spatstat (version 1.20-2)

rStrauss: Perfect Simulation of the Strauss Process

Description

Generate a random pattern of points, a simulated realisation of the Strauss process, using a perfect simulation algorithm.

Usage

rStrauss(beta, gamma = 1, R = 0, W = owin())

Arguments

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 "owin") in which to generate the random pattern. Currently this must be a rectangular window.

Value

• A point pattern (object of class "ppp").

Details

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 $$f(x_1,\dots ,x_n)=\alpha {\beta }^{n(x)}{\gamma }^{s(x)}$$ where $x_1,\ldots,x_n$ 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).

The implementation is currently experimental. 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.

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.

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 63, 467--475.

See Also

rmh

Examples

X <- rStrauss(0.05,0.2,1.5,square(141.4))
   Z <- rStrauss(100,0.7,0.05)