Aggregated point processes
The most frequent violation of the CSR hypothesis that can occur in the context of economic data is probably that due to aggregation. Aggregation of points can essentially arise because of “true contagion” of one point by another or because of “apparent contagion” between points (Arbia and Espa, 1996). When dealing with point patterns of firms, apparent contagion may arise if exogenous factors lead to the location of firms in certain specific geographical zones. For instance, firms may cluster locally in order to exploit favorable conditions within the area, such as the presence of useful infrastructure, proximity to communication routes or the possibility of benefiting from public incentives by locating in specific areas outside residential centers. On the other hand, true contagion may occur when the presence of one event in a given area stimulates the presence of other events nearby. For instance, the presence of “leader” firms could encourage the settlement of “followers” in the same area because of the working of knowledge spillovers.
Apparent contagion is related to the violation of the CSR condition of station- arity; true contagion to that of independence. Two of the main classes of models that generate aggregated point patterns are the inhomogeneous Poisson processes, leading to apparent contagion, and the Poisson cluster processes, leading to true contagion.
Inhomogeneous Poisson point processes
Apparent contagion is the result of relaxing the condition of stationarity of the homogeneous Poisson process. The lack of stationarity implies that the first- order intensity is no longer constant throughout the territory. It may be higher in certain sub-regions of the area and lower in others. As a consequence, there will be zones with a relatively high intensity of points and others with a relatively low intensity of points, and this will produce an aggregated pattern.
Apparent contagion will thus occur as a result of the fact that, although each point is located independently of each other, the presence of some zones that are more suited to accommodating points than others leads to aggregations (Arbia and Espa, 1996).
A class of point processes that describe aggregated point patterns due to apparent contagion is the class of inhomogeneous Poisson processes which can be simply defined by replacing the constant first-order intensity X of the homogeneous
Poisson point process with a non-negative function А (л) that varies on space. Therefore, following Diggle (2003), an inhomogeneous Poisson process in a finite region A, with area |A|, and characterized by a spatially varying first-order intensity A(v) is such that:
i The number of points, say n, that can be generated in A follows a Poisson distribution with mean J" Я (x) dx.
ii Given w, the n points are generated i.i.d. on A according to a distribution with probability density function proportional to A (at).
A common computational algorithm to simulate inhomogeneous Poisson processes was first suggested by Lewis and Shedler (1979). It is based on “thinning”, that is: first of all it generates a homogeneous Poisson process of intensity A<) equal to the maximum value of the function X{x) on the study region A; then it deletes each point, independently of other points, with deletion probability A(.v)/A(). By way of example, Figure 6.5 shows two partial realizations of an inhomogeneous Poisson process in a unit square, with spatially varying intensity function Я (.v) = 100exp(-3.vi), where x is the horizontal coordinate.
Figure 6.5 Two realizations of an inhomogeneous Poisson process in a unit square with intensity A(.v) = 100ехр(-3л:[) bounded by 100.
Figure 6.6 Two realizations of an inhomogeneous Poisson process conditional on 100 points in a unit square with intensity Л(.Х[,Х2) = xf + x.
If we are interested in simulations conditional on a fixed number of points «, we can follow a similar algorithm which consists of generating a point Xj =(xa,Xi2) from the uniform distribution on A and then accepting (or rejecting) it with probability A(#()/Ao. The algorithm repeats this step continuously and stops when n points have been accepted. Figure 6.6 shows two realizations of a conditional inhomogeneous Poisson process, with 100 points, characterized by A (a;i , X2) = x + x.
6.3.2.2 Cox processes
Another class of point processes characterized by apparent contagion that can be useful in the context of economic applications is represented by the Cox processes. They represent a natural extension of the inhomogeneous Poisson processes where the source of spatial inhomogeneity, that is the intensity function А(д;), rather than being deterministic, is stochastically driven by a random process. Therefore, these processes are “doubly stochastic” (Cox, 1955; Gran- dell, 1976; Daley and Vere-Jones, 2003) and allow explicit modeling of spatial intensity endogenously rather than exogenously.
Formally, following Diggle (2003), a Cox process in a finite region A can be defined as follows,
i The first-order intensity is generated by a non-negative random function Л(л:) on A.
ii Given Л(л;) = А(л;), the points are generated following an inhomogeneous Poisson process with intensity function A (at) (see Section 6.3.2.1).
Provided that Л(лт) is stationary, the first-order and second-order intensity functions of the point process are, respectively, given as
Cox processes can be straightforwardly simulated by first generating А (л;) from A (.%•) and then generating the points using the algorithm for simulating inhomogeneous Poisson processes as described in Section 6.3.2.1. Figure 6.7 shows a partial realization of a Cox process in which Л (л;) is a Gaussian random field with mean p = 100, variance ct2 = 0.25 and correlation function p(d) = exp{-«1/0.25} with d indicating the distance between locations. Also shown is the underlying intensity surface А (а;) = Л (.%•) as a grey-scale image, where the lighter areas have higher intensities.
A particularly attractive subclass of Cox processes is the so-called log- Gaussian Cox process (Moller et al., 1998). According to this type of model, A(a;) = <;a;/>{.S(at)}, where |S(a;)} is a Gaussian process with mean p$, variance
Figure 6.7 A realization of a Gaussian Cox process in a unit square with mean p = 100, variance Indeed, exploiting the moment generating function of a log-Gaussian distribution, the first-order intensity of a log-Gaussian Cox process can be written as:
For the covariance structure we have that Л(л;)Л(у) = ехр{5(л;) + 5(у)). Since S(x) + S(y) is also Gaussian with mean m = 2ц$ and variance v = 2cr^ [ 1 + ps(d) ] then £[Л(л;)Л(у)] = ехр(>и-1-г> / 2) and hence:
