The hypothesis of complete spatial randomness
Figure 6.2 illustrates three different artificial examples of spatial point patterns in a square area. In particular, the first shows local aggregations of points, which could be due to some form of clustering mechanism or to territorial variation within the considered area. Figure 6.2b depicts a pattern where points are distributed approximately regularly over the area, suggesting that a mechanism may have favored inhibition amongst points’ locations and encouraged an even spatial distribution.
The pattern in Figure 6.2c does not show any kind of systematic structure and might be considered a completely random pattern. The basic key concept, which represents the starting point for the analysis of any spatial point pattern, is the hypothesis of complete spatial randomness (Diggle, 2003; Cressie, 1993).
Figure 6.2 Paradigmatic examples of spatial point patterns: (a) aggregated pattern; (b) regular pattern; (c) random pattern.
This benchmark hypothesis for a spatial point pattern asserts, heuristically, that the points have been generated under two specific conditions:
i “stationarity”, the constant propensity to host points within the pattern: that is, the area of the pattern is homogeneous;
ii “independence”, the absence of spatial interactions amongst points: that is, each point’s location is independent of the other points’ locations.
Generally, the first step in the inferential analysis of a spatial point pattern consists of deciding whether the observed pattern is consistent with the CSR hypothesis. As in other branches of statistics, inferential analysis is primarily conducted by relying on formal stochastic mechanisms generating data. In this specific context, stochastic processes are labelled as spatial point processes.
Spatial point processes
A spatial point process is a stochastic process that generates objects in the plane forming a countable set of points Xj = (хц,Х2,-): i = 1,2,..., in which хц and хц are, respectively, the horizontal and vertical spatial coordinates of the »th object (Diggle, 2003). Point processes can be used to analyze spatial point pattern data within a model-based perspective, through comparisons between the theoretical properties of an assumed underlying point process model and their empirical counterparts estimated on the observed data.
The main properties of a generic spatial point process are the theoretical summary descriptions called “first-order” and “second-order” intensity functions. The first-order intensity function can be formally defined as follows (Diggle, 2003; Cressie, 1993):
To clarify the notation here employed (borrowed from Diggle, 2003), dx is an infinitesimally small spatial region containing the generic point x, N (dx) represents the number of points located in it and dx denotes its area.
Equation 6.1 can be interpreted as the expected number of points per unit located within an infinitesimal region centered on the generic point x. Therefore, heuristically, X(x)dx expresses the probability of finding a point around the location x. If the first-order intensity is constant throughout the space, that is if А(дг) = А, it represents the expected number of events per unitary area and the point process is stationary (Diggle, 2003).
The second-order intensity function can be similarly defined as follows (Diggle, 2003; Cressie, 1993):
where л;and у denote two distinct generic events in the area. Informally, A2 (x,y) can be interpreted as the expected number of points located in locations л: or у and hence A2 (x,y)dxdy can be interpreted as the probability that two points locate in two infinitesimal regions centered in x and у and with surface areas dx and dy respectively (Diggle et al., 2007). Therefore, Equation 6.2 is the appropriate theoretical summary description of the spatial dependence amongst points.
If a process is stationary we have A2 (л;, у) = A2 (x - у); furthermore, if a process is stationary and also isotropic, A2 (,v,_v) depends only on the distance, say d, between x and у and hence A2 (x,у) = A2 (d) (Diggle, 2003).
6.3.1 Homogeneous Poisson point process
Over the years, the spatial statistics literature has proposed a number of spatial point processes that can be used to model and analyze various kinds of point patterns. All of them are somewhat rooted in the homogeneous Poisson point process, which can essentially be seen as the basic framework for the building of more complex models. Moreover, it can be used as the reference benchmark model, since it properly represents the CSR hypothesis. Indeed, following Diggle (2003), the homogeneous Poisson process in a finite region A, with area |A|, and characterized by a first-order intensity A > 0 can be defined as fulfilling the following two conditions:
i The number of points, say n, that can be generated in A follows a Poisson distribution with mean A|A|.
ii Given n, the n points are generated i.i.d. according to the uniform distribution on A.
It can be easily noted that these conditions mirror closely the definition of CSR hypothesis. On one hand, condition i implies that the intensity is constant, and hence it corresponds exactly to the stationarity condition of CSR hypothesis; on the other, condition ii parallels the independence condition of CSR hypothesis, as it prescribes that the points are generated independently to each other.
In order to generate a point pattern as a realization of the homogeneous Poisson process on A, given X, two simulation steps can be followed. First, the simulation of the n number of points from the Poisson distribution with mean proportional to the chosen value of X is carried out. Secondly, once the random value n is returned, the n points are generated independently according to a uniform distribution on A. Figure 6.3 shows two possible realizations of the homogeneous Poisson process in a unit square, with X parameter equal to, respectively, 50 and 100.
Alternatively, instead of referring to a random value of n, we might be interested in carrying out simulations that are conditional on a fixed number of points. As we will see in Chapter 7, conditional simulated patterns have a crucial importance in inferential analysis because they can represent useful hypothetic theoretical counterfactuals of observed patterns. Conditional simulations can be carried out following the second step of the procedure mentioned above with n as a chosen constant. By way of illustration, Figure 6.4 shows two possible
Figure 6.3 Two realizations of a homogeneous Poisson process in a unit square: (a) with parameter Я = 50; (b) with parameter Я = 100.
Figure 6.4 Two realizations of a homogeneous Poisson process conditional on 100 points in a unit square.
partial realizations of a homogeneous Poisson process conditional on 100 points in a unit square.
The homogeneous Poisson process can act as a unique benchmark model because any other point process would necessarily generate point patterns that are more aggregated (as in Figure 6.2a) or more regular (as in Figure 6.2b) than a CSR pattern. Therefore, point processes can be loosely classified into “aggregated processes” and “regular processes”.