Modelling spatially correlated data via mixtures: a Bayesian approach

Carmen Fernández (St Andrews) & Peter J. Green (Bristol)
Mixture models have a long pedigree, with interest going back many decades in their role both in situations where the components of the mixture represent subgroups in a heterogeneous population, and in those where the mixture formulation is simply a convenient parsimonious form for flexible density estimation. Beyond the independent random sample setting in which basic mixture models are always formulated, attention has recently been given to using mixtures as ingredients in novel statistical models for situations in which data are more structured, such as in regression, measurement error models and the analysis of factorial experiments.

One setting in which mixtures have yet to make much impact is that of data indexed spatially, whether by points, by regions, or by cells of a lattice. In this paper, we confine attention to the case of finite, typically irregular, patterns of points or regions with prescribed spatial relationships, and to problems where it is only the weights in a mixture model that vary from one location to another.

Our specific focus is on Poisson distributed data, and applications in geographical epidemiology. We work in a Bayesian framework, with the Poisson parameters drawn from gamma priors, and the number of components being unknown. We propose two alternative models for spatially-dependent weights on the components, based on transformations of autoregressive gaussian processes: in one the mixture component labels are exchangeable, in the other they are ordered. The performance of both of these formulations is examined on both synthetic data, and real data on mortality from rare disease.

Some key words: Disease mapping, Grouped continuous model, Logistic normal, Poisson mixtures, Reversible jump MCMC.
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