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Spatial Point Patterns: Models and Statistics Adrian Baddeley

Abstract This chapter gives a brief introduction to spatial point processes, with a view to applications. The three sections focus on the construction of point process models, the simulation of point processes, and statistical inference. For further background, we recommend [Daley et al., Probability and its applications (New York). Springer, New York, 2003/2008; Diggle, Statistical analysis of spatial point patterns, 2nd edn. Hodder Arnold, London, 2003; Illian et al., Statistical analysis and modelling of spatial point patterns. Wiley, Chichester, 2008; Møller et al., Statistical inference and simulation for spatial point processes. Chapman & Hall, Boca Raton, 2004].

Introduction Spatial point patterns—data which take the form of a pattern of points in space— are encountered in many fields of research. Currently there is particular interest in point pattern analysis in radioastronomy (Fig. 3.1), epidemiology (Fig. 3.2a) and prospective geology (Fig. 3.2b). Under suitable conditions, a point pattern dataset can be modelled and analysed as a realization of a spatial point process. The main goals of point process analysis are to 1. Formulate “realistic” stochastic models for spatial point patterns 2. Analyse, predict or simulate the behaviour of the model 3. Fit models to data These three goals will be treated in three successive sections.

A. Baddeley () CSIRO, Perth, Australia e-mail: [email protected] E. Spodarev (ed.), Stochastic Geometry, Spatial Statistics and Random Fields, Lecture Notes in Mathematics 2068, DOI 10.1007/978-3-642-33305-7 3, © Springer-Verlag Berlin Heidelberg 2013

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Fig. 3.1 Sky positions of 4,215 galaxies observed in a radioastronomical survey [161]

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Fig. 3.2 Examples of point pattern data. (a) Locations of cases of cancer of the lung (plus) and larynx (filled circle), and a pollution source (oplus), in a region of England [153]. (b) Gold deposits (circle), geological faults (lines) and rock type (grey shading) in a region of Western Australia [507]

3.1 Models In this section we cover some basic notions of point processes (Sect. 3.1.1), introduce the Poisson process (Sect. 3.1.2), discuss models constructed from Poisson processes (Sect. 3.1.4), and introduce finite Gibbs point processes (Sect. 3.1.5).

3.1.1 Point Processes In one dimensional time, a point process represents the successive instants of time at which events occur, such as the clicks of a Geiger counter or the arrivals of customers at a bank. A point process in time can be characterized and analysed using several different quantities. One can use the arrival times T1 < T2 < : : : at which the events occur (Fig. 3.3a), or the waiting times Si D Ti  Ti 1 between successive arrivals (Fig. 3.3b). Alternatively one can use the counting process Nt D P 1.T  t / illustrated in Fig. 3.4, or the interval counts N.a; b D Nb  Na . i i

3 Spatial Point Patterns: Models and Statistics

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a T1

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S1

T 2 T3 S2

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T4 S4

Fig. 3.3 Arri