Generate a random pattern of points, a simulated realisation of the Penttinen process, using a perfect simulation algorithm.
rPenttinen(beta, gamma=1, R, W = owin(), expand=TRUE, nsim=1, drop=TRUE)intensity parameter (a positive number).
Interaction strength parameter (a number between 0 and 1).
disc radius (a non-negative number).
window (object of class "owin") in which to
generate the random pattern.
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.
Number of simulated realisations to be generated.
Logical. If nsim=1 and drop=TRUE (the default), the
result will be a point pattern, rather than a list
containing a point pattern.
If nsim = 1, a point pattern (object of class "ppp").
If nsim > 1, a list of point patterns.
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.
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.
# NOT RUN {
X <- rPenttinen(50, 0.5, 0.02)
# }
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