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)