Aspects of second‐order analysis of structured inhomogeneous spatio‐temporal point processes

Abstract

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 general inhomogeneous spatio‐temporal point processes and for 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 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.

Modelling, Estimation and Applications of Second-Order Spatio-Temporal Characteristics of Point Processes

“…Finally we show a sophisticated approach to the definition of spatiotemporal first-and second-order characteristics taken from Diggle et al (1995), Møller and Diaz-Avalos (2010), Møller and Ghorbani (2010), Diggle and Gabriel (2010), Ghorbani (2013), Gabriel (2013), Diggle (2013) and Cronie and Lieshout (2014) and well as the corresponding properties and non-parametric estimators. This last part of the chapter is built as an attempt to gather all the recent literature on this methodology to build the path towards the context spatio-temporal point processes.…”

“…It follows from Equation (1.41) that with probability one, for any pair of distinct points (u, s) and (v, t) from X, we have that u = v and s = l (see Møller and Ghorbani (2012)). Then we can ignore the case where the spatial and temporal component processes (say X space and X time ) have multiple points, and define them by…”

“…Gabriel and Diggle (2009) extend to the spatio-temporal setting a method proposed by Baddeley et al (2000) for inhomogeneous spatial point process data. Møller and Ghorbani (2012) consider second-order analysis based on pair correlation functions and K-functions for general inhomogeneous spatio-temporal point processes assuming spatio-temporal separability of the intensity function, but clarify different meanings of second-order spatio-temporal separability. One is second-order spatio-temporal independence, and another concerns a separable spatio-temporal covariance density.…”

“…We consider X space and X time rather than the marginal processes given by all events respective all times in X, since the later processes could not have well-defined first-and second-order properties (see Møller and Ghorbani (2012)). …”

“…We make the pragmatic working assumption that first-order effects are separable (see Møller and Ghorbani (2012)), meaning that if (u, s) ∈ R 2 × R, ρ (u, s) can be factorised as…”