Analysing spatial point patterns in R - CSIRO

210 Gibbs models for multitype point patterns
34 Gibbs models for multitype point patterns
Gibbs point process models (section 26) are also available for marked point processes, and can
be fitted to data using ppm. Currently the methods are only implemented for multitype point
processes (categorical marks), so we restrict attention to this case.
34.1 Gibbs models
Much of the theory of Gibbs models described in Section 26 carries over immediately to multitype
point processes.
34.1.1 Conditional intensity
The conditional intensity λ(u,X) of an (unmarked) point process X at a location u was defined
in section 26.5. Roughly speaking λ(u,x)du is the conditional probability of finding a point
near u, given that the rest of the point process X coincides with x.
For a marked point process Y the conditional intensity is a function λ((u,m),Y) giving a
value at a location u for each possible mark m. For a finite set of marks M, we can interpret
λ((u,m),y)du as the conditional probability finding a point with mark m near u, given the rest
of the marked point process.
The conditional intensity is related to the probability density f(y) by
λ((u,m),y) =
f(y ∪ {u})
f(y)
for (u,m) ∉ y.
For Poisson processes, the conditional intensity λ((u,m),y) coincides with the intensity
function λ(u,m) and does not depend on the configuration y. For example, the homogeneous
Poisson multitype point process or “CSRI” (Section 33.1.1) has conditional intensity
λ((u,m),y) = β m (50)
where β m ≥ 0 are constants which can be interpreted in several equivalent ways (section 26.5).
The sub-process consisting of points of type m only is Poisson with intensity β m . The process
obtained by ignoring the types, and combining all the points, is Poisson with intensity β =
∑
m β m. The marks attached to the points are i.i.d. with distribution p m = β m /β.
34.1.2 Pairwise interactions
A multitype pairwise interaction process is a Gibbs process with probability density of the form
⎡ ⎤ ⎡
⎤
n(y)
∏
f(y) = α⎣
b mi (x i ) ⎦ c mi ,m j
(x i ,x j ) ⎦ (51)
i=1
where b m (u),m ∈ M are functions determining the ‘first order trend’ for points of each type,
and c m,m ′(u,v),m,m ′ ∈ M are functions determining the interaction between a pair of points of
given types m and m ′ . The interaction functions must be symmetric, c m,m ′(u,v) = c m,m ′(v,u)
and c m,m ′ ≡ c m ′ ,m. The conditional intensity is
⎣ ∏ i