AMS2004-5: Bayesian Nonparametric Spatial Modeling With Dirichlet Processes Mixing

Alan E. Gelfand, Athanasios Kottas and Steven N. MacEachern
12/31/2004 09:00 AM
Applied Mathematics & Statistics
Customary modeling for continuous point-referenced data assumes a Gaussian process which is often taken to be stationary. When such models are fitted within a Bayesian framework, the unknown parameters of the process are assumed to be random so a random Gaussian process results. Here, we propose a novel spatial Dirichlet process mixture model to produce a random spatial process which is neither Gaussian nor stationary. We first develop a spatial Dirichlet process model for spatial data and discuss its properties. Due to familiar limitations associated with direct use of Dirichlet process models, we introduce mixing through this process against a pure error process. We then examine properties of models created through such Dirichlet process mixing. In the Bayesian framework, posterior inference is implemented using Gibbs sampling as we detail. Spatial prediction raises interesting questions but can be handled. Finally, we illustrate the approach using simulated data as well as a dataset involving precipitation measurements over the Languedoc-Roussillon region in southern France.