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:
If , then we have the following properties:
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 .
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.
- 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