ABSTRACT
In this chapter we present the essential elements of spatial models and classical analysis for point-referenced data. As mentioned in Chapter 1, the fundamental concept underlying the theory is a stochastic process {Y (s) : s ∈ D}, where D is a fixed subset of r-dimensional Euclidean space. Note that such stochastic processes have a rich presence in the time series literature, where r = 1. In the spatial context, usually we encounter r to be 2 (say, northings and eastings) or 3 (e.g., northings, eastings, and altitude above sea level). For situations where r > 1, the process is often referred to as a spatial process. For example, Y (s) may represent the level of a pollutant at site s. While it is conceptually sensible to assume the existence of a pollutant level at all possible sites in the domain, in practice the data will be a partial realization of that spatial process. That is, it will consist of measurements at a finite set of locations, say {s1, . . . , sn}, where there are monitoring stations. The problem facing the statistician is inference about the spatial process Y (s) and prediction at new locations, based upon this partial realization. The remarkable feature of the models we employ here is that, despite only seeing the process, equivalently, the spatial surface at a finite set of locations, we can infer about the surface at an uncountable number of locations. The reason is that we specify association through structured dependence which enables this broad interpolation.
