The (stationary) Geyer triplet process (Geyer, 1999)
with interaction radius \(r\) and
parameters \(\beta\) and \(\gamma\)
is the point process
in which each point contributes a factor \(\beta\) to the
probability density of the point pattern, and each triplet of close points
contributes a factor \(\gamma\) to the density.
A triplet of close points is a group of 3 points,
each pair of which is closer than \(r\) units
apart.
Thus the probability density is
$$
f(x_1,\ldots,x_n) =
\alpha \beta^{n(x)} \gamma^{s(x)}
$$
where \(x_1,\ldots,x_n\) represent the
points of the pattern, \(n(x)\) is the number of points in the
pattern, \(s(x)\) is the number of unordered triples of
points that are closer than \(r\) units apart,
and \(\alpha\) is the normalising constant.
The interaction parameter \(\gamma\) must be less than
or equal to \(1\)
so that this model describes an ``ordered'' or ``inhibitive'' pattern.
The nonstationary Triplets process is similar except that
the contribution of each individual point \(x_i\)
is a function \(\beta(x_i)\)
of location, rather than a constant beta.
The function ppm(), which fits point process models to
point pattern data, requires an argument
of class "interact" describing the interpoint interaction
structure of the model to be fitted.
The appropriate description of the Triplets process pairwise interaction is
yielded by the function Triplets(). See the examples below.
Note the only argument is the interaction radius r.
When r is fixed, the model becomes an exponential family.
The canonical parameters \(\log(\beta)\)
and \(\log(\gamma)\)
are estimated by ppm(), not fixed in
Triplets().