"""
Maximum likelihood covariance estimator.
"""
# 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 scipy import linalg
from ..base import BaseEstimator
from ..utils import check_array
from ..utils.extmath import fast_logdet
def log_likelihood(emp_cov, precision):
"""Computes the sample mean of the log_likelihood under a covariance model
computes the empirical expected log-likelihood (accounting for the
normalization terms and scaling), allowing for universal comparison (beyond
this software package)
Parameters
----------
emp_cov : 2D ndarray (n_features, n_features)
Maximum Likelihood Estimator of covariance
precision : 2D ndarray (n_features, n_features)
The precision matrix of the covariance model to be tested
Returns
-------
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.
return log_likelihood_
def empirical_covariance(X, assume_centered=False):
"""Computes the Maximum likelihood covariance estimator
Parameters
----------
X : ndarray, 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 when working with data whose mean is almost, but not exactly
zero.
If False, data are centered before computation.
Returns
-------
covariance : 2D ndarray, shape (n_features, n_features)
Empirical covariance (Maximum Likelihood Estimator).
"""
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 <covariance>`.
Parameters
----------
store_precision : bool
Specifies if the estimated precision is stored.
assume_centered : bool
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_ : 2D ndarray, shape (n_features, n_features)
Estimated covariance matrix
precision_ : 2D ndarray, shape (n_features, n_features)
Estimated pseudo-inverse matrix.
(stored only if store_precision is True)
"""
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 : 2D ndarray, 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)
else:
self.precision_ = None
def get_precision(self):
"""Getter for the precision matrix.
Returns
-------
precision_ : array-like,
The precision matrix associated to the current covariance object.
"""
if self.store_precision:
precision = self.precision_
else:
precision = linalg.pinvh(self.covariance_)
return precision
[docs] def fit(self, X, y=None):
"""Fits the Maximum Likelihood Estimator 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)
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
[docs] def score(self, X_test, y=None):
"""Computes the log-likelihood of a Gaussian data set with
`self.covariance_` as an estimator of its covariance matrix.
Parameters
----------
X_test : array-like, 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 : not used, present for API consistence purpose.
Returns
-------
res : float
The likelihood of the data set with `self.covariance_` as an
estimator of its covariance matrix.
"""
# 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):
"""Computes the Mean Squared Error between two covariance estimators.
(In the sense of the Frobenius norm).
Parameters
----------
comp_cov : array-like, shape = [n_features, n_features]
The covariance to compare with.
norm : str
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
If True (default), the squared error norm is divided by n_features.
If False, the squared error norm is not rescaled.
squared : bool
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
-------
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, observations):
"""Computes the squared Mahalanobis distances of given observations.
Parameters
----------
observations : array-like, shape = [n_observations, 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
-------
mahalanobis_distance : array, shape = [n_observations,]
Squared Mahalanobis distances of the observations.
"""
precision = self.get_precision()
# compute mahalanobis distances
centered_obs = observations - self.location_
mahalanobis_dist = np.sum(
np.dot(centered_obs, precision) * centered_obs, 1)
return mahalanobis_dist
