Testing spatial independence using a separable covariance matrix

Abstract: Spatio-temporal processes like multivariate time series and stochastic processes occur in many applications, for example the observations from functional magnetic resonance imaging (fMRl) or positron emission tomography (PET). It is interesting to test independence between k sets of the variables, that is testing spatial independence.This thesis deals with the problem of testing spatial independence for dependent observations. The sample observation matrix is assumed to follow a matrix normal distribution with a separable covariance matrix, in other words it can be written as a Kronecker product of two positive definite matrices. Instead of having a sample observation matrix with independent columns, a covariance between the columns is considered and this covariance matrix is interpreted as a temporal covariance. The main results in this thesis are the computations of the maximum likelihood estimates and the null distribution of the likelihood ratio statistic. Two cases are considered, when the temporal covariance is known and when it is unknown. When the temporal covariance is known, the maximum likelihood estimates are computed and the asymptotic null distribution is shown to be similar to the independent observation case. In the case when the temporal covariance is unknown the maximum likelihood estimates of the parameters are found by an iterative alternating algorithm.A well known fact is that when testing hypotheses for covariance matrices, distributions of quadratic forms arise. A generalization of the distribution of the multivariate quadratic form X AX', where X is a (p x n) normally distributed matrix and A is a (n x n) symmetric real matrix, is presented. It is shown that the distribution of the quadratic form is the same as the distribution of a weighted sum of noncentral Wishart distributed matrices.