R: The Triplet Point Process Model

Triplets {spatstat.core} | R Documentation

## 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

### 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,…,x_n) = alpha . beta^n(x) gamma^s(x)

where x[1],…,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()`.

### Value

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

### Author(s)

Adrian Baddeley Adrian.Baddeley@curtin.edu.au

and Rolf Turner r.turner@auckland.ac.nz

### 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

[Package spatstat.core version 2.4-4 Index]