Testing the first-order separability hypothesis for spatio-temporal point patterns

TL;DR

This paper introduces three novel statistical tests to assess the first-order separability hypothesis in spatio-temporal point processes, aiding preliminary data analysis and model selection.

Contribution

It proposes three different tests, including permutation-based, chi-squared, and reconstruction methods, for evaluating separability in both Poisson and non-Poisson processes.

Findings

Permutation test detects deviations at specific locations or lags.

Chi-squared test is computationally efficient for Poisson processes.

Reconstruction-based test applies to non-Poisson processes and is demonstrated on real data.

Abstract

First-order separability of a spatio-temporal point process plays a fundamental role in the analysis of spatio-temporal point pattern data. While it is often a convenient assumption that simplifies the analysis greatly, existing non-separable structures should be accounted for in the model construction. We propose three different tests to investigate this hypothesis as a step of preliminary data analysis. The first two tests are exact or asymptotically exact for Poisson processes. The first test based on permutations and global envelopes allows us to detect at which spatial and temporal locations or lags the data deviate from the null hypothesis. The second test is a simple and computationally cheap $\chi^2$-test. The third test is based on statistical reconstruction method and can be generally applied for non-Poisson processes. The performance of the first two tests is studied in a simulation study for Poisson and non-Poisson models. The third test is applied to the real data of the UK 2001 epidemic foot and mouth disease.