Analysing spatial point patterns in R - CSIRO

26.3 Pairwise interaction models 157
The inhomogeneous Poisson process in W with intensity function λ(u) has probability density
n∏
f(x) = α λ(x i ). (33)
i=1
where the constant α is
[∫ ]
α = exp (1 − λ(u))du .
W
The densities (32) and (33) are products of terms associated with individual points x i . This
reflects the conditional independence property (PP4) of the Poisson process.
26.3 Pairwise interaction models
In order to construct spatial point processes which exhibit interpoint interaction (stochastic
dependence between points), we need to introduce terms in the density that depend on more
than one point. The simplest are pairwise interaction models, which have probability densities
of the form
⎡ ⎤ ⎡ ⎤
n(x)
∏
f(x) = α ⎣ b(x i ) ⎦ c(x i ,x j ) ⎦ (34)
i=1
⎣ ∏ i r
0 if ||u − v|| ≤ r
(35)
where ||u − v|| denotes the distance between u and v, and r > 0 is a fixed distance, then the
density becomes
{
αβ
n(x)
if ||x
f(x) =
i − x j || > r for all i ≠ j
0 otherwise
This is the density of the Poisson process of intensity β in W conditioned on the event that no
two points of the pattern lie closer than r units apart. It is known as the (classical) hard core
process.
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