Selection of the neighborhood structure for space-time Markov random field models

A space-time, univariate dataset is assumed to have been sampled from a 3-dimensional Markov Random Field where the data dependence structure is modeled through pairwise interaction parameters. The likelihood function depends upon (1) an undirected, 3-dimensional graph, where edges connect observation points, and (2) the parameter dimension that captures possible space-time anisotropy of data interaction. Automatic model selection to discriminate both the graph and the model dimension is suggested on the basis of a penalized Pseudo-likelihood function. In most cases, the procedure can be implemented using standard statistical packages capable of GLM estimation. Weak consistency of the criterion is shown to hold under mild and easily verifiable sufficient conditions. Its performance in small samples is studied providing simulation results.