"""
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 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 <shrunk_covariance>`.
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 <shrunk_covariance>`.
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
[docs] 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 <shrunk_covariance>`.
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 <X2.T, X2>
delta_ = 0. # sum of the *squared* coefficients of <X.T, X>
# 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 (<X.T,X> - 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 <shrunk_covariance>`.
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 <shrunk_covariance>`.
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
[docs] 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 <shrunk_covariance>`.
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.
"""
[docs] 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
