Analysing spatial point patterns in R - CSIRO

158 Gibbs models
Hard core process
26.3.2 Strauss process
Generalising the hard core process, suppose we take b(u) ≡ β and
c(u,v) =
{ 1 if ||u − v|| > r
γ if ||u − v|| ≤ r
(36)
where γ is a parameter. Then the density becomes
f(x) = αβ n(x) γ s(x) (37)
where s(x) is the number of pairs of distinct points in x that lie closer than r units apart.
The parameter γ controls the ‘strength’ of interaction between points. If γ = 1 the model
reduces to a Poisson process with intensity β. If γ = 0 the model is a hard core process. For
values 0 < γ < 1, the process exhibits inhibition (negative association) between points.
Strauss(γ = 0.2) Strauss(γ = 0.7)
For γ > 1, the density (37) is not integrable. Hence the Strauss process is defined only for
0 ≤ γ ≤ 1 and is a model for inhibition between points. This is typical of most Gibbs models.
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