Perfect Simulation of the Strauss Process
Generate a random pattern of points, a simulated realisation of the Strauss process, using a perfect simulation algorithm.
rStrauss(beta, gamma = 1, R = 0, W = owin(), expand=TRUE, nsim=1, drop=TRUE)
beta |
intensity parameter (a positive number). |
gamma |
interaction parameter (a number between 0 and 1, inclusive). |
R |
interaction radius (a non-negative number). |
W |
window (object of class |
expand |
Logical. If |
nsim |
Number of simulated realisations to be generated. |
drop |
Logical. If |
This function generates a realisation of the
Strauss point process in the window W
using a ‘perfect simulation’ algorithm.
The Strauss process (Strauss, 1975; Kelly and Ripley, 1976)
is a model for spatial inhibition, ranging from
a strong ‘hard core’ inhibition to a completely random pattern
according to the value of gamma.
The Strauss process with interaction radius R and parameters beta and gamma is the pairwise interaction point process with probability density
f(x_1,…,x_n) = alpha . beta^n(x) gamma^s(x)
where x[1],…,x[n] represent the points of the pattern, n(x) is the number of points in the pattern, s(x) is the number of distinct unordered pairs of points that are closer than R units apart, and alpha is the normalising constant. Intuitively, each point of the pattern contributes a factor beta to the probability density, and each pair of points closer than r units apart contributes a factor gamma to the density.
The interaction parameter gamma must be less than or equal to 1 in order that the process be well-defined (Kelly and Ripley, 1976). This model describes an “ordered” or “inhibitive” pattern. If gamma=1 it reduces to a Poisson process (complete spatial randomness) with intensity beta. If gamma=0 it is called a “hard core process” with hard core radius R/2, since no pair of points is permitted to lie closer than R units apart.
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.
If nsim = 1, a point pattern (object of class "ppp").
If nsim > 1, a list of point patterns.
Kasper Klitgaard Berthelsen, adapted for spatstat by Adrian Baddeley Adrian.Baddeley@curtin.edu.au
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.
Kelly, F.P. and Ripley, B.D. (1976) On Strauss's model for clustering. Biometrika 63, 357–360.
Moller, J. and Waagepetersen, R. (2003). Statistical Inference and Simulation for Spatial Point Processes. Chapman and Hall/CRC.
Strauss, D.J. (1975) A model for clustering. Biometrika 62, 467–475.
X <- rStrauss(0.05,0.2,1.5,square(141.4)) Z <- rStrauss(100,0.7,0.05, nsim=2)
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