|
|
"""
|
|
|
|
est = FastICA(n_components=n_components, algorithm=algorithm,
|
|
whiten=whiten, fun=fun, fun_args=fun_args,
|
|
max_iter=max_iter, tol=tol, w_init=w_init,
|
|
random_state=random_state)
|
|
sources = est._fit(X, compute_sources=compute_sources)
|
|
|
|
if whiten:
|
|
if return_X_mean:
|
|
if return_n_iter:
|
|
return (est.whitening_, est._unmixing, sources, est.mean_,
|
|
est.n_iter_)
|
|
else:
|
|
return est.whitening_, est._unmixing, sources, est.mean_
|
|
else:
|
|
if return_n_iter:
|
|
return est.whitening_, est._unmixing, sources, est.n_iter_
|
|
else:
|
|
return est.whitening_, est._unmixing, sources
|
|
|
|
else:
|
|
if return_X_mean:
|
|
if return_n_iter:
|
|
return None, est._unmixing, sources, None, est.n_iter_
|
|
else:
|
|
return None, est._unmixing, sources, None
|
|
else:
|
|
if return_n_iter:
|
|
return None, est._unmixing, sources, est.n_iter_
|
|
else:
|
|
return None, est._unmixing, sources
|
|
|
|
|
|
class FastICA(TransformerMixin, BaseEstimator):
|
|
"""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).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, default=None
|
|
Used to initialize ``w_init`` when not specified, with a
|
|
normal distribution. Pass an int, for reproducible results
|
|
across multiple function calls.
|
|
See :term:`Glossary <random_state>`.
|
|
|
|
Attributes
|
|
----------
|
|
components_ : 2D array, shape (n_components, n_features)
|
|
The linear operator to apply to the data to get the independent
|
|
sources. This is equal to the unmixing matrix when ``whiten`` is
|
|
False, and equal to ``np.dot(unmixing_matrix, self.whitening_)`` when
|
|
``whiten`` is True.
|
|
|
|
mixing_ : array, shape (n_features, n_components)
|
|
The pseudo-inverse of ``components_``. It is the linear operator
|
|
that maps independent sources to the data.
|
|
|
|
mean_ : array, shape(n_features)
|
|
The mean over features. Only set if `self.whiten` 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.
|
|
|
|
whitening_ : array, shape (n_components, n_features)
|
|
Only set if whiten is 'True'. This is the pre-whitening matrix
|
|
that projects data onto the first `n_components` principal components.
|
|
|
|
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*
|
|
|
|
"""
|
|
@_deprecate_positional_args
|
|
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().__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)
|
|
"""
|
|
|
|
X = self._validate_data(X, copy=self.whiten, dtype=FLOAT_DTYPES,
|
|
ensure_min_samples=2).T
|
|
fun_args = {} if self.fun_args is None else self.fun_args
|
|
random_state = check_random_state(self.random_state)
|
|
|
|
alpha = fun_args.get('alpha', 1.0)
|
|
if not 1 <= alpha <= 2:
|
|
raise ValueError('alpha must be in [1,2]')
|
|
|
|
if self.fun == 'logcosh':
|
|
g = _logcosh
|
|
elif self.fun == 'exp':
|
|
g = _exp
|
|
elif self.fun == 'cube':
|
|
g = _cube
|
|
elif callable(self.fun):
|
|
def g(x, fun_args):
|
|
return self.fun(x, **fun_args)
|
|
else:
|
|
exc = ValueError if isinstance(self.fun, str) else TypeError
|
|
raise exc(
|
|
"Unknown function %r;"
|
|
" should be one of 'logcosh', 'exp', 'cube' or callable"
|
|
% self.fun
|
|
)
|
|
|
|
n_samples, n_features = X.shape
|
|
|
|
n_components = self.n_components
|
|
if not self.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_samples, n_features)
|
|
if (n_components > min(n_samples, n_features)):
|
|
n_components = min(n_samples, n_features)
|
|
warnings.warn(
|
|
'n_components is too large: it will be set to %s'
|
|
% n_components
|
|
)
|
|
|
|
if self.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(n_features)
|
|
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
|
|
|
|
w_init = self.w_init
|
|
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': self.tol,
|
|
'g': g,
|
|
'fun_args': fun_args,
|
|
'max_iter': self.max_iter,
|
|
'w_init': w_init}
|
|
|
|
if self.algorithm == 'parallel':
|
|
W, n_iter = _ica_par(X1, **kwargs)
|
|
elif self.algorithm == 'deflation':
|
|
W, n_iter = _ica_def(X1, **kwargs)
|
|
else:
|
|
raise ValueError('Invalid algorithm: must be either `parallel` or'
|
|
' `deflation`.')
|
|
del X1
|
|
|
|
if compute_sources:
|
|
if self.whiten:
|
|
S = np.dot(np.dot(W, K), X).T
|
|
else:
|
|
S = np.dot(W, X).T
|
|
else:
|
|
S = None
|
|
|
|
self.n_iter_ = n_iter
|
|
|
|
if self.whiten:
|
|
self.components_ = np.dot(W, K)
|
|
self.mean_ = X_mean
|
|
self.whitening_ = K
|
|
else:
|
|
self.components_ = W
|
|
|
|
self.mixing_ = linalg.pinv(self.components_)
|
|
self._unmixing = W
|
|
|
|
if compute_sources:
|
|
self.__sources = S
|
|
|
|
return S
|
|
|
|
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, 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.
|
|
|
|
copy : bool (optional)
|
|
If False, data passed to fit are overwritten. Defaults to True.
|
|
|
|
Returns
|
|
-------
|
|
X_new : array-like, shape (n_samples, n_components)
|
|
"""
|
|
check_is_fitted(self)
|
|
|
|
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)
|
|
|
|
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
|