""" 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.linalg.multi_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.linalg.multi_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 `. Parameters ---------- X : array-like of 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, default=None Number of components to extract. If None no dimension reduction is performed. algorithm : {'parallel', 'deflation'}, default='parallel' Apply a parallel or deflational FASTICA algorithm. whiten : bool, default=True 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 : {'logcosh', 'exp', 'cube'} or callable, 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 : dict, default=None 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, default=200 Maximum number of iterations to perform. tol : float, default=1e-04 A positive scalar giving the tolerance at which the un-mixing matrix is considered to have converged. w_init : ndarray of shape (n_components, n_components), default=None 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, 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 `. return_X_mean : bool, default=False If True, X_mean is returned too. compute_sources : bool, default=True 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, default=False Whether or not to return the number of iterations. Returns ------- K : ndarray of shape (n_components, n_features) or 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 : ndarray of 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 : ndarray of shape (n_samples, n_components) or None Estimated source matrix X_mean : ndarray of 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 `. Parameters ---------- n_components : int, default=None Number of components to use. If None is passed, all are used. algorithm : {'parallel', 'deflation'}, default='parallel' Apply parallel or deflational algorithm for FastICA. whiten : bool, default=True If whiten is false, the data is already considered to be whitened, and no whitening is performed. fun : {'logcosh', 'exp', 'cube'} or callable, 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 : dict, default=None Arguments to send to the functional form. If empty and if fun='logcosh', fun_args will take value {'alpha' : 1.0}. max_iter : int, default=200 Maximum number of iterations during fit. tol : float, default=1e-4 Tolerance on update at each iteration. w_init : ndarray of shape (n_components, n_components), default=None The mixing matrix to be used to initialize the algorithm. random_state : int, RandomState instance or None, 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 `. Attributes ---------- components_ : ndarray of 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_ : ndarray of shape (n_features, n_components) The pseudo-inverse of ``components_``. It is the linear operator that maps independent sources to the data. mean_ : ndarray of 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_ : ndarray of 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 of 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, default=False 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 : ndarray of 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, check_finite=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.linalg.multi_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_, check_finite=False) self._unmixing = W return S def fit_transform(self, X, y=None): """Fit the model and recover the sources from X. Parameters ---------- X : array-like of 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 : ndarray of 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 of 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 of 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, default=True If False, data passed to fit can be overwritten. Defaults to True. Returns ------- X_new : ndarray of shape (n_samples, n_components) """ check_is_fitted(self) X = self._validate_data(X, copy=(copy and self.whiten), dtype=FLOAT_DTYPES, reset=False) 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 of shape (n_samples, n_components) Sources, where n_samples is the number of samples and n_components is the number of components. copy : bool, default=True If False, data passed to fit are overwritten. Defaults to True. Returns ------- X_new : ndarray of 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