# rDiggleGratton

0th

Percentile

##### 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.

Keywords
spatial, datagen
##### 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 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

• 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.

rmh, DiggleGratton, rStrauss, rHardcore.
X <- rDiggleGratton(50, 0.02, 0.07)