ABSTRACT

In Chapter 2, we discussed spatial stochastic processes {Y(s) : s ∈ D ⊆ Rd} where the domain of interest was Euclidean. Turning now to a spatio-temporal domain, we consider processes

{Y(s, t) : (s, t) ∈ D ⊆ Rd × R}

that vary both as a function of the spatial location, s ∈ Rd , and time, t ∈ R. It is tempting to assume that the theory of such processes is not much different from that of spatial processes, and from a purely mathematical perspective this is indeed the case. Time can be considered an additional coordinate and, thus, from a probabilistic point of view, any spatio-temporal process can be considered a process on Rd+1 = Rd × R. In particular, all technical results on spatial covariance functions (Chapter 2) and least-squares prediction or kriging (Chapters 2 and 3) in Euclidean spaces apply to space-time problems, by applying the respective result on the domain Rd+1.