""" Maximum likelihood covariance estimator. """ # Author: Alexandre Gramfort # Gael Varoquaux # Virgile Fritsch # # License: BSD 3 clause # avoid division truncation from __future__ import division import warnings import numpy as np from scipy import linalg from ..base import BaseEstimator from ..utils import check_array from ..utils.extmath import fast_logdet def log_likelihood(emp_cov, precision): """Computes the sample mean of the log_likelihood under a covariance model computes the empirical expected log-likelihood (accounting for the normalization terms and scaling), allowing for universal comparison (beyond this software package) Parameters ---------- 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 Returns ------- sample mean of the log-likelihood """ p = precision.shape[0] log_likelihood_ = - np.sum(emp_cov * precision) + fast_logdet(precision) log_likelihood_ -= p * np.log(2 * np.pi) log_likelihood_ /= 2. return log_likelihood_ def empirical_covariance(X, assume_centered=False): """Computes the Maximum likelihood covariance estimator Parameters ---------- X : 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)) if X.shape[0] == 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) if covariance.ndim == 0: covariance = np.array([[covariance]]) return covariance class EmpiricalCovariance(BaseEstimator): """Maximum likelihood covariance estimator Read more in the :ref:`User Guide `. Parameters ---------- store_precision : bool Specifies if the estimated precision is stored. assume_centered : bool If True, data are not centered before computation. Useful when working with data whose mean is almost, but not exactly zero. If False (default), data are centered before computation. 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): self.store_precision = store_precision self.assume_centered = assume_centered def _set_covariance(self, covariance): """Saves the covariance and precision estimates Storage is done accordingly to `self.store_precision`. Precision stored only if invertible. Parameters ---------- covariance : 2D ndarray, shape (n_features, n_features) Estimated covariance matrix to be stored, and from which precision is computed. """ covariance = check_array(covariance) # set covariance self.covariance_ = covariance # set precision if self.store_precision: self.precision_ = linalg.pinvh(covariance) else: self.precision_ = None def get_precision(self): """Getter for the precision matrix. Returns ------- precision_ : array-like, The precision matrix associated to the current covariance object. """ if self.store_precision: precision = self.precision_ else: precision = linalg.pinvh(self.covariance_) return precision def fit(self, X, y=None): """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. y : not used, present for API consistence purpose. Returns ------- self : object Returns self. """ X = check_array(X) 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_covariance(covariance) return self def score(self, X_test, y=None): """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. X_test is assumed to be drawn from the same distribution than the data used in fit (including centering). y : not used, present for API consistence purpose. 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 - self.location_, assume_centered=True) # compute log likelihood res = log_likelihood(test_cov, self.get_precision()) 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") # optionally 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 squared Mahalanobis distances of given observations. Parameters ---------- observations : array-like, shape = [n_observations, n_features] The observations, the Mahalanobis distances of the which we compute. Observations are assumed to be drawn from the same distribution than the data used in fit. Returns ------- mahalanobis_distance : array, shape = [n_observations,] Squared Mahalanobis distances of the observations. """ precision = self.get_precision() # compute mahalanobis distances centered_obs = observations - self.location_ mahalanobis_dist = np.sum( np.dot(centered_obs, precision) * centered_obs, 1) return mahalanobis_dist