# rMatClust

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##### Simulate Matern Cluster Process

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

Keywords
spatial, datagen
##### 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.
##### Details

This algorithm generates a realisation of latex{Mat'ern}{Matern}'s cluster 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 placed and uniformly inside a disc of radius r centred on the parent point. The resulting point pattern is a realisation of the classical stationary Matern cluster process generated inside the window win. This point process has intensity kappa * mu.

The algorithm can also generate spatially inhomogeneous versions of the latex{Mat'ern}{Matern} 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 (2007). For a given parent point, the offspring constitute a Poisson process with intensity function equal tomu/(pi * r^2)inside the disc of radiusrcentred on the parent point, and zero intensity outside this disc. Equivalently we first generate, for each parent point, a Poisson ($M$) random number of offspring (where$M$is the maximum value ofmu) placed independently and uniformly in the disc of radiusrcentred on the parent location, and then randomly thin the offspring points, with retention probabilitymu/M.
• Both the parent points and the offspring points can be 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 r 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 latex{Mat'ern}{Matern} cluster 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 r.

The latex{Mat'ern}{Matern} cluster process model 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 latex{Mat'ern}{Matern} cluster process 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.

##### References

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

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

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

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

• rMatClust
##### 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)
Documentation reproduced from package spatstat, version 1.31-2, License: GPL (>= 2)

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