rHardcore.sphwin: Simulation of the Hard Core Process

In baddstats/spherstat: Spatial Point Pattern Analysis, Geometry and Trigonometry for the Sphere

Simulation of the Hard Core Process

Description

Generate a random pattern of points, a simulated realisation of the hard core process, using the Metropolis-Hastings simulation algorithm.

Usage

  rHardcore.sphwin(beta, R=0, p=0.5, m=100, win = sphwin(type="sphere"),
 proper=TRUE, nsim=1, drop=TRUE, expand=TRUE, as.sp=TRUE, ndim="2")

Arguments

beta	intensity parameter (a positive number).
R	interaction parameter (a number between 0 and pi*rho, inclusive, where rho is the radius of the sphere that to simulate the point pattern on.).
p	Metropolis-Hastings parameter: The conditional probability of proposing a birth given that a birth or death is proposed.
m	Metropolis-Hastings parameter: The number of proposals that should be run.
win	The window, an object of type sphwin.
proper	Logical. If TRUE, then m is ignored, and the Metropolis-Hastings algorithm continues through as many iterations as is needed to achieve a hard core process. See Details.
nsim	Number of simulated realisations to be generated.
drop	Logical. If nsim=1 and drop=TRUE (the default), the result will be a point pattern, rather than a list containing a point pattern.
expand	Logical. If TRUE (the default), the process is simulated on the entire sphere, and the output is only those points in the window defined by win. If FALSE, the process is simulated within the window defined by win.
as.sp	Logical. If TRUE, returns an object of class as defined by sp.dim. Otherwise, returns a matrix. See Value.
ndim	A string, taking value "2" or "3". Specifies whether the object should contain the locations of the points in spherical coordinates (ndim="2") or Cartesian coordinates (ndim="3").

Details

rHardcore.sphwin uses the Metropolis Hasting algorithm to generate a realisation of a hard core process inside the window win.

The hard core process is a model for spatial inhibition, in which no points can occur within a distance R of each other; with interaction radius R and parameter beta, it is the pairwise interaction point process with probability density

f(x[1], ..., x[n]) = alpha . beta^(n(x))

where x[1], ..., x[n] represent the points of the pattern, n(x) is the number of points in the pattern 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 Hardcore process may also be generated using rStrauss.sphwin, where gamma=0, but a warning message is created advising that such a process is hard core rather than Strauss.

The Metropolis-Hastings algorithm is a Markov Chain, whose states are spatial point patterns, and whose limiting distribution is the desired point process. After running the algorithm for a very large number of iterations, we may regard the state of the algorithm as a realisation from the desired point process.

However, there are difficulties in deciding whether the algorithm has run for “long enough”. The convergence of the algorithm may indeed be extremely slow. No guarantees of convergence are given! Hence, the argument proper is provided; if proper=FALSE, then rHardcore.sphwin will perform m iterations of the Metropolis-Hastings algorithm and give the resulting point pattern. completed. Otherwise, (if proper=TRUE), then rHardcore.sphwin ignores m is ignored and as many iterations of the Metropolis-Hastings algorithm are performed as is needed to attain a true hard core process (i.e. with d(x, X \ {x}) >= R for all x in X.

While it is fashionable to decry the Metropolis-Hastings algorithm for its poor convergence and other properties, it has the advantage of being easy to implement for a wide range of models.

Value

If nsim=1 and drop=FALSE then a single item as described below; otherwise a list containing nsim items.

An item is determined by the values of as.sp and ndim:

If as.sp=FALSE and ndim="2", a two column matrix giving the locations of the simulated points.

If as.sp=FALSE and ndim="3", a three column matrix giving the locations of the simulated points.

If as.sp=TRUE and ndim="2", an object of class sp2 giving the locations of the simulated points.

If as.sp=TRUE and ndim="3", an object of class sp3 giving the locations of the simulated points.

Note

This function is the analogue for point processes on the sphere of the function rmh in spatstat when used in the third example in rmh.default, which is the corresponding function for point processes in R^2. Hence elements of this help page have been taken from those for rmh and rHardcore (also in spatstat), with the permission of A. J. Baddeley. This enables the code to be highly efficient and give corresponding output to rmh, and for the information on this help page to be consistent with that for rmh and rHardcore. It is hoped that this will minimise or remove any confusion for users of both spatstat and spherstat.

Author(s)

Tom Lawrence [e-mail]

References

Lawrence, T.J. (2017) Master's Thesis, University of Western Australia.

See Also

rHardcore, rMatClust.sphwin, rMaternI.sphwin, rMaternII.sphwin, rpoispp.sphwin, rStrauss.sphwin, rThomas.sphwin

Examples

rH1 <- rHardcore.sphwin(beta=5, R=0.04*pi, p=0.5, m=100, win=sphwin())
rH1

baddstats/spherstat documentation built on Feb. 6, 2023, 1:45 a.m.