Nearest-neighbour Markov point processes on graphs with Euclidean edges

We define nearest-neighbour point processes on graphs with Euclidean edges and linear networks. They can be seen as analogues of renewal processes on the real line. We show that the Delaunay neighbourhood relation on a tree satisfies the Baddeley‒Møller consistency conditions and provide a characterisation of Markov functions with respect to this relation. We show that a modified relation defined in terms of the local geometry of the graph satisfies the consistency conditions for all graphs with Euclidean edges that do not contain triangles.

MODELLING THE POSITION OF CELL PROFILES ALLOWING FOR BOTH INHOMOGENEITY AND INTERACTION

It is of interest to consider models for point processes that allow for interaction between the points as well as for inhomogeneity in the intensity of the points. Markov point process models are very useful to describe point interaction and can also be used to describe inhomogeneity. A particular type of inhomogeneous Markov point processes obtained by transforming a homogeneous Markov point process will be considered. The position of cell proles in a 2D section of the mucous membrane in the stomach of a rat will be examined using this model.

INHOMOGENEITY IN SPATIAL COX POINT PROCESSES – LOCATION DEPENDENT THINNING IS NOT THE ONLY OPTION

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.