A multinomial logistic mixed model for the prediction of categorical spatial data

In this article, the prediction problem of categorical spatial data, that is, the estimation of class occurrence probability for (target) locations with unknown class labels given observed class labels at sample (source) locations, is analyzed in the framework of generalized linear mixed models, where intermediate, latent (unobservable) spatially correlated Gaussian variables (random effects) are assumed for the observable non-Gaussian responses to account for spatial dependence information. Within such a framework, a spatial multinomial logistic mixed model is proposed specifically to model categorical spatial data. Analogous to the dual form of kriging family, the proposed model is represented as a multinomial logistic function of spatial covariances between target and source locations. The associated inference problems, such as estimation of parameters and choice of the spatial covariance function for latent variables, and the connection of the proposed model with other methods, such as the indicator variants of the kriging family (indicator kriging and indicator cokriging) and Bayesian maximum entropy, are discussed in detail. The advantages and properties of the proposed method are illustrated via synthetic and real case studies.