Statistics for Non-sparse Spatially Homogeneous Gibbs Point Processes

Abstract

A successful approach in statistical analysis of point processes or systems of hard disks is to consider them as samples of Gibbs point processes with or without marks, where the disk radii are the marks. A practically important particular case are Gibbs processes with pair-wise interaction potentials. Such a pair potential may be a valuable statistical characteristic additionally to pair correlation functions or Mecke’s morphological functions.

The main topic of the present text is statistical estimation of such pair potentials as well as chemical activities by the Takacs-Fiksel method. The particular problem faced here is that the patterns considered are by no means sparse, which was until now the case in most statistical analyses of point patterns analysed in spatial statistics by the Gibb process approach. The data come from a series of microscopic images showing water droplets on a naphtalin-brom surface at progressing stages of condensation.