Models for Clustered and Regular Point Patterns

  In this section we discuss some point-process models whose realizations exhibit clustering and regularity respectively. Assume that the underlying point process X is stationary and isotropic (with intensity $\lambda$). Then these two kinds of deviation from complete spatial randomness of points can be studied for example by means of the reduced second-order moment measure K of X, which is uniquely determined by (2.9). Since X is not only stationary but also isotropic, K is isotropic as well and it suffices to consider the reduced second-moment function K(r)=K(b(o,r)), where b(o,r) is the circle with radius r and center at the origin o; see Section 2.2. Closely related characterstics are the product density  

$$\rho^{(2)} (r) = \frac{\lambda^2}{2\pi r} \, \frac{dK(r)}{dr}$$ (1)

and the pair-correlation function  

$$g(r) = \frac{\rho^{(2)} (r)}{\lambda^2}$$ (2)

provided that the function K(r) is differentiable; r>0. In accordance with the interpretation of $\lambda K(r)$ as the expected number of points in a circle with radius r and center at the typical point of the point process, we can interpret g(r) as 'relative frequency' of pairs of points having distance r. Note, that in the Poisson case (2.9), (2.8), (6.1) and (6.2) imply that $K(r) = \pi r^2$, $\rho^{(2)}(r) = \lambda^2$,and g(r)=1 for all $r \geq 0$.Although this extremely simple relationship will not be true for the more general point-process models discussed below, we still can determine their pair correlation functions. Thus, statistical model identification can be based on comparison of empirical pair-correlation functions with given theoretical counterparts. In particular, for small r > 0, values g(r) > 1 indicate clustering whereas values g(r) < 1 indicate inhibition of points, relative to the situation for the homogeneous Poisson process.

Suppose that a stationary and isotropic point process $X=\{X_1,X_2,\ldots\}$ with intensity $\lambda$ is observed in the convex window $B \subset \RL^2$; $0 < \nu(B) < \infty$. Then, the product density $\rho^{(2)}(r)$ defined in (6.1) can be estimated by using an edge-corrected density estimator of the form  

$$\hat{\rho}^{(2)}(r) = \sum_{n,m} \, \frac{k(\vert X_n - X_m\... ...n B, X_m \in B)}{2\pi r \nu\left( (B+X_n) \cap (B+X_m) \right)}$$ (3)

where k(s) is a nonnegative function and the summation extends over all pairs (n,m) such that $n \neq m$ and $\nu\left( (B+X_n) \cap (B+X_m) \right) \gt 0$. Usually one assumes that k(s) is a kernel function, i.e. k(s) is a probability density which is symmetric about the origin and vanishes outside a bounded interval. A suitable example of a kernel function is

$$k(s) = \left\{ \begin{array} {ll} {\D \Bigl( 1 - \frac{s^2}{... ...psilon \sqrt{5}} \\ {\D 0} & \mbox{otherwise}\end{array}\right.$$

for each fixed smoothing constant $\varepsilon \gt 0$;see Stoyan et al. (1995), Stoyan and Stoyan (1996). In Collins and Cressie (1996), local versions of the product density $\rho^{(2)}(r)$ have been considered which are associated with each individual point of the process X provided that this point lies in B. These local density functions are then grouped into bundles of similar functions. In this way the (clustering or regularity) structure of the observed point patterns can be quantified.

In Figure 14 a realization of a homogeneous Poisson process and g(r)=1 together with the estimator $\hat{g}(r)=\hat{\rho}^{(2)}(r)/ \hat{\lambda}$ are given, where $\hat{\lambda}$ is an estimator for the intensity; see Section 2.2.

  

Figure 14: Homogeneous Poisson process