scipy.stats.Covariance.from_cholesky#

static Covariance.from_cholesky(cholesky)[source]#

Representation of a covariance provided via the (lower) Cholesky factor

Parameters:

choleskyarray_like: The lower triangular Cholesky factor of the covariance matrix.

Notes

Let the covariance matrix be \(A\) and \(L\) be the lower Cholesky factor such that \(L L^T = A\). Whitening of a data point \(x\) is performed by computing \(L^{-1} x\). \(\log\det{A}\) is calculated as \(2tr(\log{L})\), where the \(\log\) operation is performed element-wise.

This Covariance class does not support singular covariance matrices because the Cholesky decomposition does not exist for a singular covariance matrix.

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 Cholesky decomposition of A and create the Covariance object.

>>> L = np.linalg.cholesky(A)
>>> cov = stats.Covariance.from_cholesky(L)

Compare the functionality of the Covariance object against reference implementation.

>>> from scipy.linalg import solve_triangular
>>> res = cov.whiten(x)
>>> ref = solve_triangular(L, x, lower=True)
>>> np.allclose(res, ref)
True
>>> res = cov.log_pdet
>>> ref = np.linalg.slogdet(A)[-1]
>>> np.allclose(res, ref)
True