"""Factor Analysis. A latent linear variable model. FactorAnalysis is similar to probabilistic PCA implemented by PCA.score While PCA assumes Gaussian noise with the same variance for each feature, the FactorAnalysis model assumes different variances for each of them. This implementation is based on David Barber's Book, Bayesian Reasoning and Machine Learning, http://www.cs.ucl.ac.uk/staff/d.barber/brml, Algorithm 21.1 """ # Author: Christian Osendorfer # Alexandre Gramfort # Denis A. Engemann # License: BSD3 import warnings from math import sqrt, log import numpy as np from scipy import linalg from ..base import BaseEstimator, TransformerMixin from ..utils import check_random_state from ..utils.extmath import fast_logdet, randomized_svd, squared_norm from ..utils.validation import check_is_fitted, _deprecate_positional_args from ..exceptions import ConvergenceWarning class FactorAnalysis(TransformerMixin, BaseEstimator): """Factor Analysis (FA). A simple linear generative model with Gaussian latent variables. The observations are assumed to be caused by a linear transformation of lower dimensional latent factors and added Gaussian noise. Without loss of generality the factors are distributed according to a Gaussian with zero mean and unit covariance. The noise is also zero mean and has an arbitrary diagonal covariance matrix. If we would restrict the model further, by assuming that the Gaussian noise is even isotropic (all diagonal entries are the same) we would obtain :class:`PPCA`. FactorAnalysis performs a maximum likelihood estimate of the so-called `loading` matrix, the transformation of the latent variables to the observed ones, using SVD based approach. Read more in the :ref:`User Guide `. .. versionadded:: 0.13 Parameters ---------- n_components : int, default=None Dimensionality of latent space, the number of components of ``X`` that are obtained after ``transform``. If None, n_components is set to the number of features. tol : float, defaul=1e-2 Stopping tolerance for log-likelihood increase. copy : bool, default=True Whether to make a copy of X. If ``False``, the input X gets overwritten during fitting. max_iter : int, default=1000 Maximum number of iterations. noise_variance_init : ndarray of shape (n_features,), default=None The initial guess of the noise variance for each feature. If None, it defaults to np.ones(n_features). svd_method : {'lapack', 'randomized'}, default='randomized' Which SVD method to use. If 'lapack' use standard SVD from scipy.linalg, if 'randomized' use fast ``randomized_svd`` function. Defaults to 'randomized'. For most applications 'randomized' will be sufficiently precise while providing significant speed gains. Accuracy can also be improved by setting higher values for `iterated_power`. If this is not sufficient, for maximum precision you should choose 'lapack'. iterated_power : int, default=3 Number of iterations for the power method. 3 by default. Only used if ``svd_method`` equals 'randomized'. rotation : {'varimax', 'quartimax'}, default=None If not None, apply the indicated rotation. Currently, varimax and quartimax are implemented. See `"The varimax criterion for analytic rotation in factor analysis" `_ H. F. Kaiser, 1958. .. versionadded:: 0.24 random_state : int or RandomState instance, default=0 Only used when ``svd_method`` equals 'randomized'. Pass an int for reproducible results across multiple function calls. See :term:`Glossary `. Attributes ---------- components_ : ndarray of shape (n_components, n_features) Components with maximum variance. loglike_ : list of shape (n_iterations,) The log likelihood at each iteration. noise_variance_ : ndarray of shape (n_features,) The estimated noise variance for each feature. n_iter_ : int Number of iterations run. mean_ : ndarray of shape (n_features,) Per-feature empirical mean, estimated from the training set. Examples -------- >>> from sklearn.datasets import load_digits >>> from sklearn.decomposition import FactorAnalysis >>> X, _ = load_digits(return_X_y=True) >>> transformer = FactorAnalysis(n_components=7, random_state=0) >>> X_transformed = transformer.fit_transform(X) >>> X_transformed.shape (1797, 7) References ---------- - David Barber, Bayesian Reasoning and Machine Learning, Algorithm 21.1. - Christopher M. Bishop: Pattern Recognition and Machine Learning, Chapter 12.2.4. See Also -------- PCA: Principal component analysis is also a latent linear variable model which however assumes equal noise variance for each feature. This extra assumption makes probabilistic PCA faster as it can be computed in closed form. FastICA: Independent component analysis, a latent variable model with non-Gaussian latent variables. """ @_deprecate_positional_args def __init__(self, n_components=None, *, tol=1e-2, copy=True, max_iter=1000, noise_variance_init=None, svd_method='randomized', iterated_power=3, rotation=None, random_state=0): self.n_components = n_components self.copy = copy self.tol = tol self.max_iter = max_iter if svd_method not in ['lapack', 'randomized']: raise ValueError('SVD method %s is not supported. Please consider' ' the documentation' % svd_method) self.svd_method = svd_method self.noise_variance_init = noise_variance_init self.iterated_power = iterated_power self.random_state = random_state self.rotation = rotation def fit(self, X, y=None): """Fit the FactorAnalysis model to X using SVD based approach Parameters ---------- X : array-like of shape (n_samples, n_features) Training data. y : Ignored Returns ------- self """ X = self._validate_data(X, copy=self.copy, dtype=np.float64) n_samples, n_features = X.shape n_components = self.n_components if n_components is None: n_components = n_features self.mean_ = np.mean(X, axis=0) X -= self.mean_ # some constant terms nsqrt = sqrt(n_samples) llconst = n_features * log(2. * np.pi) + n_components var = np.var(X, axis=0) if self.noise_variance_init is None: psi = np.ones(n_features, dtype=X.dtype) else: if len(self.noise_variance_init) != n_features: raise ValueError("noise_variance_init dimension does not " "with number of features : %d != %d" % (len(self.noise_variance_init), n_features)) psi = np.array(self.noise_variance_init) loglike = [] old_ll = -np.inf SMALL = 1e-12 # we'll modify svd outputs to return unexplained variance # to allow for unified computation of loglikelihood if self.svd_method == 'lapack': def my_svd(X): _, s, Vt = linalg.svd(X, full_matrices=False, check_finite=False) return (s[:n_components], Vt[:n_components], squared_norm(s[n_components:])) elif self.svd_method == 'randomized': random_state = check_random_state(self.random_state) def my_svd(X): _, s, Vt = randomized_svd(X, n_components, random_state=random_state, n_iter=self.iterated_power) return s, Vt, squared_norm(X) - squared_norm(s) else: raise ValueError('SVD method %s is not supported. Please consider' ' the documentation' % self.svd_method) for i in range(self.max_iter): # SMALL helps numerics sqrt_psi = np.sqrt(psi) + SMALL s, Vt, unexp_var = my_svd(X / (sqrt_psi * nsqrt)) s **= 2 # Use 'maximum' here to avoid sqrt problems. W = np.sqrt(np.maximum(s - 1., 0.))[:, np.newaxis] * Vt del Vt W *= sqrt_psi # loglikelihood ll = llconst + np.sum(np.log(s)) ll += unexp_var + np.sum(np.log(psi)) ll *= -n_samples / 2. loglike.append(ll) if (ll - old_ll) < self.tol: break old_ll = ll psi = np.maximum(var - np.sum(W ** 2, axis=0), SMALL) else: warnings.warn('FactorAnalysis did not converge.' + ' You might want' + ' to increase the number of iterations.', ConvergenceWarning) self.components_ = W if self.rotation is not None: self.components_ = self._rotate(W) self.noise_variance_ = psi self.loglike_ = loglike self.n_iter_ = i + 1 return self def transform(self, X): """Apply dimensionality reduction to X using the model. Compute the expected mean of the latent variables. See Barber, 21.2.33 (or Bishop, 12.66). Parameters ---------- X : array-like of shape (n_samples, n_features) Training data. Returns ------- X_new : ndarray of shape (n_samples, n_components) The latent variables of X. """ check_is_fitted(self) X = self._validate_data(X, reset=False) Ih = np.eye(len(self.components_)) X_transformed = X - self.mean_ Wpsi = self.components_ / self.noise_variance_ cov_z = linalg.inv(Ih + np.dot(Wpsi, self.components_.T)) tmp = np.dot(X_transformed, Wpsi.T) X_transformed = np.dot(tmp, cov_z) return X_transformed def get_covariance(self): """Compute data covariance with the FactorAnalysis model. ``cov = components_.T * components_ + diag(noise_variance)`` Returns ------- cov : ndarray of shape (n_features, n_features) Estimated covariance of data. """ check_is_fitted(self) cov = np.dot(self.components_.T, self.components_) cov.flat[::len(cov) + 1] += self.noise_variance_ # modify diag inplace return cov def get_precision(self): """Compute data precision matrix with the FactorAnalysis model. Returns ------- precision : ndarray of shape (n_features, n_features) Estimated precision of data. """ check_is_fitted(self) n_features = self.components_.shape[1] # handle corner cases first if self.n_components == 0: return np.diag(1. / self.noise_variance_) if self.n_components == n_features: return linalg.inv(self.get_covariance()) # Get precision using matrix inversion lemma components_ = self.components_ precision = np.dot(components_ / self.noise_variance_, components_.T) precision.flat[::len(precision) + 1] += 1. precision = np.dot(components_.T, np.dot(linalg.inv(precision), components_)) precision /= self.noise_variance_[:, np.newaxis] precision /= -self.noise_variance_[np.newaxis, :] precision.flat[::len(precision) + 1] += 1. / self.noise_variance_ return precision def score_samples(self, X): """Compute the log-likelihood of each sample Parameters ---------- X : ndarray of shape (n_samples, n_features) The data Returns ------- ll : ndarray of shape (n_samples,) Log-likelihood of each sample under the current model """ check_is_fitted(self) X = self._validate_data(X, reset=False) Xr = X - self.mean_ precision = self.get_precision() n_features = X.shape[1] log_like = -.5 * (Xr * (np.dot(Xr, precision))).sum(axis=1) log_like -= .5 * (n_features * log(2. * np.pi) - fast_logdet(precision)) return log_like def score(self, X, y=None): """Compute the average log-likelihood of the samples Parameters ---------- X : ndarray of shape (n_samples, n_features) The data y : Ignored Returns ------- ll : float Average log-likelihood of the samples under the current model """ return np.mean(self.score_samples(X)) def _rotate(self, components, n_components=None, tol=1e-6): "Rotate the factor analysis solution." # note that tol is not exposed implemented = ("varimax", "quartimax") method = self.rotation if method in implemented: return _ortho_rotation(components.T, method=method, tol=tol)[:self.n_components] else: raise ValueError("'method' must be in %s, not %s" % (implemented, method)) def _ortho_rotation(components, method='varimax', tol=1e-6, max_iter=100): """Return rotated components.""" nrow, ncol = components.shape rotation_matrix = np.eye(ncol) var = 0 for _ in range(max_iter): comp_rot = np.dot(components, rotation_matrix) if method == "varimax": tmp = comp_rot * np.transpose((comp_rot ** 2).sum(axis=0) / nrow) elif method == "quartimax": tmp = 0 u, s, v = np.linalg.svd( np.dot(components.T, comp_rot ** 3 - tmp)) rotation_matrix = np.dot(u, v) var_new = np.sum(s) if var != 0 and var_new < var * (1 + tol): break var = var_new return np.dot(components, rotation_matrix).T