Analysing spatial point patterns in R - CSIRO

160 Gibbs models
26.5 Conditional intensity
The main tool for analysing a Gibbs point process is its conditional intensity λ(u,X). Intuitively
this determines the conditional probability of finding a point of the process at the location u given
complete information about the rest of the process. For formal definitions see [31]. Informally,
the conditional probability of finding a point of the process inside an infinitesimal neighbourhood
of the location u, given the complete point pattern at all other locations, is λ(u,X)du.
u
For point processes in a bounded window, the conditional intensity at a location u given the
configuration x is related to the probability density f by
λ(u,x) =
f(x ∪ {u})
f(x)
(39)
(for u ∉ x), the ratio of the probability densities for the configuration x with and without the
point u added.
The homogeneous Poisson process with intensity λ has conditional intensity
λ(u,x) = λ
while the inhomogeneous Poisson process with intensity function λ(u) has conditional intensity
λ(u,x) = λ(u)
. The conditional intensity for a Poisson process does not depend on the configuration x, because
the points of a Poisson process are independent.
For the general pairwise interaction process (34) the conditional intensity is
For the hard core process,
n(x)
∏
λ(u,x) = b(u) c(u,x i ). (40)
i=1
{ β if ||u − xi || > r for all i
λ(u,x) =
0 otherwise
(41)
which has the nice interpretation that a point u is either ‘permitted’ or ‘not permitted’ depending
on whether it satisfies the hard core requirement.
For the Strauss process
λ(u,x) = βγ t(u,x) (42)
where t(u,x) = s(x ∪ {u}) − s(x) is the number of points of x that lie within a distance r of the
location u. For γ < 1, this has the interpretation that a random point is less likely to occur at
the location u if there are many points in the neighbourhood.
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