""" Maximum likelihood covariance estimator. """ # Author: Alexandre Gramfort # Gael Varoquaux # Virgile Fritsch # # License: BSD 3 clause # avoid division truncation import warnings import numpy as np from scipy import linalg from .. import config_context from ..base import BaseEstimator from ..utils import check_array from ..utils.extmath import fast_logdet from ..metrics.pairwise import pairwise_distances def log_likelihood(emp_cov, precision): """Compute the sample mean of the log_likelihood under a covariance model. Computes the empirical expected log-likelihood, allowing for universal comparison (beyond this software package), and accounts for normalization terms and scaling. Parameters ---------- emp_cov : ndarray of shape (n_features, n_features) Maximum Likelihood Estimator of covariance. precision : ndarray of shape (n_features, n_features) The precision matrix of the covariance model to be tested. Returns ------- log_likelihood_ : float 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_ /= 2.0 return log_likelihood_ def empirical_covariance(X, *, assume_centered=False): """Compute the Maximum likelihood covariance estimator. Parameters ---------- X : ndarray of shape (n_samples, n_features) Data from which to compute the covariance estimate. assume_centered : bool, default=False If `True`, data will not be centered before computation. Useful when working with data whose mean is almost, but not exactly zero. If `False`, data will be centered before computation. Returns ------- covariance : ndarray of shape (n_features, n_features) Empirical covariance (Maximum Likelihood Estimator). Examples -------- >>> from sklearn.covariance import empirical_covariance >>> X = [[1,1,1],[1,1,1],[1,1,1], ... [0,0,0],[0,0,0],[0,0,0]] >>> empirical_covariance(X) array([[0.25, 0.25, 0.25], [0.25, 0.25, 0.25], [0.25, 0.25, 0.25]]) """ 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, default=True Specifies if the estimated precision is stored. assume_centered : bool, default=False 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 ---------- location_ : ndarray of shape (n_features,) Estimated location, i.e. the estimated mean. covariance_ : ndarray of shape (n_features, n_features) Estimated covariance matrix precision_ : ndarray of shape (n_features, n_features) Estimated pseudo-inverse matrix. (stored only if store_precision is True) n_features_in_ : int Number of features seen during :term:`fit`. .. versionadded:: 0.24 feature_names_in_ : ndarray of shape (`n_features_in_`,) Names of features seen during :term:`fit`. Defined only when `X` has feature names that are all strings. .. versionadded:: 1.0 See Also -------- EllipticEnvelope : An object for detecting outliers in a Gaussian distributed dataset. GraphicalLasso : Sparse inverse covariance estimation with an l1-penalized estimator. LedoitWolf : LedoitWolf Estimator. MinCovDet : Minimum Covariance Determinant (robust estimator of covariance). OAS : Oracle Approximating Shrinkage Estimator. ShrunkCovariance : Covariance estimator with shrinkage. Examples -------- >>> import numpy as np >>> from sklearn.covariance import EmpiricalCovariance >>> from sklearn.datasets import make_gaussian_quantiles >>> real_cov = np.array([[.8, .3], ... [.3, .4]]) >>> rng = np.random.RandomState(0) >>> X = rng.multivariate_normal(mean=[0, 0], ... cov=real_cov, ... size=500) >>> cov = EmpiricalCovariance().fit(X) >>> cov.covariance_ array([[0.7569..., 0.2818...], [0.2818..., 0.3928...]]) >>> cov.location_ array([0.0622..., 0.0193...]) """ _parameter_constraints: dict = { "store_precision": ["boolean"], "assume_centered": ["boolean"], } 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 : array-like of 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, check_finite=False) else: self.precision_ = None def get_precision(self): """Getter for the precision matrix. Returns ------- precision_ : array-like of shape (n_features, n_features) The precision matrix associated to the current covariance object. """ if self.store_precision: precision = self.precision_ else: precision = linalg.pinvh(self.covariance_, check_finite=False) return precision def fit(self, X, y=None): """Fit the maximum likelihood covariance estimator 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 Not used, present for API consistency by convention. Returns ------- self : object Returns the instance itself. """ self._validate_params() X = self._validate_data(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): """Compute the log-likelihood of `X_test` under the estimated Gaussian model. The Gaussian model is defined by its mean and covariance matrix which are represented respectively by `self.location_` and `self.covariance_`. Parameters ---------- X_test : array-like of 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 : Ignored Not used, present for API consistency by convention. Returns ------- res : float The log-likelihood of `X_test` with `self.location_` and `self.covariance_` as estimators of the Gaussian model mean and covariance matrix respectively. """ X_test = self._validate_data(X_test, reset=False) # 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): """Compute the Mean Squared Error between two covariance estimators. Parameters ---------- comp_cov : array-like of shape (n_features, n_features) The covariance to compare with. norm : {"frobenius", "spectral"}, default="frobenius" 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, default=True If True (default), the squared error norm is divided by n_features. If False, the squared error norm is not rescaled. squared : bool, default=True 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 ------- result : float 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, X): """Compute the squared Mahalanobis distances of given observations. Parameters ---------- X : array-like of shape (n_samples, 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 ------- dist : ndarray of shape (n_samples,) Squared Mahalanobis distances of the observations. """ X = self._validate_data(X, reset=False) precision = self.get_precision() with config_context(assume_finite=True): # compute mahalanobis distances dist = pairwise_distances( X, self.location_[np.newaxis, :], metric="mahalanobis", VI=precision ) return np.reshape(dist, (len(X),)) ** 2