Risks of Classification of the Gaussian Markov Random Field Observations

Abstract

Given the spatial lattice endowed with particular neighborhood structure, the problem of classifying a scalar Gaussian Markov random field (GMRF) observation into one of two populations specified by different regression coefficients and special parametric covariance (precision) matrix is considered. Classification rule based on the plug-in Bayes discriminant function with inserted ML estimators of regression coefficients, spatial dependence and scale parameters is studied. The novel closed-form expression for the actual risk and the approximation of the expected risk (AER) associated with the aforementioned classifier are derived. This is the extension of the previous study of GMRF classification to the case of complete parametric uncertainty. Derived AER is used as the main performance measure for the considered classifier. GMRF sampled on a regular 2-dimensional unit spacing lattice endowed with neighborhood structure based on the Euclidean distance between sites is used for a simulation experiment. The sampling properties of ML estimators and the accuracy of the derived AER for various values of spatial dependence parameters and Mahalanobis distance are studied. The influence of the neighborhood size on the accuracy of the proposed AER is examined as well.