Quasi-Likelihood for Spatial Point Processes

Abstract

Summary Fitting regression models for intensity functions of spatial point processes is of great interest in ecological and epidemiological studies of association between spatially referenced events and geographical or environmental covariates. When Cox or cluster process models are used to accommodate clustering not accounted for by the available covariates, likelihood based inference becomes computationally cumbersome due to the complicated nature of the likelihood function and the associated score function. It is therefore of interest to consider alternative more easily computable estimating functions. We derive the optimal estimating function in a class of first-order estimating functions. The optimal estimating function depends on the solution of a certain Fredholm integral equation which in practise is solved numerically. The derivation of the optimal estimating function has close similarities to the derivation of quasi-likelihood for standard data sets. The approximate solution is further equivalent to a quasi-likelihood score for binary spatial data. We therefore use the term quasi-likelihood for our optimal estimating function approach. We demonstrate in a simulation study and a data example that our quasi-likelihood method for spatial point processes is both statistically and computationally efficient.

Testing First-Order Spherical Symmetry of Spatial Point Processes

“…This approach has been widely adopted in many previous articles. Examples include Guan and Loh (2007), Guan and Shen (2010), Waagepetersen andGuan (2009), Guan, Jalilian, andWaagepetersen (2015), Prekešová and Jensen (2013), and Schoenberg (2005).…”

A general central limit theorem and a subsampling variance estimator for α‐mixing point processes

“…The case p = 1 is relevant if interest is focused on estimation of the intensity function . Several papers have discussed choices of h and studied asymptotic properties of for the case p = 1 (see, for instance, Guan et al, ; Guan & Shen, ; Waagepetersen, ). A popular and simple choice is h θ ( u ) = ∇ θ λ θ ( u )/ λ θ ( u ), where ∇ θ denotes the gradient with respect to θ .…”

“…One approach uses Bernstein's blocking technique and a telescoping argument that goes back to Ibragimov and Linnik (, Chapter 18, Section 4). This approach has been used in a number of papers such as Guan and Sherman (); Guan and Loh (); Prokešová and Jensen (); Guan, Jalilian, and Waagepetersen (); and Xu, Waagepetersen, and Guan (). The other approach is due to Bolthausen () who considered stationary random fields and whose proof was later generalised to nonstationary random fields by Guyon () and Karácsony ().…”

Some Recent Developments in Statistics for Spatial Point Patterns

“…Besag, 1977;Fiksel, 1984;Takacs, 1986;Jensen & Møller, 1991;Billiot, 1997;Baddeley & Turner, 2000;Billiot et al, 2008;Coeurjolly et al, 2012) and various types of minimum contrast, composite likelihood, Palm likelihood and estimating functions for Cox and cluster processes and DPPs (e.g. Schoenberg, 2005;Guan, 2006;Waagepetersen, 2007;Tanaka et al, 2008;Waagepetersen & Guan, 2009;Prokešová & Jensen, 2013;Prokešová et al, 2014;Guan et al, 2015;Zhuang, 2015;Lavancier et al, 2014Lavancier et al, , 2015. As discussed in Sections 4.1-4.3, all of these methods belong to a common framework of estimating functions based on signed innovation measures Waagepetersen, 2005;Zhuang, 2006).…”