The second-order analysis of marked spatio-temporal point processes, with an application to earthquake data

Abstract: To analyse interaction in marked spatio-temporal point processes (MSTPPs), we introduce marked (cross) second-order reduced moment measures and K-functions for general inhomogeneous second-order intensity reweighted stationary MSTPPs. These summary statistics, which allow us to quantify dependence between different mark categories of the points, are depending on the specific mark space and mark reference measure chosen. We also look closer at how the summary statistics reduce under assumptions such as the MSTPP being multivariate and/or stationary. A new test for independent marking is devised and unbiased minus-sampling estimators are derived for all statistics considered. In addition, we treat Voronoi intensity estimators for MSTPPs and indicate their unbiasedness. These new statistics are finally employed to analyse the well-known Andaman sea earthquake dataset. We find that clustering takes place between main and fore-/aftershocks at virtually all space and time scales. In addition, we find evidence that, conditionally on the space-time locations of the earthquakes, the magnitudes do not behave like an iid sequence.