"""Estimate the shrunk Ledoit-Wolf covariance matrix."""

# 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()), 0.0

n_features = X.shape[1]

# 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 = (1.0 - shrinkage) * emp_cov

shrunk_cov.flat[:: n_features + 1] += shrinkage * mu

"""Estimate covariance with the Oracle Approximating Shrinkage algorithm.

The formulation is based on [1]_.

[1] "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.

https://arxiv.org/pdf/0907.4698.pdf

"""

if len(X.shape) == 2 and X.shape[1] == 1:

# for only one feature, the result is the same whatever the shrinkage

if not assume_centered:

X = X - X.mean()

return np.atleast_2d((X**2).mean()), 0.0

n_samples, n_features = X.shape

emp_cov = empirical_covariance(X, assume_centered=assume_centered)

# The shrinkage is defined as:

# shrinkage = min(

# trace(S @ S.T) + trace(S)**2) / ((n + 1) (trace(S @ S.T) - trace(S)**2 / p), 1

# )

# where n and p are n_samples and n_features, respectively (cf. Eq. 23 in [1]).

# The factor 2 / p is omitted since it does not impact the value of the estimator

# for large p.

# Instead of computing trace(S)**2, we can compute the average of the squared

# elements of S that is equal to trace(S)**2 / p**2.

# See the definition of the Frobenius norm:

# https://en.wikipedia.org/wiki/Matrix_norm#Frobenius_norm

alpha = np.mean(emp_cov**2)

mu = np.trace(emp_cov) / n_features

mu_squared = mu**2

# The factor 1 / p**2 will cancel out since it is in both the numerator and

# denominator

num = alpha + mu_squared

den = (n_samples + 1) * (alpha - mu_squared / n_features)

shrinkage = 1.0 if den == 0 else min(num / den, 1.0)

# The shrunk covariance is defined as:

# (1 - shrinkage) * S + shrinkage * F (cf. Eq. 4 in [1])

# where S is the empirical covariance and F is the shrinkage target defined as

# F = trace(S) / n_features * np.identity(n_features) (cf. Eq. 3 in [1])

shrunk_cov = (1.0 - shrinkage) * emp_cov

shrunk_cov.flat[:: n_features + 1] += shrinkage * mu

"""Calculate covariance matrices shrunk on the diagonal.

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

Parameters

----------

emp_cov : array-like of shape (..., n_features, n_features)

Covariance matrices to be shrunk, at least 2D ndarray.

shrinkage : float, default=0.1

Coefficient in the convex combination used for the computation

of the shrunk estimate. Range is [0, 1].

Returns

-------

shrunk_cov : ndarray of shape (..., n_features, n_features)

Shrunk covariance matrices.

Notes

-----

The regularized (shrunk) covariance is given by::

(1 - shrinkage) * cov + shrinkage * mu * np.identity(n_features)

where `mu = trace(cov) / n_features`.

Examples

--------

>>> import numpy as np

>>> from sklearn.datasets import make_gaussian_quantiles

>>> from sklearn.covariance import empirical_covariance, shrunk_covariance

>>> 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)

>>> shrunk_covariance(empirical_covariance(X))

array([[0.73..., 0.25...],

[0.25..., 0.41...]])

"""

emp_cov = check_array(emp_cov, allow_nd=True)

n_features = emp_cov.shape[-1]

shrunk_cov = (1.0 - shrinkage) * emp_cov

mu = np.trace(emp_cov, axis1=-2, axis2=-1) / n_features

mu = np.expand_dims(mu, axis=tuple(range(mu.ndim, emp_cov.ndim)))

shrunk_cov += shrinkage * mu * np.eye(n_features)

"""Covariance estimator with shrinkage.

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 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.

shrinkage : float, default=0.1

Coefficient in the convex combination used for the computation

of the shrunk estimate. Range is [0, 1].

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)

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.

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).

OAS : Oracle Approximating Shrinkage Estimator.

Notes

-----

The regularized covariance is given by:

(1 - shrinkage) * cov + shrinkage * mu * np.identity(n_features)

where mu = trace(cov) / n_features

Examples

--------

>>> import numpy as np

>>> from sklearn.covariance import ShrunkCovariance

>>> 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 = ShrunkCovariance().fit(X)

>>> cov.covariance_

array([[0.7387..., 0.2536...],

[0.2536..., 0.4110...]])

>>> cov.location_

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

"""

_parameter_constraints: dict = {

**EmpiricalCovariance._parameter_constraints,

"shrinkage": [Interval(Real, 0, 1, closed="both")],

}

def __init__(self, *, store_precision=True, assume_centered=False, shrinkage=0.1):

super().__init__(

store_precision=store_precision, assume_centered=assume_centered

)

self.shrinkage = shrinkage

@_fit_context(prefer_skip_nested_validation=True)

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

"""Fit the shrunk covariance model 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)

# 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)

"""Estimate the shrunk Ledoit-Wolf covariance matrix.

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

Parameters

----------

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

Data from which to compute the Ledoit-Wolf shrunk covariance shrinkage.

assume_centered : bool, default=False

If True, data will not be centered before computation.

Useful to work with data whose mean is significantly equal to

zero but is not exactly zero.

If False, data will be centered before computation.

block_size : int, default=1000

Size of 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

Examples

--------

>>> import numpy as np

>>> from sklearn.covariance import ledoit_wolf_shrinkage

>>> real_cov = np.array([[.4, .2], [.2, .8]])

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

>>> X = rng.multivariate_normal(mean=[0, 0], cov=real_cov, size=50)

>>> shrinkage_coefficient = ledoit_wolf_shrinkage(X)

>>> shrinkage_coefficient

0.23...

"""

X = check_array(X)

# for only one feature, the result is the same whatever the shrinkage

if len(X.shape) == 2 and X.shape[1] == 1:

return 0.0

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_ = 0.0 # sum of the coefficients of <X2.T, X2>

delta_ = 0.0 # sum of the *squared* coefficients of <X.T, X>

# starting block computation

for i in range(n_splits):

for j in range(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 range(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 = 1.0 / (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_ - 2.0 * 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 would invert

# the value of covariances

beta = min(beta, delta)

# finally get shrinkage

shrinkage = 0 if beta == 0 else beta / delta

"""Estimate the shrunk Ledoit-Wolf covariance matrix.

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

Parameters

----------

X : array-like 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 to work with data whose mean is significantly equal to

zero but is not exactly zero.

If False, data will be centered before computation.

block_size : int, default=1000

Size of blocks into which the covariance matrix will be split.

This is purely a memory optimization and does not affect results.

Returns

-------

shrunk_cov : ndarray of 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

Examples

--------

>>> import numpy as np

>>> from sklearn.covariance import empirical_covariance, ledoit_wolf

>>> real_cov = np.array([[.4, .2], [.2, .8]])

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

>>> X = rng.multivariate_normal(mean=[0, 0], cov=real_cov, size=50)

>>> covariance, shrinkage = ledoit_wolf(X)

>>> covariance

array([[0.44..., 0.16...],

[0.16..., 0.80...]])

>>> shrinkage

0.23...

"""

estimator = LedoitWolf(

assume_centered=assume_centered,

block_size=block_size,

store_precision=False,

).fit(X)

"""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 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.

block_size : int, default=1000

Size of 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_ : 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 :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.

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.

MinCovDet : Minimum Covariance Determinant

(robust estimator of covariance).

OAS : Oracle Approximating Shrinkage Estimator.

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 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.

Examples

--------

>>> import numpy as np

>>> from sklearn.covariance import LedoitWolf

>>> real_cov = np.array([[.4, .2],

... [.2, .8]])

>>> np.random.seed(0)

>>> X = np.random.multivariate_normal(mean=[0, 0],

... cov=real_cov,

... size=50)

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

>>> cov.covariance_

array([[0.4406..., 0.1616...],

[0.1616..., 0.8022...]])

>>> cov.location_

array([ 0.0595... , -0.0075...])

"""

_parameter_constraints: dict = {

**EmpiricalCovariance._parameter_constraints,

"block_size": [Interval(Integral, 1, None, closed="left")],

}

def __init__(self, *, store_precision=True, assume_centered=False, block_size=1000):

super().__init__(

store_precision=store_precision, assume_centered=assume_centered

)

self.block_size = block_size

@_fit_context(prefer_skip_nested_validation=True)

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

"""Fit the Ledoit-Wolf shrunk covariance model 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.

"""

# Not calling the parent object to fit, to avoid computing the

# covariance matrix (and potentially the precision)

X = self._validate_data(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)

"""Estimate covariance with the Oracle Approximating Shrinkage.

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

Parameters

----------

X : array-like 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 to work with data whose mean is significantly equal to

zero but is not exactly zero.

If False, data will be centered before computation.

Returns

-------

shrunk_cov : array-like of 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 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] :arxiv:`"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.

<0907.4698>`

Examples

--------

>>> import numpy as np

>>> from sklearn.covariance import oas

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

>>> real_cov = [[.8, .3], [.3, .4]]

>>> X = rng.multivariate_normal(mean=[0, 0], cov=real_cov, size=500)

>>> shrunk_cov, shrinkage = oas(X)

>>> shrunk_cov

array([[0.7533..., 0.2763...],

[0.2763..., 0.3964...]])

>>> shrinkage

0.0195...

"""

estimator = OAS(

assume_centered=assume_centered,

).fit(X)

"""Oracle Approximating Shrinkage Estimator.

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 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 :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.

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] :arxiv:`"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.

<0907.4698>`

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_

0.0195...

"""

@_fit_context(prefer_skip_nested_validation=True)

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

"""Fit the Oracle Approximating Shrinkage covariance model 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)

# 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)