ABSTRACT
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Statistical modeling of continuous spatial data is often based on Gaussian processes. This typically facilitates prediction, but normality is not necessarily an adequate modeling assumption for the data at hand. This has led some authors to propose data transformations before using a Gaussian model: in particular, De Oliveira, Kedem, and Short (1997) propose to use the Box-Cox family of power transformations. An approach based on generalized linear models for spatial data is presented in Diggle, Tawn, and Moyeed (1998). In this chapter, we present some flexible ways of modeling that allow the data to inform us on an appropriate distributional assumption. There are two broad classes of approaches we consider: first, we present a purely parametric modeling framework, which is wider than the Gaussian family, with the latter being a limiting case. This is achieved by scale mixing a Gaussian process with another process, and is particularly aimed at accommodating heavy tails. In fact, this approach allows us to identify spatial heteroscedasticity, and leads to relatively simple inference and prediction procedures. A second class of models is based on Bayesian nonparametric procedures. Most of the approaches discussed fall within the family of stick-breaking priors, which we will discuss briefly. These models are very flexible, in that they do not assume a single parametric family, but allow for highly non-Gaussian behavior. A perhaps even more important property of the models discussed in this chapter is that they accommodate nonstationary behavior.
