Some interaction models for clustered point patterns

Abstract

We introduce a class of spatial point processes, interacting neighbour processes, where the density of the process can be written by means of local interactions between a point and subsets of its neighbourhood but where the processes are not Markov processes with respect to this neighbourhood in the Ripley-Kelly sense. However, we show that the processes are nearest neighbour Markov processes as introduced by Baddeley and Moller (1989). Furthermore, we introduce a subclass of interacting neighbour processes, full neighbourhood interaction processes, where instead of subsets of the neighbourhood all neighbours of a point affect it simultaneously. A simulation study is presented to show that some simple full neighbourhood interaction models can produce clustered patterns of great variety. Finally, an empirical example is given.