Matrix normal distribution
|Parameters||scale (positive-definite real matrix)|
|Variance||(among-row) and (among-column)|
The probability density function for the random matrix X (n × p) that follows the matrix normal distribution has the form:
where denotes trace and M is n × p, U is n × n and V is p × p.
The matrix normal is related to the multivariate normal distribution in the following way:
if and only if
The equivalence between the above matrix normal and multivariate normal density functions can be shown using several properties of the trace and Kronecker product, as follows. We start with the argument of the exponent of the matrix normal PDF:
which is the argument of the exponent of the multivariate normal PDF. The proof is completed by using the determinant property:
The mean, or expected value is:
and we have the following second-order expectations:
where denotes trace.
More generally, for appropriately dimensioned matrices A,B,C:
Linear transform: let D (r-by-n), be of full rank r ≤ n and C (p-by-s), be of full rank s ≤ p, then:
Let's imagine a sample of n independent p-dimensional random variables identically distributed according to a multivariate normal distribution:
When defining the n × p matrix for which the ith row is , we obtain:
where each row of is equal to , that is , is the n × n identity matrix, that is the rows are independent, and .
Maximum Likelihood Parameter Estimation
Given k matrices, each of size n × p, denoted , which we assume have been sampled i.i.d. from a matrix normal distribution, the maximum likelihood estimate of the parameters can be obtained by maximizing:
The solution for the mean has a closed form, namely
but the covariance parameters do not. However, these parameters can be iteratively maximized by zero-ing their gradients at:
See for example  and references therein. It should be noted that the covariance parameters are non-identifiable in the sense that for any scale factor, s>0, we have:
Drawing values from the distribution
Sampling from the matrix normal distribution is a special case of the sampling procedure for the multivariate normal distribution. Let be an n by p matrix of np independent samples from the standard normal distribution, so that
where A and B can be chosen by Cholesky decomposition or a similar matrix square root operation.
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 matrix t-distribution, but uses different notation from that employed here.
- A K Gupta; D K Nagar (22 October 1999). "Chapter 2: MATRIX VARIATE NORMAL DISTRIBUTION". Matrix Variate Distributions. CRC Press. ISBN 978-1-58488-046-2. Retrieved 23 May 2014.
- Ding, Shanshan; R. Dennis Cook (2014). "DIMENSION FOLDING PCA AND PFC FOR MATRIX- VALUED PREDICTORS". Statistica Sinica 24 (1): 463–492.
- Glanz, Hunter; Carvalho, Luis. "An Expectation-Maximization Algorithm for the Matrix Normal Distribution". Retrieved 18 March 2015.
- 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. MR 614963.
- Dutilleul, P (1999). "The MLE algorithm for the matrix normal distribution". Journal of Statistical Computation and Simulation 64 (2): 105–123. doi:10.1080/00949659908811970.
- Arnold, S.F. (1981), The theory of linear models and multivariate analysis, New York: John Wiley & Sons, ISBN 0471050652