Deviation test construction and power comparison for marked spatial point patterns

TL;DR

This paper reviews the construction of deviation tests for marked spatial point patterns, emphasizing the influence of summary function choice and scaling on test power through examples and simulations.

Contribution

It provides a detailed analysis of deviation test construction for spatial point patterns and compares their power under different configurations.

Findings

Choice of summary function significantly affects test power.

Scaling of differences impacts the sensitivity of deviation tests.

Simulation results highlight the importance of test construction choices.

Abstract

The deviation test belong to core tools in point process statistics, where hypotheses are typically tested considering differences between an empirical summary function and its expectation under the null hypothesis, which depend on a distance variable r. This test is a classical device to overcome the multiple comparison problem which appears since the functional differences have to be considered for a range of distances r simultaneously. The test has three basic ingredients: (i) choice of a suitable summary function, (ii) transformation of the summary function or scaling of the differences, and (iii) calculation of a global deviation measure. We consider in detail the construction of such tests both for stationary and finite point processes and show by two toy examples and a simulation study for the case of the random labelling hypothesis that the points (i) and (ii) have great influence on the power of the tests.