`scipy.stats._multivariate:matrix_t_gen`

source: /scipy/stats/_multivariate.py :1579

Signature

class matrix_t_gen ( seed = None )

Members

Summary

A matrix t-random variable.

Extended Summary

The mean keyword specifies the mean. The row_spread keyword specifies the row-wise spread matrix. The col_spread keyword specifies the column-wise spread matrix.

Parameters

%(_matt_doc_default_callparams)s
%(_doc_random_state)s

Methods

pdf(x, mean=None, row_spread=None, col_spread=None): Probability density function.
logpdf(x, mean=None, row_spread=None, col_spread=None): Log of the probability density function.
rvs(mean=None, row_spread=1, col_spread=1, df=1, size=1, random_state=None): Draw random samples.

Notes

%(_matt_doc_callparams_note)s

The spread matrices specified by row_spread and col_spread must be (symmetric) positive definite. If the samples in X have shape (m,n) then row_spread must have shape (m,m) and col_spread must have shape (n,n). Spread matrices must be full rank.

The probability density function for matrix_t is

f (X ∣ M, Σ, Ω, df) = \frac{Γ _{n} ( \frac{df + m + n - 1}{2} ) ( det ( I _{n} + ( X - M ) ^{T} Σ ^{- 1} ( X - M ) Ω ^{- 1} ) ) ^{- \frac{df + m + n - 1}{2}}}{Γ _{n} ( \frac{df + n - 1}{2} ) π ^{mn /2} ( det Σ ) ^{n /2} ( det Ω ) ^{m /2}}

or, alternatively,

f (X ∣ M, Σ, Ω, df) = \frac{Γ _{m} ( \frac{df + m + n - 1}{2} ) ( det ( I _{m} + Σ ^{- 1} ( X - M ) Ω ^{- 1} ( X - M ) ^{T} ) ) ^{- \frac{df + m + n - 1}{2}}}{Γ _{m} ( \frac{df + n - 1}{2} ) π ^{mn /2} ( det Σ ) ^{n /2} ( det Ω ) ^{m /2}}

where $M$ is the mean, $Σ$ is the row-wise spread matrix, $Ω$ is the column-wise matrix, $df$ is the degrees of freedom, and $Γ_{n}$ is the multivariate gamma function.

These equivalent formulations come from the identity

det (I_{m} + A B) = det (I_{n} + B A)

for $m \times n$ arrays $A$ and $B^{T}$ and the fact that $γ_{n} (df + m) / γ_{n} (df)$ is equal to $γ_{m} (df + n) / γ_{m} (df)$ , where

γ_{m} (df) = 2^{m (m - 1) /2} Γ_{m} ((df + m - 1) /2)

denotes a normalized multivariate gamma function.

When $df = 1$ this distribution is known as the matrix variate Cauchy.

Examples

import numpy as np
from scipy.stats import matrix_t
M = np.arange(6).reshape(3,2)
M
Sigma = np.diag([1,2,3])
Sigma
Omega = 0.3*np.identity(2)

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Omega

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X = M + 0.1

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df = 3

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matrix_t.pdf(X, mean=M, row_spread=Sigma, col_spread=Omega, df=df)

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

rv = matrix_t(mean=None, row_spread=1, col_spread=1, df=1)

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Aliases

scipy.stats._multivariate.matrix_t_gen