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"""
Covariance estimators using shrinkage.
Shrinkage corresponds to regularising `cov` using a convex combination:
shrunk_cov = (1-shrinkage)*cov + shrinkage*structured_estimate.
"""
# Author: Alexandre Gramfort <alexandre.gramfort@inria.fr>
# Gael Varoquaux <gael.varoquaux@normalesup.org>
# Virgile Fritsch <virgile.fritsch@inria.fr>
#
# 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
|
