Local characteristics of functional marked point processes with applications to seismic data

We present a family of local inhomogeneous mark-weighted summary statistics for general marked point processes. These capture various types of local dependence structures depending on the specified involved weight function. We use them to propose a local random labeling test. This procedure enables us to identify points and thus regions where the random labeling assumption does not hold, for example, when the (functional) marks are spatially dependent. We further present an application to a seismic point pattern with functional marks provided by seismic waveforms. Indeed, despite the relatively long history of point process theory, few approaches to analyzing spatial point patterns where the features of interest are functions (i.e. curves) rather than qualitative or quantitative variables have been developed. Forest patterns with associated functional data, curves representing the incidence of an epidemic over time, and the evolution of distinct economic parameters such as unemployment and price rates, all for distinct spatial locations, are examples of point patterns with associated functional data.