Matrix normal distribution

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Matrix normal distribution
parameters: \mathbf{M}\, - mean
\mathbf{\Omega} \, - row covariance
\mathbf{\Sigma} \, - column covariance.
Parameters are matrices (all of them).
support: \mathbf{W}\! is a matrix
pdf: (see Notebox below)
cdf:
mean: \mathbf{M}
median:
mode:
variance:
skewness:
kurtosis:
entropy:
mgf:
cf:


Notebox Probability density function:
\frac{\exp\left( -\frac{1}{2} \mbox{tr}\left[ {\boldsymbol \Omega}^{-1} (\mathbf{X} - \mathbf{M})^{T} {\boldsymbol \Sigma}^{-1} (\mathbf{X} - \mathbf{M}) \right] \right)}{(2\pi)^{np/2} |{\boldsymbol \Omega}|^{n/2} |{\boldsymbol \Sigma}|^{p/2}}

The matrix normal distribution is a probability distribution that is a generalization of the normal distribution to matrix-valued random variables.

Contents

[edit] Definition

The probability density function for the random matrix X (n × p) that follows the matrix normal distribution has the form:

p(\mathbf{X}|\mathbf{M}, {\boldsymbol \Omega}, {\boldsymbol \Sigma})= \frac{\exp\left( -\frac{1}{2} \mbox{tr}\left[ {\boldsymbol \Omega}^{-1} (\mathbf{X} - \mathbf{M})^{T} {\boldsymbol \Sigma}^{-1} (\mathbf{X} - \mathbf{M}) \right] \right)}{(2\pi)^{np/2} |{\boldsymbol \Omega}|^{n/2} |{\boldsymbol \Sigma}|^{p/2}}

where M is n × p, Ω is p × p and Σ is n × n. There are several ways to define the two covariance matrices. One possibility is

{\boldsymbol \Sigma} = E[ (\mathbf{X} - \mathbf{M})(\mathbf{X} - \mathbf{M})^{T}]\;,\;
{\boldsymbol \Omega} = E[ (\mathbf{X} - \mathbf{M})^{T} (\mathbf{X} - \mathbf{M})]/c

where c is a constant which depends on Σ and ensures appropriate power normalization.

The matrix normal is related to the multivariate normal distribution in the following way:

\mathbf{X} \sim MN_{n\times p}(\mathbf{M}, {\boldsymbol \Omega}, {\boldsymbol \Sigma}),

if and only if

\mathrm{vec}\;\mathbf{X} \sim N_{np}(\mathrm{vec}\;\mathbf{M}, {\boldsymbol \Omega}\otimes{\boldsymbol \Sigma}),

where \otimes denotes the Kronecker product and \mathrm{vec}\;\mathbf{M} denotes the vectorization of \mathbf{M}.

[edit] Relation to other distributions

Dawid (1981) provides a discussion of the relation of the matrix-valued normal distribution to other distributions, including the Wishart distribution, Inverse Wishart distribution and a "matrix-t distribution", but uses different notation from that employed here.

[edit] See also

[edit] References

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Probability distributions
 
Multivariate (joint)
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