Matrix normal distribution
| Parameters | - mean - row covariance - column covariance.Parameters are matrices (all of them). |
|---|---|
| Support | is a matrix |
| (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 |
[edit] 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
.
[edit] 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
.
[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
- 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.
- mean
- row covariance
- column covariance.
is a matrix![\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}}](http://upload.wikimedia.org/wikipedia/en/math/c/a/4/ca4163caa07249c845de0fcd77b2a195.png)
![{\boldsymbol \Sigma} = E[ (\mathbf{X} - \mathbf{M})(\mathbf{X} - \mathbf{M})^{T}]\;,\;](http://upload.wikimedia.org/wikipedia/en/math/5/d/c/5dc27a83d8884c09e195fd8d4eb2f668.png)
![{\boldsymbol \Omega} = E[ (\mathbf{X} - \mathbf{M})^{T} (\mathbf{X} - \mathbf{M})]/c](http://upload.wikimedia.org/wikipedia/en/math/2/c/c/2cc64aff7198fb712d52ac76971a6d7c.png)

