Bayesian Thinking in Spatial Statistics

In the sections below we review basic motivations for spatial statistical analysis, review three general categories of data structure and associated inferential questions, and describe Bayesian methods for achieving inference. Our goal is to highlight similarities across spatial analytic methods, particularly with regards to how hierarchical probability structures often link approaches developed in one area of spatial analysis to components within other areas. By choosing to highlight similarities, we focus on general concepts in spatial inference, and often defer details of several interesting and current threads of development to the relevant literature. We conclude with a listing of some of these developing areas of interest and references for further reading. 1 Why spatial statistics? Data are always collected at particular locations whether these be in a forest, at a particular street address, in a laboratory, or in a particular position on a gene expression array. In many cases, the location may provide additional insight into circumstances associated with the data item, in short “where” we collect a measurement may inform on “what” or “how much” we measure. The field of spatial statistics involves statistical methods utilizing location and distance in inference. Some methods are extensions of familiar techniques such as regression, generalized linear models, and time series, while others derive from particular models of stochastic processes in space. Historically, spatial issues lurk in the background of many classical statistical settings, including the design and analysis of agricultural field trials. The recognition of spatial gradients in soil properties impacting yield led to various randomization schemes to effectively remove the spatial effect from consideration. Explicit models of spatial pattern often drew motivation from similar methods in time series, with the recognition that, in space, one no longer has a simple ordering of observations. This loss of ordering complicates the extension of popular temporal models to the spatial domain, for example, Markov processes and autoregressive schemes, while important, take on additional complexity in higher dimensions, as we will see below. Key early work in spatial analysis appears in the work of Moran (1948, 1950), Whittle (1954), and Bartlett (1964, 1975). These are followed by foundational work in spatial prediction (Matheron 1963, Gandin 1963), spatial autoregressive models (Besag 1974), and spatial point processes (Ripley 1977, Diggle 1983). One measure of the incresing interest in and development of the field is a comparison of Ripley’s (1981) 252-page and Cressie’s (1993) 900-page texts addressing spatial statistics as a whole, in addition to the recent and rapid increase in texts addressing particular areas of application and/or theory (Stein 1999, Chilès and Delfiner 1999, Lawson 2001, Lawson and Denison 2002, Webster and Oliver 2001, Waller and Gotway 2004). Much of the recent explosion in the use of spatial statistics can be tied to the relatively simultaneous increases in both spatial data availability, including accurate location