`scipy.stats._multivariate:matrix_normal_gen`

source: /scipy/stats/_multivariate.py :1113

Signature

class matrix_normal_gen ( seed = None )

Members

Summary

A matrix normal random variable.

Extended Summary

The mean keyword specifies the mean. The rowcov keyword specifies the among-row covariance matrix. The 'colcov' keyword specifies the among-column covariance matrix.

Parameters

%(_matnorm_doc_default_callparams)s
%(_doc_random_state)s

Methods

pdf(X, mean=None, rowcov=1, colcov=1): Probability density function.
logpdf(X, mean=None, rowcov=1, colcov=1): Log of the probability density function.
rvs(mean=None, rowcov=1, colcov=1, size=1, random_state=None): Draw random samples.
entropy(rowcol=1, colcov=1): Differential entropy.

Notes

%(_matnorm_doc_callparams_note)s

The covariance matrices specified by rowcov and colcov must be (symmetric) positive definite. If the samples in X are $m \times n$ , then rowcov must be $m \times m$ and colcov must be $n \times n$ . mean must be the same shape as X.

The probability density function for matrix_normal is

f (X) = (2 π)^{- \frac{mn}{2}} ∣ U ∣^{- \frac{n}{2}} ∣ V ∣^{- \frac{m}{2}} exp (- \frac{1}{2} Tr [U^{- 1} (X - M) V^{- 1} (X - M)^{T}]),

where $M$ is the mean, $U$ the among-row covariance matrix, $V$ the among-column covariance matrix.

The allow_singular behaviour of the multivariate_normal distribution is not currently supported. Covariance matrices must be full rank.

The matrix_normal distribution is closely related to the multivariate_normal distribution. Specifically, $Vec (X)$ (the vector formed by concatenating the columns of $X$ ) has a multivariate normal distribution with mean $Vec (M)$ and covariance $V \otimes U$ (where $\otimes$ is the Kronecker product). Sampling and pdf evaluation are $O (m^{3} + n^{3} + m^{2} n + m n^{2})$ for the matrix normal, but $O (m^{3} n^{3})$ for the equivalent multivariate normal, making this equivalent form algorithmically inefficient.

Examples

import numpy as np
from scipy.stats import matrix_normal

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M = np.arange(6).reshape(3,2); M
U = np.diag([1,2,3]); U

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V = 0.3*np.identity(2); V
X = M + 0.1; X
matrix_normal.pdf(X, mean=M, rowcov=U, colcov=V)

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from scipy.stats import multivariate_normal
vectorised_X = X.T.flatten()
equiv_mean = M.T.flatten()
equiv_cov = np.kron(V,U)

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multivariate_normal.pdf(vectorised_X, mean=equiv_mean, cov=equiv_cov)

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Alternatively, the object may be called (as a function) to fix the mean and covariance parameters, returning a "frozen" matrix normal random variable:

rv = matrix_normal(mean=None, rowcov=1, colcov=1)

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Aliases

scipy.stats._multivariate.matrix_normal_gen