ABSTRACT
In this chapter, we consider probability models for a spatial variable that varies over a continuous domain of interest, D ⊆ Rd , where the spatial dimension is typically d = 2 or d = 3. Our approach relies on the notion of a spatial stochastic process {Y(s) : s ∈ D ⊆ Rd}, in the sense that
Y(s) = Y(s, ω) (2.1)
spatial location chance s ∈ D ⊆ Rd ω ∈ Ω
is a collection of random variables with a well-defined joint distribution. At any single spatial location s ∈ D, we think of Y(s) as a random variable that can more fully be written as Y(s; ω), where the elementary event ω lies in some abstract sample space, Ω . If we restrict attention to any fixed, finite set of spatial locations {s1, . . . , sn} ⊂ D, then
(Y(s1), . . . , Y(sn))T (2.2)
is a random vector, whose multivariate distribution reflects the spatial dependencies in the variable of interest. Each component corresponds to a spatial site. Conversely, if we fix any elementary event ω ∈ Ω , then
{Y(s, ω) : s ∈ D ⊆ Rd} and (y1, . . . , yn)T = (Y(s1, ω), . . . , Y(sn, ω))T
are realizations of the spatial stochastic process (2.1) and the induced random vector (2.2), respectively. The observed data are considered but one such realization. A generalization
to be discussed in Chapter 28 is that of a multivariate spatial stochastic process, for which Y(s) is a random vector rather than just a random variable.
