Mark variograms for spatio-temporal point processes

Abstract

A spatio-temporal point process can be transformed into two marked point processes by taking the times or locations as marks. For these marked point processes mark variograms can be defined, opening a new way for the statistical analysis of spatio-temporal point processes. We present first the theoretical forms of the variograms for some models of spatio-temporal point processes, in order to help understanding empirical variograms. Then we study statistical problems. The used ratio-estimator for the variograms has a form that makes it possible to partially ignore inhomogeneities and edge-effects. Two data examples which were already studied with traditional methods in the point-process literature are reanalysed with the two mark variograms, showing their explorative power.