spatstat (version 1.41-1)

Triplets: The Triplet Point Process Model

Description

Creates an instance of Geyer's triplet interaction point process model which can then be fitted to point pattern data.

Usage

Triplets(r)

Arguments

r

The interaction radius of the Triplets process

Value

• An object of class "interact" describing the interpoint interaction structure of the Triplets process with interaction radius $r$.

Details

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 $$f(x_1,\dots ,x_n)=\alpha {\beta }^{n(x)}{\gamma }^{s(x)}$$ 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().

References

Geyer, C.J. (1999) Likelihood Inference for Spatial Point Processes. Chapter 3 in O.E. Barndorff-Nielsen, W.S. Kendall and M.N.M. Van Lieshout (eds) Stochastic Geometry: Likelihood and Computation, Chapman and Hall / CRC, Monographs on Statistics and Applied Probability, number 80. Pages 79--140.

See Also

ppm, triplet.family, ppm.object

Examples

Triplets(r=0.1)
   # prints a sensible description of itself

   ppm(cells, ~1, Triplets(r=0.2))
   # fit the stationary Triplets process to `cells'

   ppm(cells, ~polynom(x,y,3), Triplets(r=0.2))
   # fit a nonstationary Triplets process with log-cubic polynomial trend