Functional marked point processes: a natural structure to unify spatio-temporal frameworks and to analyse dependent functional data

Abstract

This paper treats functional marked point processes (FMPPs), which are defined asmarked point processes where the marks are random elements in some (Polish) func-tion space. Such marks may represent, for example, spatial paths or functions of time.To be able to consider, for example, multivariate FMPPs, we also attach an additional,Euclidean, mark to each point. We indicate how the FMPP framework quite naturallyconnects the point process framework with both the functional data analysis frame-work and the geostatistical framework. We further show that various existing stochasticmodels fit well into the FMPP framework. To be able to carry out nonparametric sta-tistical analyses for FMPPs, we study characteristics such as product densities andPalm distributions, which are the building blocks for many summary statistics. Weproceed to defining a new family of summary statistics, so-called weighted markedreduced moment measures, together with their nonparametric estimators, in order tostudy features of the functional marks. We further show how other summary statisticsmay be obtained as special cases of these summary statistics. We finally apply thesetools to analyse population structures, such as demographic evolution and sex ratioover time, in Spanish provinces.