# rDiggleGratton

##### Perfect Simulation of the Diggle-Gratton Process

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

##### Usage

```
rDiggleGratton(beta, delta, rho, kappa=1, W = owin(),
expand=TRUE, nsim=1, drop=TRUE)
```

##### Arguments

- beta
intensity parameter (a positive number).

- delta
hard core distance (a non-negative number).

- rho
interaction range (a number greater than

`delta`

).- kappa
interaction exponent (a non-negative number).

- W
window (object of class

`"owin"`

) in which to generate the random pattern. Currently this must be a rectangular window.- 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
Diggle-Gratton point process in the window `W`

using a ‘perfect simulation’ algorithm.

Diggle and Gratton (1984, pages 208-210) introduced the pairwise interaction point process with pair potential \(h(t)\) of the form $$ h(t) = \left( \frac{t-\delta}{\rho-\delta} \right)^\kappa \quad\quad \mbox{ if } \delta \le t \le \rho $$ with \(h(t) = 0\) for \(t < \delta\) and \(h(t) = 1\) for \(t > \rho\). Here \(\delta\), \(\rho\) and \(\kappa\) are parameters.

Note that we use the symbol \(\kappa\) where Diggle and Gratton (1984) use \(\beta\), since in spatstat we reserve the symbol \(\beta\) for an intensity parameter.

The parameters must all be nonnegative, and must satisfy \(\delta \le \rho\).

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.

Diggle, P.J. and Gratton, R.J. (1984)
Monte Carlo methods of inference for implicit statistical models.
*Journal of the Royal Statistical Society, series B*
**46**, 193 -- 212.

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

##### See Also

##### Examples

```
# NOT RUN {
X <- rDiggleGratton(50, 0.02, 0.07)
# }
```

*Documentation reproduced from package spatstat, version 1.57-1, License: GPL (>= 2)*