## Statistical Inference for Cox Processes

This chapter is concerned with statistical inference for a large class of point process models, which were studied in a seminal paper by Cox (1955) under the name doubly stochastic Poisson processes, but are today usually called Cox processes. Much of the literature on Cox processes is concerned with point processes deﬁned on the real line R, but we pay attention to the general d-dimensional Euclidean case of Rd, and in particular to the planar case d = 2 (which covers most cases of spatial applications). However, the theory does not make much use of the special properties of Rd, and extensions to other state spaces are rather obvious. We discuss in some detail how various Cox process models can be constructed and simulated, study nonparametric as well as parametric analysis (with a particular emphasis on minimum contrast estimation), and relate the methods to simulated and real datasets of aggregated spatial point patterns. Further material on Cox processes can be found in the references mentioned in the sequel and in Grandell (1976), Diggle (1983), Daley and Vere-Jones (1988), Stoyan et al. (1995), and the references therein. To explain brieﬂy what is meant by a Cox process, consider the spatial

point patterns in Figure 3.1. As demonstrated in Section 3.3, each point pattern in Figure 3.1 is more aggregated than can be expected under a homogeneous Poisson process (the reference model in statistics for spatial point patterns). The aggregation is in fact caused by a realization z = {z(x) : x ∈ R2} of an underlying nonnegative spatial process Z, which is shown in gray scale in Figure 3.1. If the conditional distribution of a point process X given Z = z is an inhomogeneous Poisson process with intensity function z, we call X a Cox process with random intensity surface Z (for details, see Section 3.3). Note that points of X are most likely to occur in areas where Z is large, cf. Figure 3.1. In many applications we can think of Z as an underlying “environmental” process. Aggregation in a spatial point processX may indeed be due to other sources, including (i) clustering of the points in X around the points of another point process C, and (ii) interaction between the points in X. For certain models, (i) is equivalent to

a Cox process model (see Section 3.5.1). This is in fact the case for the upper left point pattern in Figure 3.1, where the cluster centres C = {c1, . . . , cm} are also shown. The case (ii) is not considered in this contribution, but we refer the interested reader to the literature on Markov point processes, see, for example, Møller (1999) and van Lieshout (2000). General deﬁnitions and descriptions of Poisson and Cox processes are

given in Sections 3.2-3.3, while Section 3.4 provides some background on nonparametric analysis. Section 3.5 concerns certain parametric models for Cox processes. Speciﬁcally, Section 3.5.1 considers the case where NeymanScott processes (Neyman and Scott 1958) are Cox processes, Section 3.5.2

deals with log Gaussian Cox processes (Coles and Jones 1991, Møller et al. 1998), and Section 3.5.3 with shot noise G Cox processes (Brix 1999). The latter class of models includes the Poisson/gamma processes (Wolpert and Ickstadt 1998). As explained in more detail later, the point patterns in Figure 3.1 are realizations of a certain Neyman-Scott process, a log Gaussian Cox process (LGCP), a shot-noise G Cox process (SNGCP), and a certain “logistic process”, respectively. In most applications with an aggregated point pattern modeled by a

Cox process X, the underlying environmental process Z is unobserved. Further, only X ∩W is observed, where W is a bounded region contained in the area where the points in X occur. In Section 3.6 we discuss various approaches to estimation in parametric models for Cox processes and focus in particular on minimum contrast estimation. In Section 3.7 we discuss how Z and X \W can be predicted under the various models from Section 3.5. Section 3.8 contains some concluding remarks. The discussion in Sections 3.5-3.7 will be related to the dataset in Fig-

ure 3.2, which shows the positions of 378 weed plants (Veronica spp./ speedwell). This point pattern is a subset of a much larger dataset analyzed in Brix and Møller (2001) and Brix and Chadoeuf (2000) where several weed species at diﬀerent sampling dates were considered. Note that we have rotated the design 90◦ in Figure 3.2. The 45 frames are of size 30× 20 cm2, and they are organized in 9 groups each containing 5 frames, where the vertical and horizontal distances between two neighbouring groups are 1 m and 1.5 m, respectively. The size of the experimental area is 7.5 × 5 m2. The observation window W is given by the union of the 45 frames.