Some interaction models for clustered point patterns

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

A JJ-function for Inhomogeneous Spatio-temporal Point Processes

We propose a new summary statistic for inhomogeneous intensity-reweighted moment stationarity spatio-temporal point processes. The statistic is defined in terms of the $n$-point correlation functions of the point process, and it generalizes the $J$-function when stationarity is assumed. We show that our statistic can be represented in terms of the generating functional and that it is related to the spatio-temporal $K$-function. We further discuss its explicit form under some specific model assumptions and derive ratio-unbiased estimators. We finally illustrate the use of our statistic in practice