Optimal classification of multivariate GRF observations

The problem of classifying a multivariate Gaussian random field (GRF) single observation into one of two populations specified by different parametric mean models and common intrinsic covariogram is considered. This paper concerns with classification procedures associated with the linear Bayes Discriminant Function (BDF) under the deterministic spatial sampling design. In the case of parametric uncertainty, the ML estimators of unknown parameters are plugged in the BDF. The actual risk and the approximation of the expected risk associated with aforementioned plug-in BDF (PBDF) are derived. This is the extension of the previous one to the case of general loss function and for complete parametric uncertainty, i.e. when mean and covariance functions parameters are unknown. The values of the derived approximation are examined for various combinations of parameters for the bivariate, stationary geometric anisotropic Gaussian random field with exponential covariance function sampled on regular 2-dimensional lattice.