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 pointx
.>>> 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 theCovariance
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