Section: New Results

Modelling aggregation and regularity in spatial point pattern datasets

Participant : Frédéric Lavancier.

In the spatial point process literature, analysis of spatial point pattern datasets are often classified into three main cases: i/ regularity (or inhibition or repulsiveness), modelled by Gibbs point processes, hard core processes like Matern hard core models, and determinantal point processes; ii/ complete spatial randomness, modelled by Poisson point processes; iii/ aggregation (or clustering), modelled by Poisson cluster processes and Cox processes. For applications the classification i/-iii/ can be too simplistic, and there is a lack of useful spatial point process models with, loosely speaking, aggregation on the large scale and regularity on the small scale. For instance, we may be interested in such a model for the repartition of the centres of vesicles in a cell, that exhibit some spatial clustering at large scales while having a minimal distance between them.

In [22] , we have considered a dependent thinning of a regular point process with the aim of obtaining aggregation on the large scale and regularity on the small scale in the resulting target point process of retained points. Various parametric models for the underlying processes are suggested and the properties of the target point process are studied. Simulation and inference procedures have been discussed when a realization of the target point process is observed, depending on whether the thinned points are also observed or not. Some typical simulations of the target processes are shown in Fig. 11 .

Reference:  [22]

Collaborator: Jesper Møller (Department of Mathematical Sciences, Aalborg University, Denmark).

Figure 11. Examples of simulations with aggregation on the large scale and regularity on the small scale.