scipy.stats.Covariance.from_eigendecomposition#

static Covariance.from_eigendecomposition(eigendecomposition)[source]#

Representation of a covariance provided via eigendecomposition

Parameters:

eigendecompositionsequence: A sequence (nominally a tuple) containing the eigenvalue and eigenvector arrays as computed by scipy.linalg.eigh or numpy.linalg.eigh.

Notes

Let the covariance matrix be \(A\), let \(V\) be matrix of eigenvectors, and let \(W\) be the diagonal matrix of eigenvalues such that V W V^T = A.

When all of the eigenvalues are strictly positive, whitening of a data point \(x\) is performed by computing \(x^T (V W^{-1/2})\), where the inverse square root can be taken element-wise. \(\log\det{A}\) is calculated as \(tr(\log{W})\), where the \(\log\) operation is performed element-wise.

This Covariance class supports singular covariance matrices. When computing _log_pdet, non-positive eigenvalues are ignored. Whitening is not well defined when the point to be whitened does not lie in the span of the columns of the covariance matrix. The convention taken here is to treat the inverse square root of non-positive eigenvalues as zeros.

Examples

Prepare a symmetric positive definite covariance matrix A and a data point x.

>>> import numpy as np
>>> from scipy import stats
>>> rng = np.random.default_rng()
>>> n = 5
>>> A = rng.random(size=(n, n))
>>> A = A @ A.T  # make the covariance symmetric positive definite
>>> x = rng.random(size=n)

Perform the eigendecomposition of A and create the Covariance object.

>>> w, v = np.linalg.eigh(A)
>>> cov = stats.Covariance.from_eigendecomposition((w, v))

Compare the functionality of the Covariance object against reference implementations.

>>> res = cov.whiten(x)
>>> ref = x @ (v @ np.diag(w**-0.5))
>>> np.allclose(res, ref)
True
>>> res = cov.log_pdet
>>> ref = np.linalg.slogdet(A)[-1]
>>> np.allclose(res, ref)
True