Inhomogeneity in Spatial Cox Point Processes – Location Dependent Thinning Is Not the Only Option

Abstract

In the literature on point processes the by far most popular option for introducing inhomogeneity into a point process model is the location dependent thinning (resulting in a second-order intensity-reweighted stationary point process). This produces a very tractable model and there are several fast estimation procedures available. Nevertheless, this model dilutes the interaction (or the geometrical structure) of the original homogeneous model in a special way. When concerning the Markov point processes several alternative inhomogeneous models were suggested and investigated in the literature. But it is not so for the Cox point processes, the canonical models for clustered point patterns. In the contribution we discuss several other options how to define inhomogeneous Cox point process models that result in point patterns with different types of geometric structure. We further investigate the possible parameter estimation procedures for such models.

Hidden Second‐order Stationary Spatial Point Processes

“…Examples of point processes that do not satisfy the assumption of reweighted second‐order stationarity are processes with location dependent hard core distance and cluster processes with non‐stationary parent process; see also Prokešová (). Further details are given in Section S4 of the Supporting information.…”

“…Another example is cluster processes with non‐stationary parent process, cf. Hellmund et al (), Prokešová () and Mrkvička ().…”

On the local approximation of mean densities of random closed sets

“…We remind that for the binomial process we have λ(x) = m/H d (W ) and g(x, y) = m(m − 1)/(H d (W )) 2 ; whereas for a Matèrn cluster process in R 2 in which the parent process is a uniform Poisson process with intensity α, and each cluster consists of N ∼ Poisson(m) points independently and uniformly distributed in the ball B r (x), where x is the centre of the cluster, we have λ = mα, and g(x, y) = α 2 m 2 + αm 2 H 2 (B r (x) ∩ B r (y))/(π 2 r 4 ) ≤ α 2 m 2 + αm 2 /(πr 2 ). Other examples of processes with bounded intensity and second moment density are considered for instance in [23]. These, together with Example 1, which gives an insight into the validity of assumption (A1), provide simple examples where all the assumptions (A1)-(A3) hold.…”

“…. Other examples of processes with bounded intensity and second moment density are considered for instance in [23]. These, together with Example 1, which gives an insight into the validity of assumption (A1), provide simple examples where all the assumptions (A1)-(A3) hold.…”