# Regular point processes

A probably less frequent, but still relevant, form of violation of the CSR hypothesis in the context of economic data is the spatial “inhibition” that leads to regular or dispersed point patterns. Regularity in the location of economic activities, such as stores, can occur, for example, because the presence of a store in a given geographical area forces back competing stores due to the locally differentiated monopolistic or oligopolistic nature of the market. In a case like this, a store “inhibits” the location of other stores within the surrounding area. A common theoretical inhibitory model in the field of economic geography is the “central place” model introduced by Christaller (1933) and subsequently used by Losch (1954), Isard (1956) and others in more recent years.

The first statistical formulation of an inhibitory process is historically traced back to Matern (1960), who suggested constructing an inhibitory process starting from a realization of a homogeneous Poisson process with given intensity p and then deleting all pairs of generated points that are separated by a distance less than a specified threshold 5, defined as “inhibition distance”. The remaining points represent a realization of the Matern model I inhibition process.

Matern (1960) has also proposed a somewhat “dynamic” version of this process in which the points generated by the homogeneous Poisson process are marked with “birth times”, obtained as independent realizations of a uniform distribution in [0,1]. Each point is then removed if it is located within a distance 8 of another point that was born earlier, whether or not the latter has itself previously been retained or removed. The remaining points constitute a realization of the Matern model II inhibition process. Figure 6.12 shows two realizations of, respectively, Matern model I and Matern model II in a unit square both with p = 50 and 8 = 0.08. To make the two realizations directly comparable, the same random number seed has been used. It can be noted that Matern model II tends to have a higher final first-order intensity for the same parameter values.

A further natural variation of the scheme of Matern inhibition processes is the simple sequential inhibition (SSI) process (Diggle, 2003) which is based on the rationale that a point that has been removed cannot inhibit the occurrence of newer points. Specifically, the SSI process generates points sequentially one-by- one until a given number *n* of points has been reached. Each point is generated according to the uniform distribution in the region of interest independently of the previously generated points. If the newly generated point is located within a distance 8 of an existing point, then it is removed; otherwise, it is retained. In order to illustrate the sequential nature of the SSI process, Figure 6.13 shows three phases of the building of a realization in a unit square with *n =* 50 and 8 = 0.08.

Matern model I, Matern model II and SSI are sometimes labelled “hard-core” inhibition processes since they strictly prevent the occurrence of points located at

*Figure 6.12* Two realizations of Matern inhibition processes in a unit square with p = 50 and (5 = 0.08 : (a) Matern model I; (b) Matern model II.

*Figure 6.13* Three phases of the development of a simple sequential inhibition process in a unit square with *n* = 50 and 8* =* 0.08.

a distance smaller than the inhibition distance. This lack of flexibility makes the hard-core processes unrealistic in describing many empirical patterns. Indeed, to describe properly numerous regular patterns, especially those of economic phenomena, a less strict inhibition criterion, be it static or dynamic, may be necessary. For example, in real circumstances it is not uncommon to find two or more stores competing with each other but being located very close to one another.

The inhibition processes that are more flexible in this respect are often labelled “soft-core” processes in that they generate patterns where the number of neighbor points within the inhibition distance *8* tends to be smaller with respect to CSR patterns, but it is not zero. A popular soft-core process is the Strauss process (Strauss, 1975; Kelly and Ripley, 1976), which is characterized by three parameters, namely the spatial intensity /?, the inhibition distance <5 and the interaction parameter y. A Strauss process generates *n* points of a pattern, say according to the following probability density function:

where *s* represents the number of distinct pairs of points that are separated by a distance lower than *8* and *a* is a normalizing constant. In practice, the Strauss process generates CSR patterns with given first-order intensity */**8**,* but in which the points have a probability equal to 1-y of being deleted if they are located closer than *8* to other points.

The parameter у regulates the level of strictness of the inhibition mechanism, higher values of у imply more flexibility. In particular, if y=0 the inhibition process is hard-core as it does not allow distances among points that are lower than *8**;* on the other hand, if y= 1 it reduces to a homogeneous Poisson process with first-order intensity */3.* It is necessary to respect the condition y< 1 in order to avoid the process exploding and generating an infinite number of points (Kelly and Ripley, 1976). Figure 6.14 illustrates the role of у by showing three realizations, obtained using the same random number seed, of a Strauss process in a unit square with *[3 =* 50, 5 = 0.08 and y= 0.1,0.5 and 0.8, respectively. The increase in the level of flexibility is clear.

*Figure 6.14* Three realizations of the Strauss process in a unit square with *[3 =* 50*,8* = 0.08 and (a) y= 0.1; (b) y= 0.5; (c) y= 0.8.