""" 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 3 clause # avoid division truncation from __future__ import division import warnings import numpy as np from .empirical_covariance_ import empirical_covariance, EmpiricalCovariance from ..externals.six.moves import xrange from ..utils import check_array # ShrunkCovariance estimator def shrunk_covariance(emp_cov, shrinkage=0.1): """Calculates a covariance matrix shrunk on the diagonal Read more in the :ref:`User Guide `. 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 = check_array(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 Read more in the :ref:`User Guide `. Parameters ---------- store_precision : boolean, default True Specify if the estimated precision is stored assume_centered : boolean, 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, data are centered before computation. shrinkage : float, 0 <= shrinkage <= 1, default 0.1 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, assume_centered=False, shrinkage=0.1): super(ShrunkCovariance, self).__init__(store_precision=store_precision, assume_centered=assume_centered) self.shrinkage = shrinkage def fit(self, X, y=None): """ 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. y : not used, present for API consistence purpose. Returns ------- self : object Returns self. """ X = check_array(X) # Not calling the parent object to fit, to avoid a potential # matrix inversion when setting the precision 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) covariance = shrunk_covariance(covariance, self.shrinkage) self._set_covariance(covariance) return self # Ledoit-Wolf estimator def ledoit_wolf_shrinkage(X, assume_centered=False, block_size=1000): """Estimates the shrunk Ledoit-Wolf covariance matrix. Read more in the :ref:`User Guide `. Parameters ---------- X : array-like, shape (n_samples, n_features) Data from which to compute the Ledoit-Wolf shrunk covariance shrinkage. assume_centered : Boolean If True, data are not centered before computation. Useful to work with data whose mean is significantly equal to zero but is not exactly zero. If False, data are centered before computation. block_size : int Size of the blocks into which the covariance matrix will be split. Returns ------- shrinkage : float Coefficient in the convex combination used for the computation of the shrunk estimate. Notes ----- The regularized (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: return 0. 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") n_samples, n_features = X.shape # optionally center data if not assume_centered: X = X - X.mean(0) # A non-blocked version of the computation is present in the tests # in tests/test_covariance.py # number of blocks to split the covariance matrix into n_splits = int(n_features / block_size) X2 = X ** 2 emp_cov_trace = np.sum(X2, axis=0) / n_samples mu = np.sum(emp_cov_trace) / n_features beta_ = 0. # sum of the coefficients of delta_ = 0. # sum of the *squared* coefficients of # starting block computation for i in xrange(n_splits): for j in xrange(n_splits): rows = slice(block_size * i, block_size * (i + 1)) cols = slice(block_size * j, block_size * (j + 1)) beta_ += np.sum(np.dot(X2.T[rows], X2[:, cols])) delta_ += np.sum(np.dot(X.T[rows], X[:, cols]) ** 2) rows = slice(block_size * i, block_size * (i + 1)) beta_ += np.sum(np.dot(X2.T[rows], X2[:, block_size * n_splits:])) delta_ += np.sum( np.dot(X.T[rows], X[:, block_size * n_splits:]) ** 2) for j in xrange(n_splits): cols = slice(block_size * j, block_size * (j + 1)) beta_ += np.sum(np.dot(X2.T[block_size * n_splits:], X2[:, cols])) delta_ += np.sum( np.dot(X.T[block_size * n_splits:], X[:, cols]) ** 2) delta_ += np.sum(np.dot(X.T[block_size * n_splits:], X[:, block_size * n_splits:]) ** 2) delta_ /= n_samples ** 2 beta_ += np.sum(np.dot(X2.T[block_size * n_splits:], X2[:, block_size * n_splits:])) # use delta_ to compute beta beta = 1. / (n_features * n_samples) * (beta_ / n_samples - delta_) # delta is the sum of the squared coefficients of ( - mu*Id) / p delta = delta_ - 2. * mu * emp_cov_trace.sum() + n_features * mu ** 2 delta /= n_features # get final beta as the min between beta and delta # We do this to prevent shrinking more than "1", which whould invert # the value of covariances beta = min(beta, delta) # finally get shrinkage shrinkage = 0 if beta == 0 else beta / delta return shrinkage def ledoit_wolf(X, assume_centered=False, block_size=1000): """Estimates the shrunk Ledoit-Wolf covariance matrix. Read more in the :ref:`User Guide `. Parameters ---------- X : array-like, shape (n_samples, n_features) Data from which to compute the covariance estimate assume_centered : boolean, default=False If True, data are not centered before computation. Useful to work with data whose mean is significantly equal to zero but is not exactly zero. If False, data are centered before computation. block_size : int, default=1000 Size of the blocks into which the covariance matrix will be split. This is purely a memory optimization and does not affect results. 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 regularized (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 # get Ledoit-Wolf shrinkage shrinkage = ledoit_wolf_shrinkage( X, assume_centered=assume_centered, block_size=block_size) emp_cov = empirical_covariance(X, assume_centered=assume_centered) mu = np.sum(np.trace(emp_cov)) / n_features 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. Read more in the :ref:`User Guide `. Parameters ---------- store_precision : bool, default=True Specify 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. block_size : int, default=1000 Size of the blocks into which the covariance matrix will be split during its Ledoit-Wolf estimation. This is purely a memory optimization and does not affect results. 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 shrinkage 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 __init__(self, store_precision=True, assume_centered=False, block_size=1000): super(LedoitWolf, self).__init__(store_precision=store_precision, assume_centered=assume_centered) self.block_size = block_size def fit(self, X, y=None): """ 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. y : not used, present for API consistence purpose. Returns ------- self : object Returns self. """ # Not calling the parent object to fit, to avoid computing the # covariance matrix (and potentially the precision) X = check_array(X) if self.assume_centered: self.location_ = np.zeros(X.shape[1]) else: self.location_ = X.mean(0) covariance, shrinkage = ledoit_wolf(X - self.location_, assume_centered=True, block_size=self.block_size) self.shrinkage_ = shrinkage self._set_covariance(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. Useful 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 The formula we used to implement the OAS does not correspond to the one given in the article. It has been taken from the MATLAB program available from the author's webpage (http://tbayes.eecs.umich.edu/yilun/covestimation). """ 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 = 1. if den == 0 else 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 Read more in the :ref:`User Guide `. 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 program available from the authors' webpage (http://tbayes.eecs.umich.edu/yilun/covestimation). In the original article, formula (23) states that 2/p is multiplied by Trace(cov*cov) in both the numerator and denominator, this operation is omitted in the author's MATLAB program because for a large p, the value of 2/p is so small that it doesn't affect the value of the estimator. Parameters ---------- store_precision : bool, default=True Specify 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 ---------- 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 shrinkage 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, y=None): """ 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. y : not used, present for API consistence purpose. Returns ------- self : object Returns self. """ X = check_array(X) # Not calling the parent object to fit, to avoid computing the # covariance matrix (and potentially the precision) if self.assume_centered: self.location_ = np.zeros(X.shape[1]) else: self.location_ = X.mean(0) covariance, shrinkage = oas(X - self.location_, assume_centered=True) self.shrinkage_ = shrinkage self._set_covariance(covariance) return self