Non‐ and semi‐parametric estimation of interaction in inhomogeneous point patterns

Abstract

We develop methods for analysing the`interaction' or dependence between points in a spatial point pattern, when the pattern is spatially inhomogeneous. Completely non-parametric study of interactions is possible using an analogue of the K -function. Alternatively one may assume a semi-parametric model in which a (parametrically speci®ed) homogeneous Markov point process is subjected to (non-parametric) inhomogeneous independent thinning. The effectiveness of these approaches is tested on datasets representing the positions of trees in forests.

Thinning spatial point processes into Poisson processes

“…Figure 4b shows the resulting thinned process, and Figure 4c displays a centered version of the estimated inhomogeneous L-function (Baddeley et al, 2000;Veen and Schoenberg, 2005), confirming that the process in Figure 4c is an inhomogeneous Poisson process. L_inhom(d)−d * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * …”

“…Sections 2.4 and 3.3 show specific examples, using the L-function to check if the thinned process is Poisson. The L-function is a standardized version of Ripley's K-function (Ripley, 1976(Ripley, , 1977 and both can be extended to the inhomogeneous case (Baddeley, Møller and Waagepetersen, 2000), though many other techniques for checking whether a point process is Poisson have been developed; see e.g. Cressie (1993) and Stoyan, Kendall and Mecke (1995).…”

Assessing Spatial Point Process Models Using Weighted K-functions: Analysis of California Earthquakes

“…We focus here on Ripley's K-function, in particular on a modified version of the statistic which we call the weighted K-function, K W , and which was first introduced as the inhomogeneous K-function in [BMW00]. It may be used to test a quite general class of null hypothesis models for the point process under consideration and it provides a direct test for goodness-of-fit, without having to assume homogeneity or to transform the points using residual analysis, the latter of which often introduces problems of highly irregular boundaries and large sampling variability when the conditional intensity in question is highly variable (see [Sch03]).…”

Fast block variance estimation procedures for inhomogeneous spatial point processes

“…For a Borel set B ⊂ R 2 , let |B| denote the Lebesgue measure of B, and let N (B) denote the number of events of N in B. Let λ(s) and λ(s 1 , s 2 ) denote the first-and second-order intensity functions (Diggle, 2003, p. 43) E{N (ds 1 )N (ds 2 )} |ds 1 ||ds 2 | , where ds is an infinitesimal region containing s. We will focus on a flexible class of spatial point processes called second-order intensity reweighted stationary processes (Baddeley et al, 2000). We assume that λ(s 1 , s 2 ) = λ(s 1 )λ(s 2 )g(s 1 − s 2 ) for some function g (·), where g(·) is called the pair correlation function (Møller & Waagepetersen, 2004, p. 31).…”