Simulate Thomas Process

Description

Generate a random point pattern, a realisation of the Thomas cluster process.

Usage

 rThomas(kappa, sigma, mu, win = owin(c(0,1),c(0,1)))

Arguments

Details

This algorithm generates a realisation of the (‘modified’) Thomas process, a special case of the Neyman-Scott process, inside the window win.

In the simplest case, where kappa and mu are single numbers, the algorithm generates a uniform Poisson point process of “parent” points with intensity kappa. Then each parent point is replaced by a random cluster of “offspring” points, the number of points per cluster being Poisson (mu) distributed, and their positions being isotropic Gaussian displacements from the cluster parent location. The resulting point pattern is a realisation of the classical “stationary Thomas process” generated inside the window win. This point process has intensity kappa * mu.

The algorithm can also generate spatially inhomogeneous versions of the Thomas process:

• The parent points can be spatially inhomogeneous. If the argument kappa is a function(x,y) or a pixel image (object of class "im"), then it is taken as specifying the intensity function of an inhomogeneous Poisson process that generates the parent points.

• The offspring points can be inhomogeneous. If the argument mu is a function(x,y) or a pixel image (object of class "im"), then it is interpreted as the reference density for offspring points, in the sense of Waagepetersen (2007). For a given parent point, the offspring constitute a Poisson process with intensity function equal to mu * f, where f is the Gaussian probability density centred at the parent point. Equivalently we first generate, for each parent point, a Poisson (mumax) random number of offspring (where M is the maximum value of mu) with independent Gaussian displacements from the parent location, and then randomly thin the offspring points, with retention probability mu/M.

• Both the parent points and the offspring points can be spatially inhomogeneous, as described above.

Note that if kappa is a pixel image, its domain must be larger than the window win. This is because an offspring point inside win could have its parent point lying outside win. In order to allow this, the simulation algorithm first expands the original window win by a distance 4 * sigma and generates the Poisson process of parent points on this larger window. If kappa is a pixel image, its domain must contain this larger window.

The intensity of the Thomas process is kappa * mu if either kappa or mu is a single number. In the general case the intensity is an integral involving kappa, mu and f.

The Thomas process with homogeneous parents (i.e. where kappa is a single number) can be fitted to data using kppm or related functions. Currently it is not possible to fit the Thomas model with inhomogeneous parents.

Value

The simulated point pattern (an object of class "ppp").

Additionally, some intermediate results of the simulation are returned as attributes of this point pattern. See rNeymanScott.

Author(s)

Adrian Baddeley Adrian.Baddeley@uwa.edu.au http://www.maths.uwa.edu.au/~adrian/ and Rolf Turner r.turner@auckland.ac.nz

References

Diggle, P. J., Besag, J. and Gleaves, J. T. (1976) Statistical analysis of spatial point patterns by means of distance methods. Biometrics 32 659–667.

Thomas, M. (1949) A generalisation of Poisson's binomial limit for use in ecology. Biometrika 36, 18–25.

Waagepetersen, R. (2007) An estimating function approach to inference for inhomogeneous Neyman-Scott processes. Biometrics 63, 252–258.

See Also

rpoispp, rMatClust, rGaussPoisson, rNeymanScott, thomas.estK, thomas.estpcf, kppm

Examples

  #homogeneous
  X <- rThomas(10, 0.2, 5)
  #inhomogeneous
  Z <- as.im(function(x,y){ 5 * exp(2 * x - 1) }, owin())
  Y <- rThomas(10, 0.2, Z)