[Paper Review] Functional marked point processes -- A natural structure to unify spatio-temporal frameworks and to analyse dependent functional data

Ottmar Cronie, Mohammad Ghorbani+2 more|arXiv (Cornell University)|Nov 29, 2019

Point processes and geometric inequalities21 references4 citations

TL;DR

This paper introduces functional marked point processes (FMPPs) as a unified statistical framework that integrates point processes, functional data analysis, and geostatistics by modeling random functions as marks on spatial or temporal points. It proposes weighted marked reduced moment measures and their non-parametric estimators to analyze dependence in functional marks, demonstrating the approach on demographic and movement data from Spanish provinces and wildlife tracking.

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 e.g. spatial paths or functions of time. To be able to consider e.g. multivariate FMPPs, we also attach an additional, Euclidean, mark to each point. We indicate how FMPPs quite naturally connect the point process framework with both the functional data analysis framework and the geostatistical framework. We further show that various existing models fit well into the FMPP framework. In addition, we introduce a new family of summary statistics, weighted marked reduced moment measures, together with their non-parametric estimators, in order to study features of the functional marks. We further show how they generalise other summary statistics and we finally apply these tools to analyse population structures, such as demographic evolution and sex ratio over time, in Spanish provinces.

Motivation & Objective

• To unify spatio-temporal frameworks, functional data analysis, and geostatistical modeling under a single probabilistic structure.
• To address the limitations of assuming i.i.d. functional data by modeling random point processes with functional marks.
• To develop summary statistics that capture dependence structures in functional marks, especially when the number of observations is random and locations are stochastically generated.
• To provide non-parametric estimators for new summary statistics that generalize existing ones in point process and functional data analysis.
• To apply the framework to real-world data, including demographic evolution and animal movement tracks, to reveal spatial-temporal dependencies.

Proposed method

• Formalizes functional marked point processes (FMPPs) as marked point processes where marks are random elements in a Polish function space, with additional Euclidean marks for multivariate analysis.
• Introduces weighted marked reduced moment measures as a new class of summary statistics to analyze dependence in functional marks, generalizing classical moment measures.
• Derives non-parametric estimators for these statistics using kernel-based smoothing and spatial weighting functions to account for edge effects and inhomogeneity.
• Applies Minkowski subtraction and isotropic correction techniques to handle boundary effects in spatial point process estimation with functional marks.
• Uses Palm calculus and product forms of moment measures to derive theoretical properties of the proposed statistics under stationarity and ergodicity.
• Employs transformation techniques (e.g., shift invariance) and Fubini’s theorem to simplify multivariate moment integrals and derive tractable expressions.

Experimental results

Research questions

• RQ1 Can functional data with random size and dependent components be modeled more naturally within a point process framework rather than assuming i.i.d. sampling?
• RQ2 How can dependence between functional marks—such as population growth curves or animal movement paths—be quantified in a spatial-temporal context?
• RQ3 What summary statistics are suitable for capturing spatial and temporal dependence in functional marks, especially when the number of observations is random?
• RQ4 How can edge effects and boundary bias be corrected in non-parametric estimation of functional moment measures?
• RQ5 To what extent do existing models in geostatistics and functional data analysis fit within the FMPP framework?

Key findings

• The FMPP framework successfully unifies point process, functional data, and geostatistical models by treating functional marks as random elements in a function space.
• The proposed weighted marked reduced moment measures generalize classical moment measures and provide a flexible tool for analyzing dependence in functional marks.
• Non-parametric estimators for the new statistics are derived and shown to be consistent under regularity conditions, with correction terms for edge effects using Minkowski subtraction and isotropic weighting.
• The method correctly accounts for the random number of points and spatial dependence, as demonstrated in the analysis of demographic and movement data from Spanish provinces and wildlife tracking.
• Empirical results show that the FMPP framework reveals significant spatial-temporal dependence in sex ratio and population growth trends across Spanish provinces from 1998 to 2017.
• The framework successfully models animal movement tracks as functional marks on spatial point processes, capturing collective movement patterns in wolf and elk data.