spatstat (version 1.23-1)

rMatClust: Simulate Matern Cluster Process

Description

Generate a random point pattern, a simulated realisation of the Mat'ern Cluster Process.

Usage

rMatClust(kappa, r, mu, win = owin(c(0,1),c(0,1)))

Arguments

kappa

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

r

Radius parameter of the clusters.

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.

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.

Details

This algorithm generates a realisation of Mat'ern's cluster process inside the window win. The process is constructed by first generating a Poisson point process of ``parent'' points with intensity kappa. Then each parent point is replaced by a random cluster of points, the number of points in each cluster being random with a Poisson (mu) distribution, and the points being placed independently and uniformly inside a disc of radius r centred on the parent point.

In this implementation, parent points are not restricted to lie in the window; the parent process is effectively the uniform Poisson process on the infinite plane.

This classical model can be fitted to data by the method of minimum contrast, using matclust.estK or kppm. The algorithm can also generate spatially inhomogeneous versions of the Mat'ern cluster process:

• The parent points can be spatially inhomogeneous. If the argumentkappais afunction(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 argumentmuis afunction(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 (2006). For a given parent point, the offspring constitute a Poisson process with intensity function equal to theaveragevalue ofmuinside the disc of radiusrcentred on the parent point, and zero intensity outside this disc.

When the parents are homogeneous (kappa is a single number) and the offspring are inhomogeneous (mu is a function or pixel image), the model can be fitted to data using kppm, or using matclust.estK applied to the inhomogeneous $K$ function.

References

Mat'ern, B. (1960) Spatial Variation. Meddelanden fraan Statens Skogsforskningsinstitut, volume 59, number 5. Statens Skogsforskningsinstitut, Sweden.

Mat'ern, B. (1986) Spatial Variation. Lecture Notes in Statistics 36, Springer-Verlag, New York.

Waagepetersen, R. (2006) An estimating function approach to inference for inhomogeneous Neyman-Scott processes. Submitted for publication.

See Also

rpoispp, rThomas, rGaussPoisson, rNeymanScott, matclust.estK, kppm.

Examples

# homogeneous
 X <- rMatClust(10, 0.05, 4)
 # inhomogeneous
 Z <- as.im(function(x,y){ 4 * exp(2 * x - 1) }, owin())
 Y <- rMatClust(10, 0.05, Z)