Second-order analysis of structured inhomogeneous spatio-temporal point processes

Statistical methodology for spatio-temporal point processes is in its infancy.
We consider second-order analysis based on pair correlation functions
and K-functions for first general inhomogeneous spatio-temporal
point processes and second inhomogeneous spatio-temporal Cox processes.
Assuming spatio-temporal separability of the intensity function, we clarify
different meanings of second-order spatio-temporal separability. One is
second-order spatio-temporal independence and relates e.g. to log-Gaussian
Cox processes with an additive covariance structure of the underlying
spatio-temporal Gaussian process. Another concerns shot-noise Cox processes
with a separable spatio-temporal covariance density. We propose diagnostic
procedures for checking hypotheses of second-order spatio-temporal
separability, which we apply on simulated and real data (the UK 2001 epidemic
foot and mouth disease data).