# Triplets

##### The Triplet Point Process Model

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

.

##### Value

- An object of class
`"interact"`

describing the interpoint interaction structure of the Triplets process with interaction radius $r$.

##### 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

##### Examples

```
Triplets(r=0.1)
# prints a sensible description of itself
ppm(cells, ~1, Triplets(r=0.1))
# fit the stationary Triplets process to `cells'
ppm(cells, ~polynom(x,y,3), Triplets(r=0.1))
# fit a nonstationary Triplets process with log-cubic polynomial trend
```

*Documentation reproduced from package spatstat, version 1.31-3, License: GPL (>= 2)*