Prediction of a Multivariate Spatial Random Field with Continuous, Count and Ordinal Outcomes

As most georeferenced data sets are multivariate and concern variables of different kinds, spatial mapping methods must be able to deal with such data. The main difficulties are the prediction of non-Gaussian variables and the dependence modelling between processes. The aim of this paper is to present a new approach that permits simultaneous modelling of Gaussian, count and ordinal spatial processes. We consider a hierarchical model implemented within a Bayesian framework. The method used for Gaussian and count variables is based on the generalized linear model. Ordinal variable is taken into account through a generalization of the ordinal probit model. We use the moving average approach of Ver Hoef and Barry to model the dependencies between the processes. INTRODUCTION The prediction of multivariate spatial processes from collected data is a major issue in many research areas including biological sciences (McBratney et al., 2000), epidemiology (Golam Kibria et al., 2002) and economics (Chica-Olmo, 2007; Gelfand et al., 2007). In most cases few data are available as they are expensive to collect. Moreover, available data are often of different nature. For example, in geological studies, concentrations of elements (continuous variable), granularity (ordinal variable) and coloration (nominal variable) are classically measured for soil characterization (Epron et al., 2006). Spatial mapping methods thus have to be able to handle related data of different nature. This raises two difficulties: the prediction of multivariate discrete random fields and the modelling of the dependence between continuous and discrete spatial processes.