"""Compute the sample mean of the log_likelihood under a covariance model.

Computes the empirical expected log-likelihood, allowing for universal

comparison (beyond this software package), and accounts for normalization

terms and scaling.

Parameters

----------

emp_cov : ndarray of shape (n_features, n_features)

Maximum Likelihood Estimator of covariance.

precision : ndarray of shape (n_features, n_features)

The precision matrix of the covariance model to be tested.

Returns

-------

log_likelihood_ : float

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_ /= 2.0

"""Compute the Maximum likelihood covariance estimator.

Parameters

----------

X : ndarray of shape (n_samples, n_features)

Data from which to compute the covariance estimate.

assume_centered : bool, default=False

If `True`, data will not be centered before computation.

Useful when working with data whose mean is almost, but not exactly

zero.

If `False`, data will be centered before computation.

Returns

-------

covariance : ndarray of shape (n_features, n_features)

Empirical covariance (Maximum Likelihood Estimator).

Examples

--------

>>> from sklearn.covariance import empirical_covariance

>>> X = [[1,1,1],[1,1,1],[1,1,1],

... [0,0,0],[0,0,0],[0,0,0]]

>>> empirical_covariance(X)

array([[0.25, 0.25, 0.25],

[0.25, 0.25, 0.25],

[0.25, 0.25, 0.25]])

"""

X = check_array(X, ensure_2d=False, force_all_finite=False)

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]])

"""Maximum likelihood covariance estimator.

Read more in the :ref:`User Guide <covariance>`.

Parameters

----------

store_precision : bool, default=True

Specifies 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

----------

location_ : ndarray of shape (n_features,)

Estimated location, i.e. the estimated mean.

covariance_ : ndarray of shape (n_features, n_features)

Estimated covariance matrix

precision_ : ndarray of shape (n_features, n_features)

Estimated pseudo-inverse matrix.

(stored only if store_precision is True)

n_features_in_ : int

Number of features seen during :term:`fit`.

.. versionadded:: 0.24

feature_names_in_ : ndarray of shape (`n_features_in_`,)

Names of features seen during :term:`fit`. Defined only when `X`

has feature names that are all strings.

.. versionadded:: 1.0

See Also

--------

EllipticEnvelope : An object for detecting outliers in

a Gaussian distributed dataset.

GraphicalLasso : Sparse inverse covariance estimation

with an l1-penalized estimator.

LedoitWolf : LedoitWolf Estimator.

MinCovDet : Minimum Covariance Determinant

(robust estimator of covariance).

OAS : Oracle Approximating Shrinkage Estimator.

ShrunkCovariance : Covariance estimator with shrinkage.

Examples

--------

>>> import numpy as np

>>> from sklearn.covariance import EmpiricalCovariance

>>> from sklearn.datasets import make_gaussian_quantiles

>>> real_cov = np.array([[.8, .3],

... [.3, .4]])

>>> rng = np.random.RandomState(0)

>>> X = rng.multivariate_normal(mean=[0, 0],

... cov=real_cov,

... size=500)

>>> cov = EmpiricalCovariance().fit(X)

>>> cov.covariance_

array([[0.7569..., 0.2818...],

[0.2818..., 0.3928...]])

>>> cov.location_

array([0.0622..., 0.0193...])

"""

_parameter_constraints: dict = {

"store_precision": ["boolean"],

"assume_centered": ["boolean"],

}

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 : array-like of 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, check_finite=False)

else:

self.precision_ = None

def get_precision(self):

"""Getter for the precision matrix.

Returns

-------

precision_ : array-like of shape (n_features, n_features)

The precision matrix associated to the current covariance object.

"""

if self.store_precision:

precision = self.precision_

else:

precision = linalg.pinvh(self.covariance_, check_finite=False)

return precision

@_fit_context(prefer_skip_nested_validation=True)

def fit(self, X, y=None):

"""Fit the maximum likelihood covariance estimator to X.

Parameters

----------

X : array-like of shape (n_samples, n_features)

Training data, where `n_samples` is the number of samples and

`n_features` is the number of features.

y : Ignored

Not used, present for API consistency by convention.

Returns

-------

self : object

Returns the instance itself.

"""

X = self._validate_data(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

def score(self, X_test, y=None):

"""Compute the log-likelihood of `X_test` under the estimated Gaussian model.

The Gaussian model is defined by its mean and covariance matrix which are

represented respectively by `self.location_` and `self.covariance_`.

Parameters

----------

X_test : array-like of 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 : Ignored

Not used, present for API consistency by convention.

Returns

-------

res : float

The log-likelihood of `X_test` with `self.location_` and `self.covariance_`

as estimators of the Gaussian model mean and covariance matrix respectively.

"""

X_test = self._validate_data(X_test, reset=False)

# 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):

"""Compute the Mean Squared Error between two covariance estimators.

Parameters

----------

comp_cov : array-like of shape (n_features, n_features)

The covariance to compare with.

norm : {"frobenius", "spectral"}, default="frobenius"

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, default=True

If True (default), the squared error norm is divided by n_features.

If False, the squared error norm is not rescaled.

squared : bool, default=True

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

-------

result : float

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, X):

"""Compute the squared Mahalanobis distances of given observations.

Parameters

----------

X : array-like of shape (n_samples, 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

-------

dist : ndarray of shape (n_samples,)

Squared Mahalanobis distances of the observations.

"""

X = self._validate_data(X, reset=False)

precision = self.get_precision()

with config_context(assume_finite=True):

# compute mahalanobis distances

dist = pairwise_distances(

X, self.location_[np.newaxis, :], metric="mahalanobis", VI=precision

)