Gibbs Point Processes

There are many different classes or models of point processes that have arisen in the literature over the years and this section considers the class referred to as Gibbs point processes. They are a development of the idea in Section 1.1.6: that new point processes can be obtained from old ones by transforming their distributions via a probability density. Following Daley & Vere-Jones (1988) and Stoyanet al.(1995) the construction of Gibbs processes is described below. The origins of such processes stem from statistical physics, being related to the so-calledGibbs distributions

which describe the equilibrium states of subsystems of very large closed physical systems. Such processes are described by means of forces acting on and between the particles. The total potential energy of a given configuration of particles or individuals is assumed to be decomposable into terms representing the interactions between the particles taken in pairs, triples, and so on. First-order terms representing the presence of an external force can also be included. Thus Gibbs processes can be thought of as processes generated byinteraction potentials.

The fundamental ingredient in specifying a Gibbs process X is an underlying basic or weight

processX0 with distributionQ(this is usually taken to be a Poisson process). The distributionPof

a Gibbs process can then be defined by means of a densityf: P(B) =

Z

B

f(x)Q(dx), B ∈ B(Xe). (1.12)

Rather than specifying a distributionPfor the Gibbs process and then checking absolute continuity PQ, the usual trick is to do the reverse. An integrable functionf (with respect toQ) is specified and the distribution P is then defined via Eq. (1.12). The form of f is often determined by the field of application: it can be chosen conveniently to model interactions between individuals of the process. This approach is straightforward if the process contains only finitely many points confined to a bounded regionW. More generally, for point processes in all ofX, the density idea must be applied to conditional distributions confined to bounded regions.

Gibbs Processes in Bounded Regions

In this section we consider how to define Gibbs point processes in a bounded regionW ⊂ X. In order to do so there are two different cases to consider:

Canonical Ensemble: Here the process contains a fixed number n of particles, all contained in

W. This is of great practical importance since one often conditions on the number of points observed in someW. The form off is usuallyf(x) =f(ξ1, . . . , ξn) = e

−U(x)

Z , whereZ is a

normalizing constant called theconfigurational partition function, andU : Xe → R∪ {∞}

is the energy function. Frequently U is chosen to have a specialized form: an interaction potentialU(x) = P

y⊂xV (y); or a sum ofpair potentialsU(x) =

P P

1≤i,j≤nθ(kξi−ξjk).

The functionθ is referred to as the pair potential, in homage to the origins of the subject in Physics, andk · kis a norm onX.

Grand Canonical Ensemble: Here the total number of pointsN is random but all assumed to be within W. One approach is to define a sequence of “conditional densities” fn. The Gibbs

process is then obtained by first arrangingN to have some distribution and then (conditional on the value ofN) using the Canonical Ensemble construction. That is givenN = n then

points are distributed inW using the joint densityfn. Another approach is to define a density

directly onB(We), whereWeis the exponential space (Section 1.1.1) ofW.

Stationary Gibbs Processes

When considering Gibbs processes on unbounded spacesX, a more sophisticated method than just specifying the density on boundedW ⊂ X is required. Additionally, the distribution of the process restricted to the observation windowW must beconditionedon the process outsideW.

Formally one considers the family oflocal specificationsπW (· | ·)for bounded BorelW. These

represent the probability that the Gibbs processXonW belongs to the setB ∈ B(Xe), given that the

process takes on some configurationxoutsideW, ie.πW(B |x) = P[X∩W ∈B |X∩Wc=x],

wherex∈ {y∈ Xe; y(B) = 0}. Then a point processX with distributionPis said to be aGibbs

after Dobrushin, Lanford & Ruelle (Stoyanet al.1995), holds: DLR-Equation: P(B) =

Z

πW(B |x∩Wc)P(dx). (1.13)

Examples of Gibbs Point Processes

Any process with density of the form given in Canonical Ensemble is a Gibbs process; some exam- ples are given below and several more in Section 1.1.11, as examples of Markov point processes.

1. Hard Core Process: This is a process where no two points are allowed to be within distance

R > 0of each other, so that the density is given byf(x)∝λn(x)Q

{ξ,η:ξ6=η}1{kξ−ηk>2R}. 2. Strauss Process: This model has density with respect to a Poisson(1) given by f(x) ∝

λn(x)_eβ sR(x)_{. The parameter λ > 0 represents the underlying Poisson intensity and s}

R(x)

denotes the number of pairs ξ, η ∈ xwhich are closer than distance R >0. This is an exam- ple of apair-wiseinteraction Gibbs process, since the density depends only on the number of

R-close pairs. The case when β = 0corresponds to a Poisson(λ)process; whenβ → −∞

the Strauss process converges to the hard core model since eβ sR(x) _{will be non-zero only if}

sR(x) = 0. The Strauss process can be considered as asoft coreprocess since the density is

weighted by the number ofR-close pairs; the case when the weight is non-zero only for zero

R-close pairs corresponds to a hard core process.

3. Geyer (1999) remarks that in practical applications, “it is likely that no process model in the existing literature would be of scientific interest and a model specific to the application would be invented”. Furthermore, he illustrates the ease with which one can ‘invent’ new point processes and do statistical inference; the triplets andsaturation processes are described as examples of two new point processes.