Simulate Thomas Process

Description

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

Usage

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  rThomas(kappa, scale, mu, win = owin(c(0,1),c(0,1)),
          nsim=1, drop=TRUE, 
          saveLambda=FALSE, expand = 4*scale, ...,
          poisthresh=1e-6)

Arguments

kappa

Intensity of the Poisson process of cluster centres. A single positive number, a function, or a pixel image.

scale

Standard deviation of random displacement (along each coordinate axis) of a point from its cluster centre.

mu

Mean number of points per cluster (a single positive number) or reference intensity for the cluster points (a function or a pixel image).

win

Window in which to simulate the pattern. An object of class "owin" or something acceptable to as.owin.

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.

saveLambda

Logical. If TRUE then the random intensity corresponding to the simulated parent points will also be calculated and saved, and returns as an attribute of the point pattern.

expand

Numeric. Size of window expansion for generation of parent points. Has a sensible default.

...

Passed to clusterfield to control the image resolution when saveLambda=TRUE and to clusterradius when expand is missing.

poisthresh

Numerical threshold below which the model will be treated as a Poisson process. See Details.

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 expand 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. Currently it is not possible to fit the Thomas model with inhomogeneous parents.

If the pair correlation function of the model is very close to that of a Poisson process, deviating by less than poisthresh, then the model is approximately a Poisson process, and will be simulated as a Poisson process with intensity kappa * mu, using rpoispp. This avoids computations that would otherwise require huge amounts of memory.

Value

A point pattern (an object of class "ppp") if nsim=1, or a list of point patterns if nsim > 1.

Additionally, some intermediate results of the simulation are returned as attributes of this point pattern (see rNeymanScott). Furthermore, the simulated intensity function is returned as an attribute "Lambda", if saveLambda=TRUE.

Author(s)

\adrian

and \rolf

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, rCauchy, rVarGamma, rNeymanScott, rGaussPoisson, kppm, clusterfit.

Examples

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  #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)

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