Testing the weak stationarity of a spatio-temporal point process

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.…”

“…Møller and Ghorbani (2012) define the spatio-temporal inhomogeneous K-function in a more natural form, this definition involves the spatio-temporal pair correlation function g 0 which, according to Gabriel and Diggle (2009), can be informally interpreted as the standardized probability density that an event occurs in each of two small volumes. Note further that for a spatiotemporal Poisson process, g 0 = 1 and K(r, t) = 2πr 2 t. In Møller and Ghorbani (2012), Ghorbani (2013), and Gabriel (2013) alternative approximately unbiased non-parametric estimators for both the spatio-temporal inhomogeneous K-function and the intensity function are given.…”

“…The pair (r, ϕ) denotes the point with polar coordinates r and ϕ. Note that this definition slightly differs from that of K(r, t) even in the SOIRS stationary and isotropic case, see Ghorbani (2013). The reduced second moment measure K studied in Møller and Toftaker (2012), is given by K(r, t) = K(r, t, 2π) = 2K(r, t, π).…”

“…, n. For a pair of temporal observed events s i , s j ∈ T where s j > s i , w 2 (s i , s j ) is the temporal edge-correction factor which is equal to one if both ends of the interval of length 2|s i − s j | and center s i lie within T , and two otherwise (Diggle et al (1995)). An unbiased estimator for ρ 2 is n(n − 1)/(|W ||T |) 2 , see Ghorbani (2013).…”

“…One is second-order spatio-temporal independence, and another concerns a separable spatio-temporal covariance density. Ghorbani (2013) suggests a weak stationarity of a spatio-temporal point process test and Gabriel (2013) builds a rigorous simulation study to show the efficiency of the second-order estimators on differents scenarios for various spatio-temporal edge-corrections. Cronie and Lieshout (2014) extend and estimate the J-function for inhomogeneous spatio-temporal point processes.…”