# 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())`

##### 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.

##### Details

This function generates a realisation of the
Diggle-Gratton point process in the window `W`

using a

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

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

The simulation algorithm used to generate the point pattern
is `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

- A point pattern (object of class
`"ppp"`

).

##### 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

`X <- rDiggleGratton(50, 0.02, 0.07)`

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