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Inzynierka/Lib/site-packages/sklearn/covariance/_shrunk_covariance.py

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"""
Covariance estimators using shrinkage.
Shrinkage corresponds to regularising `cov` using a convex combination:
shrunk_cov = (1-shrinkage)*cov + shrinkage*structured_estimate.
"""
# 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
import warnings
from numbers import Real, Integral
import numpy as np
from . import empirical_covariance, EmpiricalCovariance
from .._config import config_context
from ..utils import check_array
from ..utils._param_validation import Interval
def _oas(X, *, assume_centered=False):
"""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
return shrunk_cov, shrinkage
###############################################################################
# Public API
# ShrunkCovariance estimator
def shrunk_covariance(emp_cov, shrinkage=0.1):
"""Calculate a covariance matrix 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 matrix to be shrunk.
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.
Notes
-----
The regularized (shrunk) covariance is given by::
(1 - shrinkage) * cov + shrinkage * mu * np.identity(n_features)
where `mu = trace(cov) / n_features`.
"""
emp_cov = check_array(emp_cov)
n_features = emp_cov.shape[0]
mu = np.trace(emp_cov) / n_features
shrunk_cov = (1.0 - shrinkage) * emp_cov
shrunk_cov.flat[:: n_features + 1] += shrinkage * mu
return shrunk_cov
class ShrunkCovariance(EmpiricalCovariance):
"""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
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.
"""
self._validate_params()
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)
return self
# Ledoit-Wolf estimator
def ledoit_wolf_shrinkage(X, assume_centered=False, block_size=1000):
"""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
"""
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
return shrinkage
def ledoit_wolf(X, *, assume_centered=False, block_size=1000):
"""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
"""
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:
if not assume_centered:
X = X - X.mean()
return np.atleast_2d((X**2).mean()), 0.0
if X.ndim == 1:
X = np.reshape(X, (1, -1))
warnings.warn(
"Only one sample available. You may want to reshape your data array"
)
n_features = X.size
else:
_, n_features = X.shape
# 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
return shrunk_cov, shrinkage
class LedoitWolf(EmpiricalCovariance):
"""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
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.
"""
self._validate_params()
# 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)
with config_context(assume_finite=True):
covariance, shrinkage = ledoit_wolf(
X - self.location_, assume_centered=True, block_size=self.block_size
)
self.shrinkage_ = shrinkage
self._set_covariance(covariance)
return self
# OAS estimator
def oas(X, *, assume_centered=False):
"""Estimate covariance with the Oracle Approximating Shrinkage as proposed in [1]_.
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>`
"""
X = np.asarray(X)
# 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
if X.ndim == 1:
X = np.reshape(X, (1, -1))
warnings.warn(
"Only one sample available. You may want to reshape your data array"
)
n_samples = 1
n_features = X.size
else:
n_samples, n_features = X.shape
emp_cov = empirical_covariance(X, assume_centered=assume_centered)
mu = np.trace(emp_cov) / n_features
# formula from Chen et al.'s **implementation**
alpha = np.mean(emp_cov**2)
num = alpha + mu**2
den = (n_samples + 1.0) * (alpha - (mu**2) / n_features)
shrinkage = 1.0 if den == 0 else min(num / den, 1.0)
shrunk_cov = (1.0 - shrinkage) * emp_cov
shrunk_cov.flat[:: n_features + 1] += shrinkage * mu
return shrunk_cov, shrinkage
class OAS(EmpiricalCovariance):
"""Oracle Approximating Shrinkage Estimator as proposed in [1]_.
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...
"""
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.
"""
self._validate_params()
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)
return self
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