Analysing spatial point patterns in R - CSIRO

156 Gibbs models
26 Gibbs models
One way to construct a statistical model (in any field of statistics) is to write down its probability
density. Advantages of doing this are:
the functional form of the density reflects its probabilistic properties.
terms or factors in the density often have an interpretation as ‘components’ of the model.
it is easy to introduce terms that represent the dependence of the model on covariates,
etc.
This approach is useful provided the density can be written down, and provided the density
is tractable.
Spatial point process models that are constructed by writing down their probability densities
are called ‘Gibbs processes’. Good references on Gibbs point processes are [63, 31].
26.1 Probability densities
It is possible to define probability densities for spatial point processes that live inside a bounded
window W.
The probability density will be a function f(x) defined for each finite configuration x =
{x 1 ,... ,x n } of points x i ∈ W for any n ≥ 0. Notice that the number of points n is not fixed,
and may be zero. Apart from this peculiarity, probability densities for point processes behave
much like probability densities in more familiar contexts.
That’s all you need to know for applications. If you’re interested in the mathematical
technicalities, read on; otherwise, skip to section 26.2.
A point process X inside W is defined to have probability density f if and only if, for any
nonnegative integrable function h,
∑
∞ ∫ ∫
E[h(X)] = e −|W | h(∅)f(∅) + e −|W | 1
· · · h({x 1 ,...,x n })f({x 1 ,...,x n })dx 1 · · · dx n
n!
n=1 W W
(31)
where |W | denotes the area of W.
In particular, the probability that X contains exactly n points is
p n = P{n(X) = n} = e−|W | ∫
n!
W
∫
· · · f({x 1 ,... ,x n })dx 1 · · · dx n
W
for n ≥ 1 and p 0 = P{n(X) = 0} = e −|W | f(∅). Given that there are exactly n points, the
conditional joint density of the locations x 1 ,... ,x n is f({x 1 ,... ,x n })/p n .
26.2 Poisson processes
The uniform Poisson process with intensity 1 has probability density f(x) ≡ 1.
The uniform Poisson process in W with intensity λ has probability density
f(x) = α λ n(x) (32)
where n(x) is the number of points in the configuration x, and the constant α is
α = e (1−λ)|W | .
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