scipy.stats.matrix_normal — SciPy v1.15.2 Manual (original) (raw)

scipy.stats.matrix_normal = <scipy.stats._multivariate.matrix_normal_gen object>[source]#

A matrix normal random variable.

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:

meanarray_like, optional

Mean of the distribution (default: None)

rowcovarray_like, optional

Among-row covariance matrix of the distribution (default: 1)

colcovarray_like, optional

Among-column covariance matrix of the distribution (default: 1)

seed{None, int, np.random.RandomState, np.random.Generator}, optional

Used for drawing random variates. If seed is None, the RandomState singleton is used. If seed is an int, a new RandomState instance is used, seeded with seed. If seed is already a RandomState or Generator instance, then that object is used. Default is None.

Notes

If mean is set to None then a matrix of zeros is used for the mean. The dimensions of this matrix are inferred from the shape of rowcov and_colcov_, if these are provided, or set to 1 if ambiguous.

rowcov and colcov can be two-dimensional array_likes specifying the covariance matrices directly. Alternatively, a one-dimensional array will be be interpreted as the entries of a diagonal matrix, and a scalar or zero-dimensional array will be interpreted as this value times the identity matrix.

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

\[f(X) = (2 \pi)^{-\frac{mn}{2}}|U|^{-\frac{n}{2}} |V|^{-\frac{m}{2}} \exp\left( -\frac{1}{2} \mathrm{Tr}\left[ U^{-1} (X-M) V^{-1} (X-M)^T \right] \right),\]

where \(M\) is the mean, \(U\) the among-row covariance matrix,\(V\) the among-column covariance matrix.

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

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

Added in version 0.17.0.

Examples

import numpy as np from scipy.stats import matrix_normal

M = np.arange(6).reshape(3,2); M array([[0, 1], [2, 3], [4, 5]]) U = np.diag([1,2,3]); U array([[1, 0, 0], [0, 2, 0], [0, 0, 3]]) V = 0.3*np.identity(2); V array([[ 0.3, 0. ], [ 0. , 0.3]]) X = M + 0.1; X array([[ 0.1, 1.1], [ 2.1, 3.1], [ 4.1, 5.1]]) matrix_normal.pdf(X, mean=M, rowcov=U, colcov=V) 0.023410202050005054

Equivalent multivariate normal

from scipy.stats import multivariate_normal vectorised_X = X.T.flatten() equiv_mean = M.T.flatten() equiv_cov = np.kron(V,U) multivariate_normal.pdf(vectorised_X, mean=equiv_mean, cov=equiv_cov) 0.023410202050005054

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)

Frozen object with the same methods but holding the given

mean and covariance fixed.

Methods