606 lines
20 KiB
Python
606 lines
20 KiB
Python
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"""
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Covariance estimators using shrinkage.
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Shrinkage corresponds to regularising `cov` using a convex combination:
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shrunk_cov = (1-shrinkage)*cov + shrinkage*structured_estimate.
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"""
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# Author: Alexandre Gramfort <alexandre.gramfort@inria.fr>
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# Gael Varoquaux <gael.varoquaux@normalesup.org>
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# Virgile Fritsch <virgile.fritsch@inria.fr>
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#
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# License: BSD 3 clause
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# avoid division truncation
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import warnings
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import numpy as np
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from . import empirical_covariance, EmpiricalCovariance
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from ..utils import check_array
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from ..utils.validation import _deprecate_positional_args
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# ShrunkCovariance estimator
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def shrunk_covariance(emp_cov, shrinkage=0.1):
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"""Calculates a covariance matrix shrunk on the diagonal
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Read more in the :ref:`User Guide <shrunk_covariance>`.
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Parameters
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----------
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emp_cov : array-like of shape (n_features, n_features)
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Covariance matrix to be shrunk
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shrinkage : float, default=0.1
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Coefficient in the convex combination used for the computation
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of the shrunk estimate. Range is [0, 1].
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Returns
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-------
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shrunk_cov : ndarray of shape (n_features, n_features)
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Shrunk covariance.
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Notes
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-----
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The regularized (shrunk) covariance is given by:
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(1 - shrinkage) * cov + shrinkage * mu * np.identity(n_features)
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where mu = trace(cov) / n_features
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"""
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emp_cov = check_array(emp_cov)
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n_features = emp_cov.shape[0]
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mu = np.trace(emp_cov) / n_features
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shrunk_cov = (1. - shrinkage) * emp_cov
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shrunk_cov.flat[::n_features + 1] += shrinkage * mu
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return shrunk_cov
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class ShrunkCovariance(EmpiricalCovariance):
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"""Covariance estimator with shrinkage
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Read more in the :ref:`User Guide <shrunk_covariance>`.
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Parameters
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----------
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store_precision : bool, default=True
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Specify if the estimated precision is stored
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assume_centered : bool, default=False
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If True, data will not be centered before computation.
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Useful when working with data whose mean is almost, but not exactly
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zero.
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If False, data will be centered before computation.
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shrinkage : float, default=0.1
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Coefficient in the convex combination used for the computation
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of the shrunk estimate. Range is [0, 1].
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Attributes
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----------
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covariance_ : ndarray of shape (n_features, n_features)
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Estimated covariance matrix
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location_ : ndarray of shape (n_features,)
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Estimated location, i.e. the estimated mean.
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precision_ : ndarray of shape (n_features, n_features)
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Estimated pseudo inverse matrix.
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(stored only if store_precision is True)
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Examples
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--------
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>>> import numpy as np
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>>> from sklearn.covariance import ShrunkCovariance
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>>> from sklearn.datasets import make_gaussian_quantiles
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>>> real_cov = np.array([[.8, .3],
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... [.3, .4]])
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>>> rng = np.random.RandomState(0)
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>>> X = rng.multivariate_normal(mean=[0, 0],
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... cov=real_cov,
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... size=500)
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>>> cov = ShrunkCovariance().fit(X)
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>>> cov.covariance_
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array([[0.7387..., 0.2536...],
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[0.2536..., 0.4110...]])
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>>> cov.location_
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array([0.0622..., 0.0193...])
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Notes
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-----
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The regularized covariance is given by:
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(1 - shrinkage) * cov + shrinkage * mu * np.identity(n_features)
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where mu = trace(cov) / n_features
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"""
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@_deprecate_positional_args
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def __init__(self, *, store_precision=True, assume_centered=False,
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shrinkage=0.1):
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super().__init__(store_precision=store_precision,
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assume_centered=assume_centered)
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self.shrinkage = shrinkage
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def fit(self, X, y=None):
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"""Fit the shrunk covariance model according to the given training data
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and parameters.
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Parameters
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----------
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X : array-like of shape (n_samples, n_features)
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Training data, where n_samples is the number of samples
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and n_features is the number of features.
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y : Ignored
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Not used, present for API consistency by convention.
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Returns
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-------
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self : object
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"""
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X = self._validate_data(X)
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# Not calling the parent object to fit, to avoid a potential
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# matrix inversion when setting the precision
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if self.assume_centered:
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self.location_ = np.zeros(X.shape[1])
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else:
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self.location_ = X.mean(0)
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covariance = empirical_covariance(
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X, assume_centered=self.assume_centered)
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covariance = shrunk_covariance(covariance, self.shrinkage)
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self._set_covariance(covariance)
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return self
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# Ledoit-Wolf estimator
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def ledoit_wolf_shrinkage(X, assume_centered=False, block_size=1000):
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"""Estimates the shrunk Ledoit-Wolf covariance matrix.
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Read more in the :ref:`User Guide <shrunk_covariance>`.
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Parameters
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----------
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X : array-like of shape (n_samples, n_features)
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Data from which to compute the Ledoit-Wolf shrunk covariance shrinkage.
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assume_centered : bool, default=False
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If True, data will not be centered before computation.
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Useful to work with data whose mean is significantly equal to
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zero but is not exactly zero.
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If False, data will be centered before computation.
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block_size : int, default=1000
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Size of blocks into which the covariance matrix will be split.
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Returns
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-------
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shrinkage : float
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Coefficient in the convex combination used for the computation
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of the shrunk estimate.
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Notes
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-----
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The regularized (shrunk) covariance is:
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(1 - shrinkage) * cov + shrinkage * mu * np.identity(n_features)
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where mu = trace(cov) / n_features
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"""
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X = np.asarray(X)
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# for only one feature, the result is the same whatever the shrinkage
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if len(X.shape) == 2 and X.shape[1] == 1:
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return 0.
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if X.ndim == 1:
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X = np.reshape(X, (1, -1))
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if X.shape[0] == 1:
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warnings.warn("Only one sample available. "
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"You may want to reshape your data array")
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n_samples, n_features = X.shape
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# optionally center data
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if not assume_centered:
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X = X - X.mean(0)
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# A non-blocked version of the computation is present in the tests
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# in tests/test_covariance.py
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# number of blocks to split the covariance matrix into
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n_splits = int(n_features / block_size)
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X2 = X ** 2
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emp_cov_trace = np.sum(X2, axis=0) / n_samples
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mu = np.sum(emp_cov_trace) / n_features
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beta_ = 0. # sum of the coefficients of <X2.T, X2>
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delta_ = 0. # sum of the *squared* coefficients of <X.T, X>
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# starting block computation
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for i in range(n_splits):
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for j in range(n_splits):
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rows = slice(block_size * i, block_size * (i + 1))
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cols = slice(block_size * j, block_size * (j + 1))
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beta_ += np.sum(np.dot(X2.T[rows], X2[:, cols]))
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delta_ += np.sum(np.dot(X.T[rows], X[:, cols]) ** 2)
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rows = slice(block_size * i, block_size * (i + 1))
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beta_ += np.sum(np.dot(X2.T[rows], X2[:, block_size * n_splits:]))
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delta_ += np.sum(
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np.dot(X.T[rows], X[:, block_size * n_splits:]) ** 2)
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for j in range(n_splits):
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cols = slice(block_size * j, block_size * (j + 1))
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beta_ += np.sum(np.dot(X2.T[block_size * n_splits:], X2[:, cols]))
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delta_ += np.sum(
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np.dot(X.T[block_size * n_splits:], X[:, cols]) ** 2)
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delta_ += np.sum(np.dot(X.T[block_size * n_splits:],
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X[:, block_size * n_splits:]) ** 2)
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delta_ /= n_samples ** 2
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beta_ += np.sum(np.dot(X2.T[block_size * n_splits:],
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X2[:, block_size * n_splits:]))
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# use delta_ to compute beta
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beta = 1. / (n_features * n_samples) * (beta_ / n_samples - delta_)
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# delta is the sum of the squared coefficients of (<X.T,X> - mu*Id) / p
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delta = delta_ - 2. * mu * emp_cov_trace.sum() + n_features * mu ** 2
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delta /= n_features
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# get final beta as the min between beta and delta
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# We do this to prevent shrinking more than "1", which whould invert
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# the value of covariances
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beta = min(beta, delta)
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# finally get shrinkage
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shrinkage = 0 if beta == 0 else beta / delta
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return shrinkage
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@_deprecate_positional_args
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def ledoit_wolf(X, *, assume_centered=False, block_size=1000):
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"""Estimates the shrunk Ledoit-Wolf covariance matrix.
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Read more in the :ref:`User Guide <shrunk_covariance>`.
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Parameters
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----------
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X : array-like of shape (n_samples, n_features)
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Data from which to compute the covariance estimate
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assume_centered : bool, default=False
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If True, data will not be centered before computation.
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Useful to work with data whose mean is significantly equal to
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zero but is not exactly zero.
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If False, data will be centered before computation.
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block_size : int, default=1000
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Size of blocks into which the covariance matrix will be split.
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This is purely a memory optimization and does not affect results.
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Returns
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-------
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shrunk_cov : ndarray of shape (n_features, n_features)
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Shrunk covariance.
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shrinkage : float
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Coefficient in the convex combination used for the computation
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of the shrunk estimate.
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Notes
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-----
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The regularized (shrunk) covariance is:
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(1 - shrinkage) * cov + shrinkage * mu * np.identity(n_features)
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where mu = trace(cov) / n_features
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"""
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X = np.asarray(X)
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# for only one feature, the result is the same whatever the shrinkage
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if len(X.shape) == 2 and X.shape[1] == 1:
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if not assume_centered:
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X = X - X.mean()
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return np.atleast_2d((X ** 2).mean()), 0.
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if X.ndim == 1:
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X = np.reshape(X, (1, -1))
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warnings.warn("Only one sample available. "
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"You may want to reshape your data array")
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n_features = X.size
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else:
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_, n_features = X.shape
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# get Ledoit-Wolf shrinkage
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shrinkage = ledoit_wolf_shrinkage(
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X, assume_centered=assume_centered, block_size=block_size)
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emp_cov = empirical_covariance(X, assume_centered=assume_centered)
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mu = np.sum(np.trace(emp_cov)) / n_features
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shrunk_cov = (1. - shrinkage) * emp_cov
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shrunk_cov.flat[::n_features + 1] += shrinkage * mu
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return shrunk_cov, shrinkage
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class LedoitWolf(EmpiricalCovariance):
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"""LedoitWolf Estimator
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Ledoit-Wolf is a particular form of shrinkage, where the shrinkage
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coefficient is computed using O. Ledoit and M. Wolf's formula as
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described in "A Well-Conditioned Estimator for Large-Dimensional
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Covariance Matrices", Ledoit and Wolf, Journal of Multivariate
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Analysis, Volume 88, Issue 2, February 2004, pages 365-411.
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Read more in the :ref:`User Guide <shrunk_covariance>`.
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Parameters
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----------
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store_precision : bool, default=True
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Specify if the estimated precision is stored.
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assume_centered : bool, default=False
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If True, data will not be centered before computation.
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Useful when working with data whose mean is almost, but not exactly
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zero.
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If False (default), data will be centered before computation.
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block_size : int, default=1000
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Size of blocks into which the covariance matrix will be split
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during its Ledoit-Wolf estimation. This is purely a memory
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optimization and does not affect results.
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Attributes
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----------
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covariance_ : ndarray of shape (n_features, n_features)
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Estimated covariance matrix.
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location_ : ndarray of shape (n_features,)
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Estimated location, i.e. the estimated mean.
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precision_ : ndarray of shape (n_features, n_features)
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Estimated pseudo inverse matrix.
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(stored only if store_precision is True)
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shrinkage_ : float
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Coefficient in the convex combination used for the computation
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of the shrunk estimate. Range is [0, 1].
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Examples
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||
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--------
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>>> import numpy as np
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>>> from sklearn.covariance import LedoitWolf
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>>> real_cov = np.array([[.4, .2],
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... [.2, .8]])
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>>> np.random.seed(0)
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>>> X = np.random.multivariate_normal(mean=[0, 0],
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... cov=real_cov,
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... size=50)
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>>> cov = LedoitWolf().fit(X)
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>>> cov.covariance_
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||
|
|
array([[0.4406..., 0.1616...],
|
||
|
|
[0.1616..., 0.8022...]])
|
||
|
|
>>> cov.location_
|
||
|
|
array([ 0.0595... , -0.0075...])
|
||
|
|
|
||
|
|
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.
|
||
|
|
"""
|
||
|
|
@_deprecate_positional_args
|
||
|
|
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 according to the given
|
||
|
|
training data and parameters.
|
||
|
|
|
||
|
|
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
|
||
|
|
"""
|
||
|
|
# 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)
|
||
|
|
|
||
|
|
return self
|
||
|
|
|
||
|
|
|
||
|
|
# OAS estimator
|
||
|
|
@_deprecate_positional_args
|
||
|
|
def oas(X, *, assume_centered=False):
|
||
|
|
"""Estimate covariance with the Oracle Approximating Shrinkage algorithm.
|
||
|
|
|
||
|
|
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 (shrunk) covariance is:
|
||
|
|
|
||
|
|
(1 - shrinkage) * cov + shrinkage * mu * np.identity(n_features)
|
||
|
|
|
||
|
|
where mu = trace(cov) / n_features
|
||
|
|
|
||
|
|
The formula we used to implement the OAS is slightly modified compared
|
||
|
|
to the one given in the article. See :class:`OAS` for more details.
|
||
|
|
"""
|
||
|
|
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.
|
||
|
|
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.) * (alpha - (mu ** 2) / n_features)
|
||
|
|
|
||
|
|
shrinkage = 1. if den == 0 else min(num / den, 1.)
|
||
|
|
shrunk_cov = (1. - shrinkage) * emp_cov
|
||
|
|
shrunk_cov.flat[::n_features + 1] += shrinkage * mu
|
||
|
|
|
||
|
|
return shrunk_cov, shrinkage
|
||
|
|
|
||
|
|
|
||
|
|
class OAS(EmpiricalCovariance):
|
||
|
|
"""Oracle Approximating Shrinkage Estimator
|
||
|
|
|
||
|
|
Read more in the :ref:`User Guide <shrunk_covariance>`.
|
||
|
|
|
||
|
|
OAS is a particular form of shrinkage described in
|
||
|
|
"Shrinkage Algorithms for MMSE Covariance Estimation"
|
||
|
|
Chen et al., IEEE Trans. on Sign. Proc., Volume 58, Issue 10, October 2010.
|
||
|
|
|
||
|
|
The formula used here does not correspond to the one given in the
|
||
|
|
article. In the original article, formula (23) states that 2/p 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.
|
||
|
|
|
||
|
|
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].
|
||
|
|
|
||
|
|
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...
|
||
|
|
|
||
|
|
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 References)
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
"Shrinkage Algorithms for MMSE Covariance Estimation"
|
||
|
|
Chen et al., IEEE Trans. on Sign. Proc., Volume 58, Issue 10, October 2010.
|
||
|
|
"""
|
||
|
|
|
||
|
|
def fit(self, X, y=None):
|
||
|
|
"""Fit the Oracle Approximating Shrinkage covariance model
|
||
|
|
according to the given training data and parameters.
|
||
|
|
|
||
|
|
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
|
||
|
|
"""
|
||
|
|
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
|