""" Maximum likelihood covariance estimator. """ # Author: Alexandre Gramfort # Gael Varoquaux # Virgile Fritsch # # License: BSD Style. # avoid division truncation from __future__ import division import warnings import numpy as np from scipy import linalg from ..base import BaseEstimator from ..utils import array2d from ..utils.extmath import fast_logdet def log_likelihood(emp_cov, precision): """Computes the log_likelihood of the data Params ------ emp_cov: 2D ndarray (n_features, n_features) Maximum Likelihood Estimator of covariance precision: 2D ndarray (n_features, n_features) The precision matrix of the covariance model to be tested """ return -np.sum(emp_cov * precision) + fast_logdet(precision) def empirical_covariance(X, assume_centered=False): """Computes the Maximum likelihood covariance estimator Parameters ---------- X: 2D ndarray, shape (n_samples, n_features) Data from which to compute the covariance estimate assume_centered: Boolean If True, data are not centered before computation. Useful when working with data whose mean is almost, but not exactly zero. If False, data are centered before computation. Returns ------- covariance: 2D ndarray, shape (n_features, n_features) Empirical covariance (Maximum Likelihood Estimator) """ X = np.asarray(X) if X.ndim == 1: X = np.reshape(X, (1, -1)) warnings.warn("Only one sample available. " \ "You may want to reshape your data array") if assume_centered: covariance = np.dot(X.T, X) / X.shape[0] else: covariance = np.cov(X.T, bias=1) return covariance class EmpiricalCovariance(BaseEstimator): """Maximum likelihood covariance estimator Parameters ---------- store_precision : bool Specifies if the estimated precision is stored Attributes ---------- `covariance_` : 2D ndarray, shape (n_features, n_features) Estimated covariance matrix `precision_` : 2D ndarray, shape (n_features, n_features) Estimated pseudo-inverse matrix. (stored only if store_precision is True) """ def __init__(self, store_precision=True, assume_centered=False): """ Parameters ---------- store_precision: bool Specify if the estimated precision is stored assume_centered: Boolean If True, data are not centered before computation. Useful when working with data whose mean is almost, but not exactly zero. If False, data are centered before computation. """ self.store_precision = store_precision self.assume_centered = assume_centered def _set_estimates(self, covariance): """Saves the covariance and precision estimates Storage is done accordingly to `self.store_precision`. Precision stored only if invertible. Params ------ covariance: 2D ndarray, shape (n_features, n_features) Estimated covariance matrix to be stored, and from which precision is computed. """ covariance = array2d(covariance) # set covariance self.covariance_ = covariance # set precision if self.store_precision: self.precision_ = linalg.pinv(covariance) else: self.precision_ = None def fit(self, X): """Fits the Maximum Likelihood Estimator covariance model according to the given training data and parameters. 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. Returns ------- self : object Returns self. """ if self.assume_centered: self.location_ = np.zeros(X.shape[1]) else: self.location_ = X.mean(0) covariance = empirical_covariance( X, assume_centered=self.assume_centered) self._set_estimates(covariance) return self def score(self, X_test, assume_centered=False): """Computes the log-likelihood of a gaussian data set with `self.covariance_` as an estimator of its covariance matrix. Parameters ---------- X_test : array-like, shape = [n_samples, n_features] Test data of which we compute the likelihood, where n_samples is the number of samples and n_features is the number of features. Returns ------- res : float The likelihood of the data set with `self.covariance_` as an estimator of its covariance matrix. """ # compute empirical covariance of the test set test_cov = empirical_covariance(X_test, assume_centered=assume_centered) # compute log likelihood if self.store_precision: res = log_likelihood(test_cov, self.precision_) else: res = log_likelihood(test_cov, linalg.pinv(self.covariance_)) return res def error_norm(self, comp_cov, norm='frobenius', scaling=True, squared=True): """Computes the Mean Squared Error between two covariance estimators. (In the sense of the Frobenius norm) Parameters ---------- comp_cov: array-like, shape = [n_features, n_features] The covariance to compare with. norm: str The type of norm used to compute the error. Available error types: - 'frobenius' (default): sqrt(tr(A^t.A)) - 'spectral': sqrt(max(eigenvalues(A^t.A)) where A is the error ``(comp_cov - self.covariance_)``. scaling: bool If True (default), the squared error norm is divided by n_features. If False, the squared error norm is not rescaled. squared: bool Whether to compute the squared error norm or the error norm. If True (default), the squared error norm is returned. If False, the error norm is returned. Returns ------- The Mean Squared Error (in the sense of the Frobenius norm) between `self` and `comp_cov` covariance estimators. """ # compute the error error = comp_cov - self.covariance_ # compute the error norm if norm == "frobenius": squared_norm = np.sum(error ** 2) elif norm == "spectral": squared_norm = np.amax(linalg.svdvals(np.dot(error.T, error))) else: raise NotImplementedError( "Only spectral and frobenius norms are implemented") # optionaly scale the error norm if scaling: squared_norm = squared_norm / error.shape[0] # finally get either the squared norm or the norm if squared: result = squared_norm else: result = np.sqrt(squared_norm) return result def mahalanobis(self, observations): """Computes the mahalanobis distances of given observations. The provided observations are assumed to be centered. One may want to center them using a location estimate first. Parameters ---------- observations: array-like, shape = [n_observations, n_features] The observations, the Mahalanobis distances of the which we compute. Returns ------- mahalanobis_distance: array, shape = [n_observations,] Mahalanobis distances of the observations. """ # get precision if self.store_precision: precision = self.precision_ else: precision = linalg.pinv(self.covariance_) # compute mahalanobis distances mahalanobis_dist = np.sum( np.dot(observations, precision) * observations, 1) return mahalanobis_dist