- Matrix normal distribution
-
Matrix normal distribution parameters: - mean
- row covariance
- column covariance.
Parameters are matrices (all of them).support: is a matrix
pdf: (see Notebox below) mean: Notebox Probability density function:
The matrix normal distribution is a probability distribution that is a generalization of the normal distribution to matrix-valued random variables.
Contents
Definition
The probability density function for the random matrix X (n × p) that follows the matrix normal distribution has the form:
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
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:
if and only if
where
denotes the Kronecker product and
denotes the vectorization of
.
Example
Matrix Normal random variables arise from a sample identically distributed multivariate Normal random variables with possible dependence between the vectors. For example, if
is an (n × p) matrix whose rows are independent with distribution
then
, where
is the (n × n) identity matrix. On the other hand, the columns of
are dependent but identically distributed multivariate Normal random variables. Furthermore,
, where
.
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
References
- Dawid, A.P. (1981). "Some matrix-variate distribution theory: Notational considerations and a Bayesian application". Biometrika 68 (1): 265–274. doi:10.1093/biomet/68.1.265. JSTOR 2335827. MR614963.
Categories:- Random matrices
- Continuous distributions
- Multivariate continuous distributions
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