On the local odds ratio between points and marks in marked point processes

Marked point processes are widely used stochastic models for representing a finite number of natural hazard events located in space and time and their data often associate event measurements (i.e. marks) with event locations (i.e. points). An interesting statistical problem of marked point processes is to measure and estimate the localized dependence between points and marks. To solve this problem, an approach of local odds ratio is proposed, where the local odds ratio is defined by the localized ratio of the relative risk for an event to have a small mark to the relative risk to have a large mark. To establish the approach, the article presents definition, estimation, and statistical properties. To justify the usefulness of the approach, the article presents two particular examples in natural hazards: a forest wildfire study and an earthquake study. It finds that values of local odds ratios are mostly likely low in one subarea but high in another subarea, which indicates that events with large mark values are mostly likely to appear in the former subarea but less likely to appear in the latter subarea. It is expected that the proposed approach will be widely applicable in natural hazard studies.