Generalized Spatial Dirichlet Process Models

Abstract

In many applications, spatial data are assumed to be point referenced, e.g., ob-served at geo-coded locations. Modelling for this kind of data usually introduces spatial structure in the form of spatial random effects where a term capturing residual spatial association is explicitly introduced. This pure spatial effect is customarily modelled as a mean-zero stationary Gaussian process (GP). The SDP introduced by Gelfand et al. (2005) produces a random spatial process which is neither Gaussian nor station-ary. Rather, it varies about a process that is assumed to be stationary and Gaussian. The SDP arises as a probability weighted collection of random surfaces. This can be unattractive for modelling, hence inferential purposes since it insists that a process 1 realization is one of these surfaces. In this paper, we introduce a random distribution for the spatial effects that allows different surface selection at different sites. Moreover, we can specify the model to preserve the property that the marginal distribution of the effect at each site still comes from a Dirichlet process. The development is offered con-structively, providing a multivariate extension of the stick-breaking representation of the weights. We then introduce mixing using this generalized spatial Dirichlet process (GSDP). We illustrate with a simulated dataset of independent replications and then demonstrate how to embed the GSDP within a dynamic model specification to remove the restrictive independence assumption, again providing an illustrative example. Fi-nally, the GSDP is considerably more computationally demanding to work with than the SDP and so we also offer a collection of justifications for its use over the SDP. Some key words: Dirichlet process mixing; dynamic models; latent processes; non-Gaussian; nonstationary; stick-breaking. 1