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"""
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 Style.
# avoid division truncation
from __future__ import division
import warnings
import numpy as np
from scipy import linalg
from ..base import BaseEstimator
from ..utils import array2d
from ..utils.extmath import fast_logdet
def log_likelihood(emp_cov, precision):
"""Computes the log_likelihood of the data
Params
------
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
"""
return -np.sum(emp_cov * precision) + fast_logdet(precision)
def empirical_covariance(X, assume_centered=False):
"""Computes the Maximum likelihood covariance estimator
Parameters
----------
X: 2D 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))
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)
return covariance
class EmpiricalCovariance(BaseEstimator):
"""Maximum likelihood covariance estimator
Parameters
----------
store_precision : bool
Specifies if the estimated precision is stored
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):
"""
Parameters
----------
store_precision: bool
Specify if the estimated precision is stored
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.
"""
self.store_precision = store_precision
self.assume_centered = assume_centered
def _set_estimates(self, covariance):
"""Saves the covariance and precision estimates
Storage is done accordingly to `self.store_precision`.
Precision stored only if invertible.
Params
------
covariance: 2D ndarray, shape (n_features, n_features)
Estimated covariance matrix to be stored, and from which precision
is computed.
"""
covariance = array2d(covariance)
# set covariance
self.covariance_ = covariance
# set precision
if self.store_precision:
self.precision_ = linalg.pinv(covariance)
else:
self.precision_ = None
def fit(self, X):
"""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.
Returns
-------
self : object
Returns self.
"""
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_estimates(covariance)
return self
def score(self, X_test, assume_centered=False):
"""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.
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,
assume_centered=assume_centered)
# compute log likelihood
if self.store_precision:
res = log_likelihood(test_cov, self.precision_)
else:
res = log_likelihood(test_cov, linalg.pinv(self.covariance_))
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")
# optionaly 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 mahalanobis distances of given observations.
The provided observations are assumed to be centered. One may want to
center them using a location estimate first.
Parameters
----------
observations: array-like, shape = [n_observations, n_features]
The observations, the Mahalanobis distances of the which we compute.
Returns
-------
mahalanobis_distance: array, shape = [n_observations,]
Mahalanobis distances of the observations.
"""
# get precision
if self.store_precision:
precision = self.precision_
else:
precision = linalg.pinv(self.covariance_)
# compute mahalanobis distances
mahalanobis_dist = np.sum(
np.dot(observations, precision) * observations, 1)
return mahalanobis_dist
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