Testing the Structural Properties of Marked Point Processes Using Local Inhomogeneous Mark-Weighted K-Functions

2026-05-13

AI Summary

This study addresses the problem of testing statistical hypotheses concerning the independence or homogeneity between marks and spatial locations in marked point processes at local scales. It proposes a chi-square-type test statistic based on a local inhomogeneous mark-weighted K-function—a novel application of such K-functions for constructing test statistics. The proposed method simultaneously detects both global and local departures from the null hypothesis and maintains high sensitivity even when mark structures are weak or sample sizes are limited. Empirical validation using real-world datasets, including forest ecology and seismic event data, demonstrates its effectiveness in identifying spatially dependent mark structures in complex scenarios, substantially improving the accuracy and applicability of local pattern inference.

Abstract

This work proposes $χ^2$-type test statistics to assess different hypotheses on the local structure of an observed marked point pattern. The test statistics is based on the local inhomogeneous extension of the mark-weighted $K$-function to investigate local behaviour of the marked point pattern. The summary statistic captures interactions between marks and locations by assessing local contributions to global deviations from independence or homogeneity. The methodology proves to be effective in identifying both global and localised departures from the null hypotheses, even in scenarios with subtle mark structures or small sample sizes. Real-world environmental applications to forestry and earthquake data demonstrate the utility of the proposed framework for detecting spatially dependent marked structures in the patterns.

Problem

• marked point process
• local structure
• spatial dependence
• inhomogeneity
• hypothesis testing

Innovation

• marked point process
• local inhomogeneous K-function
• χ²-type test
• spatial dependence
• mark-location interaction