A kernel-based test for the first-order separability of spatio-temporal point processes

We present an innovative statistical test designed to assess the first-order separability of a spatio-temporal point process. Our proposed test employs block permutations and a novel test statistic that incorporates a machine learning technique known as the Hilbert–Schmidt independence criterion. To enhance the practicality of the criterion, we apply the kernel trick. The block permutations are designed to maintain the second-order structure of the point pattern, disrupting it only at the block borders. This design enables the application of our test to a general spatio-temporal point process, which may exhibit small-scale clustering or regularity. We investigated the empirical level of the block permutation-based tests with the new and two previously proposed test statistics for clustered and regular point processes, represented in our study by log Gaussian Cox processes and determinantal point processes. By comparing our results with those obtained from a previously proposed permutation-based test, we confirmed the effectiveness of our method in terms of significance level, power, and notably computational cost. We applied the test to real-world datasets, namely the UK’s 2001 foot-and-mouth disease epidemic and varicella data from Valencia.