Statistical classification of multivariate conditionally autoregressive Gaussian random field observations

The problem of supervised classifying of a multivariate Gaussian Markov random field (GMRF) observation into one of two populations specified by different regression mean models is considered. We focus on a multivariate conditionally autoregressive model, a subclass of GMRF with parametrical structure proposed by Pettitt et al. (2002) and generalized by Jin et al. (2005). For complete parametric certainty and with fixed training sample locations, the formula of the Bayes error rate is derived. Plug-in Bayes discriminant function obtained by replacing the regression and scale parameters with their ML estimators in the Bayes discriminant function is investigated. The novel formulas for the actual error rate and the approximation of the expected error rate (AER) associated with plug-in Bayes discriminant function are derived. These results are multivariate generalizations of the univariate ones. A numerical analysis of the derived formulas is implemented for the bivariate conditionally autoregressive model sampling on a regular 2-dimensional unit spacing lattice with the neighborhood structure based on the Euclidean distance. A comparison of the proposed classifier performance for different parametric structures of populations is done by studying the accuracy of the derived AER. Finally, the influence of the neighborhood order on the accuracy of the derived AER is examined.