Spatial and Spatio-temporal Point Processes

The present paper gives the background to point processes in space and time. Some special types of spatial and spatio-temporal models of point processes are introduced with the emphasis to Cox processes. Their basic properties are described and some characteristics are calculated. The simulation of one of these models is shown. Introduction At present, the theory of point processes is well established (see [Daley and Vere-Jones, 1988], [Stoyan et al., 1995] or [Møller and Waagepetersen, 2003]), but most of the models are constructed for temporal or spatial point processes separately and there are only few models which consider time and space together. However, in applications there often occur events which need the spatio-temporal description, so it is also useful to develop the spatio-temporal models. The first part of this paper introduces spatial point processes and concerns their characteristics. Then, a few special models of spatial point processes are shown. In the second part, some spatio-temporal point processes, which have been developed recently, are stated. Both parts are aimed especially at Cox processes because these processes can describe many events (e.g. in natural sciences or engineering). Spatial models Consider the space R, d ≥ 2, its Borel σ-algebra B and B0 the class of bounded Borel sets. Introduce N the system of locally finite subsets of R with the σ-algebra N = σ({x ∈ N : #(x∩A) = m} : A ∈ B0,m ∈ N0) where # denotes the number of points. Then a point process Φ defined on R is a measurable mapping from some probability space (Ω,F , P ) to (N,N ). We assume Φ to be simple, i.e. each realization Φ = {xn} consists of pairwise different points. By Φ(·), the corresponding counting measure is denoted, i.e. Φ(A) is the number of points of Φ in A ∈ B0. The mean number of points in a set, EΦ(·) = λ(·), is called the intensity measure. Example The basic spatial point process is the Poisson process Φ , i.e. the process satisfying: (i) for any finite collection {An} of disjoint sets in R, the numbers of points in these sets, Φ (An), are independent random variables, (ii) for each A ⊂ R bounded, Φ (A) has Poisson distribution with parameter λ(A) where λ is the intensity measure. For any point process, if there exists a function ρ(x) for x ∈ R such that λ(A) = ∫ A ρ(x)dx then ρ(x) is called an intensity function. A point process Φ is called stationary if its distribution PΦ is invariant under translation, i.e. the processes Φ = {xn} and Φx = {xn + x} have the same distribution for all x ∈ R. For stationary point processes, the intensity function is constant, i.e. ρ(x) = ρ, called the intensity. Consider A1, . . . , An ⊂ R . The n-th factorial moment measure is defined as α(A1, . . . , An) = E 6= ∑ x1,...,xn∈Φ 1[x1∈A1] · . . . · 1[xn∈An]. where ∑ 6= denotes the sum over all pairwise disjoint points. If α(A1, . . . , An) can be written as α(A1, . . . , An) = ∫