Matrix normal distribution

Matrix normal distribution
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)
mean: \mathbf{M}
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.



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}.


Matrix Normal random variables arise from a sample identically distributed multivariate Normal random variables with possible dependence between the vectors. For example, if \mathbf{Y} is an (n × p) matrix whose rows are independent with distribution N_{p}(\mathbf{0}, {\boldsymbol \Sigma}) then \mathbf{Y} \sim MN_{n\times p}(\mathbf{0}, \mathbf{I}_n, {\boldsymbol \Sigma}), where \mathbf{I}_n is the (n × n) identity matrix. On the other hand, the columns of \mathbf{X=}{\boldsymbol \Gamma}\mathbf{Y} are dependent but identically distributed multivariate Normal random variables. Furthermore, \mathbf{X+M}\sim MN_{n\times p}(\mathbf{M}, {\boldsymbol \Omega, \Sigma}), where {\boldsymbol \Omega = \Gamma\Gamma}^T.

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.

See also


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать реферат

Look at other dictionaries:

  • Normal distribution — This article is about the univariate normal distribution. For normally distributed vectors, see Multivariate normal distribution. Probability density function The red line is the standard normal distribution Cumulative distribution function …   Wikipedia

  • Multivariate normal distribution — MVN redirects here. For the airport with that IATA code, see Mount Vernon Airport. Probability density function Many samples from a multivariate (bivariate) Gaussian distribution centered at (1,3) with a standard deviation of 3 in roughly the… …   Wikipedia

  • Complex normal distribution — In probability theory, the family of complex normal distributions consists of complex random variables whose real and imaginary parts are jointly normal.[1] The complex normal family has three parameters: location parameter μ, covariance matrix Γ …   Wikipedia

  • Matrix (mathematics) — Specific elements of a matrix are often denoted by a variable with two subscripts. For instance, a2,1 represents the element at the second row and first column of a matrix A. In mathematics, a matrix (plural matrices, or less commonly matrixes)… …   Wikipedia

  • Normal-scaled inverse gamma distribution — Normal scaled inverse gamma parameters: location (real) (real) (real) (real) support …   Wikipedia

  • Normal-gamma distribution — Normal gamma parameters: location (real) (real) (real) (real) support …   Wikipedia

  • Normal-inverse Gaussian distribution — Normal inverse Gaussian (NIG) parameters: μ location (real) α tail heavyness (real) β asymmetry parameter (real) δ scale parameter (real) support …   Wikipedia

  • normal — I. adjective Etymology: Latin normalis, from norma Date: circa 1696 1. perpendicular; especially perpendicular to a tangent at a point of tangency 2. a. according with, constituting, or not deviating from a norm, rule, or principle b. conforming… …   New Collegiate Dictionary

  • Normal (mathematics) — In mathematics, normal can have several meanings:* Surface normal, a vector (or line) that is perpendicular to a surface. * Normal component, the component of a vector that is perpendicular to a surface. ** Normal curvature, of a curve on a… …   Wikipedia

  • Inverse-Wishart distribution — In statistics, the inverse Wishart distribution, also called the inverted Wishart distribution, is a probability density function defined on matrices. In Bayesian statistics it is used as the conjugate prior for the covariance matrix of a… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”