scipy.stats.Covariance.from_diagonal#

static Covariance.from_diagonal(diagonal)[source]#

Return a representation of a covariance matrix from its diagonal.

Parameters:

diagonalarray_like: The diagonal elements of a diagonal matrix.

Notes

Let the diagonal elements of a diagonal covariance matrix \(D\) be stored in the vector \(d\).

When all elements of \(d\) are strictly positive, whitening of a data point \(x\) is performed by computing \(x \cdot d^{-1/2}\), where the inverse square root can be taken element-wise. \(\log\det{D}\) is calculated as \(-2 \sum(\log{d})\), where the \(\log\) operation is performed element-wise.

This Covariance class supports singular covariance matrices. When computing _log_pdet, non-positive elements of \(d\) 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 elements of \(d\) 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 = np.diag(rng.random(n))
>>> x = rng.random(size=n)

Extract the diagonal from A and create the Covariance object.

>>> d = np.diag(A)
>>> cov = stats.Covariance.from_diagonal(d)

Compare the functionality of the Covariance object against a reference implementations.

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