This paper develops goodness of fit statistics that can be used to formally assess Markov random field models for spatial data, when the model distributions are discrete or continuous and potentially parametric. Test statistics are formed from generalized spatial residuals which are collected over groups of nonneighboring spatial observations, called concliques. Under a hypothesized Markov model structure, spatial residuals within each conclique are shown to be independent and identically distributed as uniform variables. The information from a series of concliques can be then pooled into goodness of fit statistics. Under some conditions, large sample distributions of these statistics are explicitly derived for testing both simple and composite hypotheses, where the latter involves additional parametric estimation steps. The distributional results are verified through simulation, and a data example illustrates the method for model assessment.
"Goodness of fit tests for a class of Markov random field models." Ann. Statist. 40 (1) 104 - 130, February 2012. https://doi.org/10.1214/11-AOS948