Poisson cluster point processes

A class of point processes that describe aggregated point patterns due to true contagion is the class of Poisson cluster processes. It was introduced by Neyman and Scott (1958) as a class of models that allow the modeling of dependence among points and hence the incorporation of an explicit form of spatial clustering (Diggle, 2003). An interesting schematic way of visualizing the occurrence of an aggregated point pattern by means of a Poisson cluster process is the “leader- follower” framework proposed by Arbia and Espa (1996). Let us consider a finite region and let us suppose that a certain number of leader firms are located within its boundaries. Then let us fix a threshold distance for local related activities and, according to this threshold, define an area of influence for each leader. For ease of simplicity we consider that this area has the same extension for all leader firms. Now let us fix a certain number of followers for each leader as the realization of a random variable. Finally, allocate the followers to the leader firms on the basis of a given bivariate distribution. For example, a bivariate uniform distribution may be used in those cases where the probability within the area of influence can be taken to be constant or, alternatively, we may consider a bivariate normal distribution if we assume that the probability of locating the followers decreases exponentially with an increase in the distance from the leader. The resulting process is the Poisson cluster process shown in Figure 6.8.

More formally, Diggle (2003) defines a Poisson cluster process in a finite region A as fulfilling:

i The leader points are spatially distributed in A according to a homogeneous Poisson process with first-order intensity p.

ii The numbers of followers per leader are generated i.i.d. according to a given probability distribution (typically Poisson).

iii the locations of the followers are generated i.i.d. around their respective leaders’ locations according to a given bivariate probability density function (typically normal or uniform).

Like the homogeneous Poisson process, the Poisson cluster processes are stationary and are characterized by a first-order intensity A = pp, where p is the mean number of followers per leader.

If a Poisson cluster process is specified using a Poisson distribution with mean p for the random number of followers per leader and a radially symmetric normal distribution with standard deviation a for the location of followers around their leaders, it is said to be a Thomas cluster process (Thomas, 1949). If, instead, the dispersion of followers with respect to their leaders’ locations follows a uniform distribution inside a circle of radius R centered on the leader point, we have a Matern cluster process (Matern, 1986). The dispersion parameters a and R represent the spatial extension of the area of influence for each leader.

Figure 6.9 shows two realizations of a Poisson cluster process in a unit square with p = 25 and p = 4. In Figure 6.9a the location of each follower relative to its

Figure 6.8 The genesis of a Poisson cluster process: (a) location of leaders; (b) setting the dimensions of the areas of influence of local related activities and location of the followers; (c) the resulting point process.

leader is realized following the uniform distribution on a random circular disc with maximum radius R = 0.025. In Figure 6.9b, instead, followers are dispersed around their leaders according to a radially symmetric normal distribution with <7 = 0.025.

In Figure 6.9a, it can be clearly seen that the dispersion of followers within the cluster is homogeneous across all the area of influence. In contrast, the pattern represented by Figure 6.9b suggests that the followers intensity tends to diminish as the distance from the cluster center increases.

Conditioning on the number of leader points and number of followers per leader is straightforward. Rather than specifying the parameters of random variables, we simply need to fix constant values. By way of illustration, Figure 6.10 shows realizations of a Poisson cluster process conditional on fixed quantities. In Figure 6.10a the 25 groups of four followers are clearly identifiable. The visual identification of clusters is less straightforward in Figure 6.10b due to the coalescence between nearby clusters.

Figure 6.9 Two realizations of a Poisson cluster process in a unit square with Я = 25 and // = 4: (a) uniform dispersion of followers, with maximum radius 0.025; (b) radially symmetric normal dispersion of followers, with

Figure 6.10 Two realizations of a Poisson cluster process in a unit square with 25 leader points and 4 followers per leader: (a) uniform dispersion of followers, with maximum radius 0.025; (b) radially symmetric normal dispersion of followers, with О = 0.025.

Figure 6.11 Two realizations of a Poisson cluster process in a unit square with 100 followers randomly assigned amongst 25 leader points: (a) uniform dispersion of followers, with maximum radius 0.025; (b) radially symmetric normal dispersion of followers, with a = 0.025.

For identification purposes, in order to recreate specific paradigmatic situations, we might be interested in performing simulations conditional on the total number of events on the study region. Indeed, when the number of followers per leader is randomly generated according to a Poisson distribution, the point process can be simulated by randomly allocating a fixed number of events amongst the leader points. This is illustrated in Figure 6.11 which shows two simulated patterns where the 100 followers are randomly assigned amongst the 25 leader points.