Parametric methods for spatial point processes

(This text is submitted for the volume ‘A Handbook of Spatial Statistics' edited by A.E. Gelfand, P. Diggle, M. Fuentes, and P. Guttorp, to be published by Chapmand and Hall/CRC Press, and planned to appear as Chapter 4.4 with the title ‘Parametric methods'.)

1 Introduction

This chapter considers inference procedures for parametric spatial point process models. The widespread use of sensible but ad hoc methods based on summary statistics of the kind studied in Chapter 4.3 have through the last two decades been supplied by likelihood based methods for parametric spatial point process models. The increasing development of such likelihood based methods, whether frequentist or Bayesian, has lead to more objective and efficient statistical procedures. When checking a fitted parametric point process model, summary statistics and residual analysis (Chapter 4.5) play an important role in combination with simulation procedures. Simulation free estimation methods based on composite likelihoods or pseudo likelihoods are discussed in Section 3. Markov chain Monte Carlo (MCMC) methods have had an increasing impact on the development of simulationbased likelihood inference, where maximum likelihood inference is studied in Section 4, and Bayesian inference in Section 5. On one hand, as the development in computer technology and computational statistics continues,computationally-intensive simulation-based methods for likelihood inference probably will play a increasing role for statistical analysis of spatial point patterns. On the other hand, since larger and larger point pattern dataset are expected to be collected in the future, and the simulation free methods are 1 much faster, they may continue to be of importance, at least at a preliminary stage of a parametric spatial point process analysis, where many different parametric models may quickly be investigated. Much of this review is inspired by the monograph Møller andWaagepetersen (2003) and the discussion paper Møller andWaagepetersen (2007). Other recent textbooks related to the topic of this chapter include Baddeley, Gregori, Mateu, Stoica and Stoyan (2006), Diggle (2003), Illian, Penttinen, Stoyan and Stoyan (2008), and Van Lieshout (2000). Readers interested in background material on MCMC algorithms for spatial point processes are referred to Geyer and Møller (1994), Geyer (1999), Møller and Waagepetersen (2003), and the references therein. Notice the comments and corrections to Møller and Waagepetersen (2003) at www.math.aau.dk/~jm.