# rPenttinen

##### Perfect Simulation of the Penttinen Process

Generate a random pattern of points, a simulated realisation of the Penttinen process, using a perfect simulation algorithm.

##### Usage

`rPenttinen(beta, gamma=1, R, W = owin(), expand=TRUE, nsim=1, drop=TRUE)`

##### Arguments

- beta
intensity parameter (a positive number).

- gamma
Interaction strength parameter (a number between 0 and 1).

- R
disc radius (a non-negative number).

- W
window (object of class

`"owin"`

) in which to generate the random pattern.- expand
Logical. If

`FALSE`

, simulation is performed in the window`W`

, which must be rectangular. If`TRUE`

(the default), simulation is performed on a larger window, and the result is clipped to the original window`W`

. Alternatively`expand`

can be an object of class`"rmhexpand"`

(see`rmhexpand`

) determining the expansion method.- nsim
Number of simulated realisations to be generated.

- drop
Logical. If

`nsim=1`

and`drop=TRUE`

(the default), the result will be a point pattern, rather than a list containing a point pattern.

##### Details

This function generates a realisation of the
Penttinen point process in the window `W`

using a ‘perfect simulation’ algorithm.

Penttinen (1984, Example 2.1, page 18), citing Cormack (1979), described the pairwise interaction point process with interaction factor $$ h(d) = e^{\theta A(d)} = \gamma^{A(d)} $$ between each pair of points separated by a distance $d$. Here \(A(d)\) is the area of intersection between two discs of radius \(R\) separated by a distance \(d\), normalised so that \(A(0) = 1\).

The simulation algorithm used to generate the point pattern
is ‘dominated coupling from the past’
as implemented by
Berthelsen and Moller (2002, 2003).
This is a ‘perfect simulation’ or ‘exact simulation’
algorithm, so called because the output of the algorithm is guaranteed
to have the correct probability distribution exactly (unlike the
Metropolis-Hastings algorithm used in `rmh`

, whose output
is only approximately correct).

There is a tiny chance that the algorithm will run out of space before it has terminated. If this occurs, an error message will be generated.

##### Value

If `nsim = 1`

, a point pattern (object of class `"ppp"`

).
If `nsim > 1`

, a list of point patterns.

##### References

Berthelsen, K.K. and Moller, J. (2002)
A primer on perfect simulation for spatial point processes.
*Bulletin of the Brazilian Mathematical Society* 33, 351-367.

Berthelsen, K.K. and Moller, J. (2003)
Likelihood and non-parametric Bayesian MCMC inference
for spatial point processes based on perfect simulation and
path sampling.
*Scandinavian Journal of Statistics* 30, 549-564.

Cormack, R.M. (1979)
Spatial aspects of competition between individuals.
Pages 151--212 in *Spatial and Temporal Analysis in Ecology*,
eds. R.M. Cormack and J.K. Ord, International Co-operative
Publishing House, Fairland, MD, USA.

Moller, J. and Waagepetersen, R. (2003).
*Statistical Inference and Simulation for Spatial Point Processes.*
Chapman and Hall/CRC.

Penttinen, A. (1984)
*Modelling Interaction in Spatial Point Patterns:
Parameter Estimation by the Maximum Likelihood Method.*
Jyvaskyla Studies in Computer Science, Economics and Statistics **7**,
University of Jyvaskyla, Finland.

##### See Also

##### Examples

```
# NOT RUN {
X <- rPenttinen(50, 0.5, 0.02)
Z <- rPenttinen(50, 0.5, 0.01, nsim=2)
# }
```

*Documentation reproduced from package spatstat, version 1.63-0, License: GPL (>= 2)*