Modeling Marked Poisson Point Processes with Real-Valued Marks

Abstract

In the analysis of spatial point patterns with associated real-valued marks, standard models either rely on strong distributional assumptions about the marks to incorporate their effect on the estimated intensity, or exclude them from the fitting procedure, studying the marks through marked summary statistics. In this article, we address this issue by proposing two approaches, one parametric and one semi-parametric, to model the intensity of marked point patterns with real-valued marks in two spatial dimensions, without making any assumption on the marks distribution. Both methods allow us to estimate the effect of the mark on the intensity of the process and the density of the mark across the observed window. We show that by including mark information in the model, when the real-valued mark has an impact on the intensity, we can obtain better intensity estimates with respect to unmarked models, giving an additional layer of information about the process.