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 as marked point processes where the marks are random elements in some (Polish) function 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 naturally connects the point process framework with both the functional data analysis framework and the geostatistical framework. We further show that various existing stochastic models fit well into the FMPP framework. To be able to carry out nonparametric statistical analyses for FMPPs, we study characteristics such as product densities and Palm distributions, which are the building blocks for many summary statistics. We proceed to defining a new family of summary statistics, so-called weighted marked reduced moment measures, together with their nonparametric estimators, in order to study features of the functional marks. We further show how other summary statistics may be obtained as special cases of these summary statistics. We finally apply these tools to analyse population structures, such as demographic evolution and sex ratio over time, in Spanish provinces.

Statistical learning and cross-validation for point processes

“…. , u n ) only depends on the separation vectors u i − u j , i = j, and the intensity function is positive/bounded away from 0, the point process X is called nth-order intensity reweighted stationary (van Lieshout, 2011, Cronie and van Lieshout, 2016b, Ghorbani et al, 2020. When n = 2 this is referred to as secondorder intensity reweighted stationarity (SOIRS) (Baddeley et al, 2000), and we write…”

“…These are usually of interest when each event carries some additional piece of information, which is not directly connected to S, e.g. a label, some quantitative measurement or more abstract objects such as functions and sets (Chiu et al, 2013, Ghorbani et al, 2020. Our main interest in using marking here is related to the fact that so-called thinnings of point processes may be obtained through a particular kind of marking; our cross-validation approaches presented in Section 4.1 are based on thinning.…”

Partial characteristics for marked spatial point processes

“…We now discuss mark summary characteristics for an observed point pattern x " tx i , opx i qu n i"1 in which each point x i is augmented by a non-scalar quantity o i living on some suitable mark space M, which depends on the specificity of marks in question. Apart from the functionvalued mark setting, where M is the Hilbert/L 2 space (Comas et al, 2008(Comas et al, , 2011(Comas et al, , 2013Ghorbani et al, 2021;, this newly introduced class of marked spatial point processes also includes the cases where marks are constrained arrays or inherently structured quantities .…”

“…In an effort to study non-integer/real-valued marks, the focus is directed toward proposing novel methodologies capable of handling diverse forms of marks. More specifically, by borrowing ideas from functional data analysis (Ramsay and Silverman, 1997), extensions of Stoyan's mark correlation function (Stoyan and Stoyan, 1994) to function-valued marks are proposed by Comas et al (2011Comas et al ( , 2013 for stationary point processes, and Ghorbani et al (2021) proposed a framework for functional marked point processes together with some weighted marked reduced moment measures. Moreover, extended some summary characteristics for spatial point processes with integervalued marks to the case of multivariate point processes with multivariate function-valued marks and constrained vector-valued quantities.…”