# rMatClust

##### Simulate Matern Cluster Process

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`

.

##### 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 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 (2006). For a given parent point, the offspring constitute a Poisson process with intensity function equal to the*average*value of`mu`

inside the disc of radius`r`

centred on the parent point, and zero intensity outside this disc.

`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.
##### 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

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)
```

*Documentation reproduced from package spatstat, version 1.16-3, License: GPL (>= 2)*