OAS

classsklearn.covariance.OAS(*, store_precision=True, assume_centered=False)[source]

Oracle Approximating Shrinkage Estimator.

Read more in the User Guide.

Parameters:

store_precisionbool, default=True

Specify if the estimated precision is stored.

assume_centeredbool, 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 (default), data will be centered before computation.

Attributes:

covariance_ndarray of shape (n_features, n_features)

Estimated covariance matrix.

location_ndarray of shape (n_features,)

Estimated location, i.e. the estimated mean.

precision_ndarray of shape (n_features, n_features)

Estimated pseudo inverse matrix. (stored only if store_precision is True)

shrinkage_float

coefficient in the convex combination used for the computation of the shrunk estimate. Range is [0, 1].

n_features_in_int

Number of features seen during fit.

Added in version 0.24.

feature_names_in_ndarray of shape (n_features_in_,)

Names of features seen during fit. Defined only when X has feature names that are all strings.

Added in version 1.0.

See also

EllipticEnvelope

An object for detecting outliers in a Gaussian distributed dataset.

EmpiricalCovariance

Maximum likelihood covariance estimator.

GraphicalLasso

Sparse inverse covariance estimation with an l1-penalized estimator.

GraphicalLassoCV

Sparse inverse covariance with cross-validated choice of the l1 penalty.

LedoitWolf

LedoitWolf Estimator.

MinCovDet

Minimum Covariance Determinant (robust estimator of covariance).

ShrunkCovariance

Covariance estimator with shrinkage.

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 [1]).

The shrinkage formulation implemented here differs from Eq. 23 in [1]. In the original article, formula (23) states that 2/p (p being the number of features) is multiplied by Trace(cov*cov) in both the numerator and denominator, but this operation is omitted because for a large p, the value of 2/p is so small that it doesn’t affect the value of the estimator.

References

[1] (1,2)

“Shrinkage algorithms for MMSE covariance estimation.”, Chen, Y., Wiesel, A., Eldar, Y. C., & Hero, A. O. IEEE Transactions on Signal Processing, 58(10), 5016-5029, 2010.

Examples

>>> import numpy as np
>>> from sklearn.covariance import OAS
>>> 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)
>>> oas = OAS().fit(X)
>>> oas.covariance_
array([[0.7533, 0.2763],
       [0.2763, 0.3964]])
>>> oas.precision_
array([[ 1.7833, -1.2431 ],
       [-1.2431,  3.3889]])
>>> oas.shrinkage_
np.float64(0.0195)

See also Shrinkage covariance estimation: LedoitWolf vs OAS and max-likelihood and Ledoit-Wolf vs OAS estimation for more detailed examples.

error_norm(comp_cov, norm='frobenius', scaling=True, squared=True)[source]

Compute the Mean Squared Error between two covariance estimators.

Parameters:

comp_covarray-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_).

scalingbool, default=True

If True (default), the squared error norm is divided by n_features. If False, the squared error norm is not rescaled.

squaredbool, 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:

resultfloat

The Mean Squared Error (in the sense of the Frobenius norm) between self and comp_cov covariance estimators.

fit(X, y=None)[source]

Fit the Oracle Approximating Shrinkage covariance model to X.

Parameters:

Xarray-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.

yIgnored

Not used, present for API consistency by convention.

Returns:

selfobject

Returns the instance itself.

get_metadata_routing()[source]

Get metadata routing of this object.

Please check User Guide on how the routing mechanism works.

Returns:

routingMetadataRequest

A MetadataRequest encapsulating routing information.

get_params(deep=True)[source]

Get parameters for this estimator.

Parameters:

deepbool, default=True

If True, will return the parameters for this estimator and contained subobjects that are estimators.

Returns:

paramsdict

Parameter names mapped to their values.

get_precision()[source]

Getter for the precision matrix.

Returns:

precision_array-like of shape (n_features, n_features)

The precision matrix associated to the current covariance object.

mahalanobis(X)[source]

Compute the squared Mahalanobis distances of given observations.

For a detailed example of how outliers affects the Mahalanobis distance, see Robust covariance estimation and Mahalanobis distances relevance.

Parameters:

Xarray-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:

distndarray of shape (n_samples,)

Squared Mahalanobis distances of the observations.

score(X_test, y=None)[source]

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_testarray-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).

yIgnored

Not used, present for API consistency by convention.

Returns:

resfloat

The log-likelihood of X_test with self.location_ and self.covariance_ as estimators of the Gaussian model mean and covariance matrix respectively.

set_params(**params)[source]

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as Pipeline). The latter have parameters of the form <component>__<parameter> so that it’s possible to update each component of a nested object.

Parameters:

**paramsdict

Estimator parameters.

Returns:

selfestimator instance

Estimator instance.

Gallery examples

Normal, Ledoit-Wolf and OAS Linear Discriminant Analysis for classification

Shrinkage covariance estimation: LedoitWolf vs OAS and max-likelihood

Ledoit-Wolf vs OAS estimation