Analysing spatial point patterns in R - CSIRO

34.1 Gibbs models 211
34.1.3 Pairwise interactions not depending on marks
The simplest examples of multitype pairwise interaction processes are those in which the interaction
term c m,m ′(u,v) does not depend on the marks m,m ′ . For example, we can take any of
the interaction functions c(u,v) described in section 26.3 and use it to construct a marked point
process.
Such processes can be constructed equivalently as follows [14]:
an unmarked Gibbs process is generated with first order term b(u) = ∑ m∈M b m(u) and
pairwise interaction c(u,v).
each point x i of this unmarked process is labelled with a mark m i with probability distribution
P{m i = m} = b i (x i )/b(x i ) independent of other points.
If additionally the intensity functions are constant, b m (u) ≡ β m , then such a point process
has the random labelling property.
34.1.4 Mark-dependent pairwise interactions
Various complex kinds of behaviour can be created by postulating a pairwise interaction that
does depend on the marks.
A simple example is the multitype hard core process in which β m (u) ≡ β and
{ 1 if ||u − v|| > rm,m ′
c m,m ′(u,v) =
0 if ||u − v|| ≤ r m,m ′
(53)
where r m,m ′ = r m ′ ,m > 0 is the hard core distance for type m with type m ′ . In this process, two
points of type m and m ′ respectively can never come closer than the distance r m,m ′.
By setting r m,m ′ = 0 for a particular pair of marks m,m ′ we effectively remove the interaction
term between points of these types. If there are only two types, say M = {1,2},
then setting r 1,2 = 0 implies that the sub-processes X 1 and X 2 , consisting of points of types
1 and 2 respectively, are independent point processes. In other words the process satisfies the
independence-of-components property.
The multitype Strauss process has pairwise interaction term
{ 1 if ||u − v|| > rm,m ′
c m,m ′(u,v) =
(54)
γ m,m ′ if ||u − v|| ≤ r m,m ′
where r m,m ′ > 0 are interaction radii as above, and γ m,m ′ ≥ 0 are interaction parameters.
In contrast to the unmarked Strauss process, which is well-defined only when its interaction
parameter γ is between 0 and 1, the multitype Strauss process allows some of the interaction
parameters γ m,m ′ to exceed 1 for m ≠ m ′ , provided one of the relevant types has a hard core
(γ m,m = 0 or γ m ′ ,m ′ = 0).
If there are only two types, say M = {1,2}, then setting γ 1,2 = 1 implies that the subprocesses
X 1 and X 2 , consisting of points of types 1 and 2 respectively, are independent Strauss
processes.
The multitype Strauss-hard core process has pairwise interaction term
⎧
⎨
c m,m ′(u,v) =
⎩
0 if ||u − v|| < h m,m ′
γ m,m ′ if h m,m ′ ≤ ||u − v|| ≤ r m,m ′
1 if ||u − v|| > r m,m ′
where r m,m ′ > 0 are interaction distances and γ m,m ′ ≥ 0 are interaction parameters as above,
and h m,m ′ are hard core distances satisfying h m,m ′ = h m ′ ,m and 0 < h m,m ′ < r m,m ′.
(55)
CopyrightCSIRO 2010