Analysing spatial point patterns in R - CSIRO

26.4 Higher-order interactions 159
26.3.3 Other pairwise interaction models
Other pairwise interactions that are considered in spatstat include the Strauss-hard core interaction
(with hard core distance h > 0 and interaction distance r > h)
⎧
⎨ 0 if ||u − v|| ≤ h
c(u,v) = γ if h < ||u − v|| ≤ r ,
⎩
1 if ||u − v|| > r
the soft-core interaction (with scale σ > 0 and index 0 < κ < 1)
( )
σ 2/κ
c(u,v) =
,
||u − v||
the Diggle-Gates-Stibbard interaction (with interaction range ρ)
{ ( ) 2
sin π||u−v||
c(u,v) = 2ρ
if ||u − v|| ≤ ρ
1 if ||u − v|| > ρ ,
the Diggle-Gratton interaction (with hard core distance δ, interaction distance ρ and index κ)
⎧
⎪⎨ (
0
)
if ||u − v|| ≤ δ
κ
c(u,v) = ||u−v||−δ
ρ−δ
if δ < ||u − v|| ≤ ρ ,
⎪⎩
1 if ||u − v|| > ρ
and the general piecewise constant interaction in which c(||u − v||) is a step function of ||u − v||.
interaction
0.0 0.2 0.4 0.6 0.8 1.0
Piecewise constant interaction
0.00 0.05 0.10 0.15 0.20
26.4 Higher-order interactions
There are some useful Gibbs point process models which exhibit interactions of higher order,
that is, in which the probability density has contributions from m-tuples of points, where m > 2.
One example is the area-interaction or Widom-Rowlinson process [17] with probability density
f(x) = αβ n(x) γ −A(x) (38)
where α is the normalising constant, β > 0 is an intensity parameter, and γ > 0 is an interaction
parameter. Here A(x) denotes the area of the region obtained by drawing a disc of radius r
centred at each point x i , and taking the union of these discs. The value γ = 1 again corresponds
to a Poisson process, while γ < 1 produces a regular process and γ > 1 a clustered process.
This process has interactions of all orders. It can be used as a model for moderate regularity or
clustering.
CopyrightCSIRO 2010