Local inhomogeneous weighted summary statistics for marked point processes

TL;DR

This paper develops new local inhomogeneous weighted summary statistics for marked point processes, enabling detection of local dependence and deviations from random labelling, with applications to seismic data.

Contribution

It introduces a flexible family of local summary statistics for marked point processes and a local test for random labelling, enhancing analysis of spatial dependence.

Findings

The new statistics can capture various local dependence structures.

The local test effectively detects deviations from random labelling.

Application to earthquake data demonstrates practical utility.

Abstract

We introduce a family of local inhomogeneous mark-weighted summary statistics, of order two and higher, for general marked point processes. Depending on how the involved weight function is specified, these summary statistics capture different kinds of local dependence structures. We first derive some basic properties and show how these new statistical tools can be used to construct most existing summary statistics for (marked) point processes. We then propose a local test of random labelling. This procedure allows us to identify points, and consequently regions, where the random labelling assumption does not hold, e.g.~when the (functional) marks are spatially dependent. Through a simulation study we show that the test is able to detect local deviations from random labelling. We also provide an application to an earthquake point pattern with functional marks given by seismic waveforms.