Analysing spatial point patterns in R - CSIRO

27.2 Fitting Gibbs models in spatstat 163
all other parameters are ‘irregular’ parameters. For example, the Strauss process conditional
intensity (42) can be recast as
log λ(u,x) = log β + (log γ)t(u,x)
so that θ = (log β,log γ) are regular parameters, but the interaction distance r is an irregular
parameter (technically called a ‘bloody nuisance parameter’).
In spatstat we split the conditional intensity into first-order and higher-order terms:
log λ θ (u,x) = η · S(u) + ϕ · V (u,x). (46)
The ‘first order term’ S(u) describes spatial inhomogeneity and/or covariate effects. The ‘higher
order term’ V (u,x) describes interpoint interaction.
The model with conditional intensity (46) is fitted by calling ppm in the form
ppm(X, ~ terms, V)
The first argument X is the point pattern dataset. The second argument ~terms is a model
formula, specifying the first order term S(u) in (46), in the manner described in Section 15.
Thus the first order term S(u) in (46) may take very general forms.
The third argument V is an object of the special class "interact" which describes the
interpoint interaction term V (u,x) in (46). It may be compared to the ‘family’ argument
which determines the distribution of the responses in a linear model or generalised linear model.
Only a limited number of canned interactions are available in spatstat, because they must be
constructed carefully to ensure that the point process exists.
To fit the Strauss process to the cells data using ppm,
> data(cells)
> ppm(cells, ~1, Strauss(r = 0.1))
Stationary Strauss process
First order term:
beta
762.6005
Interaction: Strauss process
interaction distance: 0.1
Fitted interaction parameter gamma: 0.008
Relevant coefficients:
Interaction
-4.825006
Here Strauss is a special function that creates an ‘interaction’ object (class "interact")
describing the interaction structure of the Strauss process. Notice that we had to specify the
value of the irregular parameter r (more about that later).
To fit the inhomogeneous Strauss process with conditional intensity
λ(u,x) = b(u)γ t(u,x)
where, say, b(u) is loglinear in the Cartesian coordinates,
we simply type
log b((x,y)) = β 0 + β 1 x + β 2 y
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