Spatio-temporal c\`adl\`ag functional marked point processes: Unifying spatio-temporal frameworks

TL;DR

This paper introduces c extbackslash adl extbackslash ag functional marked point processes (CFMPPs), unifying spatio-temporal point process and random field frameworks, with applications to Boolean models and statistical inference tools.

Contribution

It generalizes marked spatio-temporal point processes to include c extbackslash adl extbackslash ag stochastic process marks, connecting point processes with random fields and developing statistical tools.

Findings

Defined CFMPPs and their properties

Constructed spatio-temporal Boolean models using CFMPPs

Developed statistical inference tools like likelihood estimation

Abstract

This paper defines the class of c\`adl\`ag functional marked point processes (CFMPPs). These are (spatio-temporal) point processes marked by random elements which take values in a c\`adl\`ag function space, i.e. the marks are given by c\`adl\`ag stochastic processes. We generalise notions of marked (spatio-temporal) point processes and indicate how this class, in a sensible way, connects the point process framework with the random fields framework. We also show how they can be used to construct a class of spatio-temporal Boolean models, how to construct different classes of these models by choosing specific mark functions, and how c\`adl\`ag functional marked Cox processes have a double connection to random fields. We also discuss finite CFMPPs, purely temporally well-defined CFMPPs and Markov CFMPPs. Furthermore, we define characteristics such as product densities, Palm distributions and conditional intensities, in order to develop statistical inference tools such as likelihood estimation schemes.