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

TL;DR

This paper introduces chi-squared test statistics based on local inhomogeneous mark-weighted K-functions to evaluate hypotheses about the local structure of marked point patterns, effectively detecting subtle dependencies.

Contribution

It develops a novel local inhomogeneous mark-weighted K-function framework for hypothesis testing in marked point processes, with demonstrated effectiveness in real-world environmental data.

Findings

Effective in identifying global and local departures from independence or homogeneity.

Capable of detecting subtle mark structures and small sample effects.

Proven useful in forestry and earthquake spatial data analysis.

Abstract

This work proposes $\chi^2$-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.