Discussion of ‘Marked Spatial Point Processes: Current State and Extensions to Point Processes on Linear Networks’ by Matthias Eckardt and Mehdi Moradi

Abstract

We congratulate Professors Eckardt and Moradi on their fine paper in which they survey the state of the art of summary statistics for marked point processes. The authors focus on non-stationary point processes and distinguish between discrete, real-valued and objectvalued marks. The case of point processes in an Euclidean space with discrete marks is by far the most well studied. Here, centre stage is taken by the cross K , H and J statistics. After a brief detour into the frequency domain, the authors consider point processes on linear networks. The concept of intensity-reweighted moment pseudo-stationarity (IRMPS) is introduced and used to define counterparts of the cross statistics just mentioned.From a measure-theoretic point of view, point processes are counting measures. It is therefore interesting to note that the cross K , H and J statistics have been generalised to multivariate random measures as well (Van Lieshout 2018). The basic ideas are to replace λ (m) in (3) by p m , the m-point coverage function, and to let the role of the generating functional be taken over by the Laplace transform. For example,under coverage-reweighted moment stationarity assumptions that are satisfied, for instance, by the lognormal random field models of Ballani et al. (2012) for which