""" Covariance estimators using shrinkage. Shrinkage corresponds to regularising `cov` using a convex combination: shrunk_cov = (1-shrinkage)*cov + shrinkage*structured_estimate. """ # Author: Alexandre Gramfort # Gael Varoquaux # Virgile Fritsch # # License: BSD Style. # avoid division truncation from __future__ import division import warnings import numpy as np from .empirical_covariance_ import empirical_covariance, EmpiricalCovariance from ..utils import array2d ############################################################################### # ShrunkCovariance estimator def shrunk_covariance(emp_cov, shrinkage=0.1): """Calculates a covariance matrix shrunk on the diagonal Parameters ---------- emp_cov: array-like, shape (n_features, n_features) Covariance matrix to be shrunk shrinkage: float, 0 <= shrinkage <= 1 coefficient in the convex combination used for the computation of the shrunk estimate. Returns ------- shrunk_cov: array-like shrunk covariance Notes ----- The regularized (shrunk) covariance is given by (1 - shrinkage)*cov + shrinkage*mu*np.identity(n_features) where mu = trace(cov) / n_features """ emp_cov = array2d(emp_cov) n_features = emp_cov.shape[0] mu = np.trace(emp_cov) / n_features shrunk_cov = (1. - shrinkage) * emp_cov shrunk_cov.flat[::n_features + 1] += shrinkage * mu return shrunk_cov class ShrunkCovariance(EmpiricalCovariance): """Covariance estimator with shrinkage Parameters ---------- store_precision : bool Specify if the estimated precision is stored shrinkage: float, 0 <= shrinkage <= 1 coefficient in the convex combination used for the computation of the shrunk estimate. Attributes ---------- `covariance_` : array-like, shape (n_features, n_features) Estimated covariance matrix `precision_` : array-like, shape (n_features, n_features) Estimated pseudo inverse matrix. (stored only if store_precision is True) `shrinkage`: float, 0 <= shrinkage <= 1 coefficient in the convex combination used for the computation of the shrunk estimate. Notes ----- The regularized covariance is given by (1 - shrinkage)*cov + shrinkage*mu*np.identity(n_features) where mu = trace(cov) / n_features """ def __init__(self, store_precision=True, shrinkage=0.1): self.store_precision = store_precision self.shrinkage = shrinkage def fit(self, X, assume_centered=False): """ Fits the shrunk 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. assume_centered: Boolean If True, data are not centered before computation. Usefull to work with data whose mean is significantly equal to zero but is not exactly zero. If False, data are centered before computation. Returns ------- self : object Returns self. """ empirical_cov = empirical_covariance( X, assume_centered=assume_centered) covariance = shrunk_covariance(empirical_cov, self.shrinkage) self._set_estimates(covariance) return self ############################################################################### # Ledoit-Wolf estimator def ledoit_wolf(X, assume_centered=False): """Estimates the shrunk Ledoit-Wolf covariance matrix. Parameters ---------- X: array-like, shape (n_samples, n_features) Data from which to compute the covariance estimate assume_centered: Boolean If True, data are not centered before computation. Usefull to work with data whose mean is significantly equal to zero but is not exactly zero. If False, data are centered before computation. Returns ------- shrunk_cov: array-like, shape (n_features, n_features) Shrunk covariance shrinkage: float coefficient in the convex combination used for the computation of the shrunk estimate. Notes ----- The regularised (shrunk) covariance is: (1 - shrinkage)*cov + shrinkage * mu * np.identity(n_features) where mu = trace(cov) / n_features """ X = np.asarray(X) # for only one feature, the result is the same whatever the shrinkage if len(X.shape) == 2 and X.shape[1] == 1: if not assume_centered: X = X - X.mean() return np.atleast_2d((X ** 2).mean()), 0. if X.ndim == 1: X = np.reshape(X, (1, -1)) warnings.warn("Only one sample available. " \ "You may want to reshape your data array") n_samples = 1 n_features = X.size else: n_samples, n_features = X.shape # optionaly center data if not assume_centered: X = X - X.mean(0) emp_cov = empirical_covariance(X, assume_centered=assume_centered) mu = np.trace(emp_cov) / n_features delta_ = emp_cov.copy() delta_.flat[::n_features + 1] -= mu delta = (delta_ ** 2).sum() / n_features X2 = X ** 2 beta_ = 1. / (n_features * n_samples) \ * np.sum(np.dot(X2.T, X2) / n_samples - emp_cov ** 2) beta = min(beta_, delta) shrinkage = beta / delta shrunk_cov = (1. - shrinkage) * emp_cov shrunk_cov.flat[::n_features + 1] += shrinkage * mu return shrunk_cov, shrinkage class LedoitWolf(EmpiricalCovariance): """LedoitWolf Estimator Ledoit-Wolf is a particular form of shrinkage, where the shrinkage coefficient is computed using O.Ledoit and M.Wolf's formula as described in "A Well-Conditioned Estimator for Large-Dimensional Covariance Matrices", Ledoit and Wolf, Journal of Multivariate Analysis, Volume 88, Issue 2, February 2004, pages 365-411. Parameters ---------- store_precision : bool Specify if the estimated precision is stored Attributes ---------- `covariance_` : array-like, shape (n_features, n_features) Estimated covariance matrix `precision_` : array-like, shape (n_features, n_features) Estimated pseudo inverse matrix. (stored only if store_precision is True) `shrinkage_`: float, 0 <= shrinkage <= 1 coefficient in the convex combination used for the computation of the shrunk estimate. Notes ----- The regularised covariance is:: (1 - shrinkage)*cov + shrinkage*mu*np.identity(n_features) where mu = trace(cov) / n_features and shinkage is given by the Ledoit and Wolf formula (see References) References ---------- "A Well-Conditioned Estimator for Large-Dimensional Covariance Matrices", Ledoit and Wolf, Journal of Multivariate Analysis, Volume 88, Issue 2, February 2004, pages 365-411. """ def fit(self, X, assume_centered=False): """ Fits the Ledoit-Wolf shrunk 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. assume_centered: Boolean If True, data are not centered before computation. Usefull to work with data whose mean is significantly equal to zero but is not exactly zero. If False, data are centered before computation. Returns ------- self : object Returns self. """ covariance, shrinkage = ledoit_wolf(X, assume_centered=assume_centered) self.shrinkage_ = shrinkage self._set_estimates(covariance) return self ############################################################################### # OAS estimator def oas(X, assume_centered=False): """Estimate covariance with the Oracle Approximating Shrinkage algorithm. Parameters ---------- X: array-like, shape (n_samples, n_features) Data from which to compute the covariance estimate assume_centered: boolean If True, data are not centered before computation. Usefull to work with data whose mean is significantly equal to zero but is not exactly zero. If False, data are centered before computation. Returns ------- shrunk_cov: array-like, shape (n_features, n_features) Shrunk covariance shrinkage: float coefficient in the convex combination used for the computation of the shrunk estimate. Notes ----- The regularised (shrunk) covariance is: (1 - shrinkage)*cov + shrinkage * mu * np.identity(n_features) where mu = trace(cov) / n_features """ X = np.asarray(X) # for only one feature, the result is the same whatever the shrinkage if len(X.shape) == 2 and X.shape[1] == 1: if not assume_centered: X = X - X.mean() return np.atleast_2d((X ** 2).mean()), 0. if X.ndim == 1: X = np.reshape(X, (1, -1)) warnings.warn("Only one sample available. " \ "You may want to reshape your data array") n_samples = 1 n_features = X.size else: n_samples, n_features = X.shape emp_cov = empirical_covariance(X, assume_centered=assume_centered) mu = np.trace(emp_cov) / n_features # formula from Chen et al.'s **implementation** alpha = np.mean(emp_cov ** 2) num = alpha + mu ** 2 den = (n_samples + 1.) * (alpha - (mu ** 2) / n_features) shrinkage = min(num / den, 1.) shrunk_cov = (1. - shrinkage) * emp_cov shrunk_cov.flat[::n_features + 1] += shrinkage * mu return shrunk_cov, shrinkage class OAS(EmpiricalCovariance): """ Oracle Approximating Shrinkage Estimator OAS is a particular form of shrinkage described in "Shrinkage Algorithms for MMSE Covariance Estimation" Chen et al., IEEE Trans. on Sign. Proc., Volume 58, Issue 10, October 2010. The formula used here does not correspond to the one given in the article. It has been taken from the matlab programm available from the authors webpage (https://tbayes.eecs.umich.edu/yilun/covestimation). Parameters ---------- store_precision : bool Specify if the estimated precision is stored Attributes ---------- `covariance_` : array-like, shape (n_features, n_features) Estimated covariance matrix `precision_` : array-like, shape (n_features, n_features) Estimated pseudo inverse matrix. (stored only if store_precision is True) `shrinkage_`: float, 0 <= shrinkage <= 1 coefficient in the convex combination used for the computation of the shrunk estimate. Notes ----- The regularised covariance is:: (1 - shrinkage)*cov + shrinkage*mu*np.identity(n_features) where mu = trace(cov) / n_features and shinkage is given by the OAS formula (see References) References ---------- "Shrinkage Algorithms for MMSE Covariance Estimation" Chen et al., IEEE Trans. on Sign. Proc., Volume 58, Issue 10, October 2010. """ def fit(self, X, assume_centered=False): """ Fits the Oracle Approximating Shrinkage 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. assume_centered: boolean If True, data are not centered before computation. Usefull to work with data whose mean is significantly equal to zero but is not exactly zero. If False, data are centered before computation. Returns ------- self : object Returns self. """ covariance, shrinkage = oas(X, assume_centered=assume_centered) self.shrinkage_ = shrinkage self._set_estimates(covariance) return self