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

This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. 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. 1. Introduction Categorical spatial data, such as land use classes, vegetation species types, or categories of socioeconomic status, are all important information sources in spatial analysis, resource management, decision support systems, and planning in general. Such data types, which can be nominal, ordinal, or interval, exhibit spatial or spatiotemporal patterns with sharp boundaries and complex geometrical characteristics. This discrete (non-Gaussian) nature limits the applications of successful statistical methods that have been widely used for continuous variables including the celebrated kriging family of methods (Chilès and Delfiner 1999). A key task in statistical modeling of categorical spatial data is to estimate the joint probability mass function of a set of geo-referenced (spatially …