# rMatClust

##### Simulate Matern Cluster Process

Generate a random point pattern, a simulated realisation of the

##### 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
`win`

.

In the simplest case, where `kappa`

and `mu`

are single numbers, the algorithm
generates a uniform Poisson point process of `kappa`

. Then each parent point is
replaced by a random cluster of `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
`win`

.
This point process has intensity `kappa * mu`

.

The algorithm can also generate spatially inhomogeneous versions of
the

- 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/(pi * r^2)`

inside the disc of radius`r`

centred 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 of`mu`

) placed independently and uniformly in the disc of radius`r`

centred on 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 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 `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 `kappa`

is a single number)
can be fitted to data using `kppm`

or related functions.
Currently it is not possible to fit the

##### 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

*Spatial Variation*.
Meddelanden

*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.

##### See Also

`rpoispp`

,
`rThomas`

,
`rGaussPoisson`

,
`rNeymanScott`

,
`matclust.estK`

,
`matclust.estpcf`

,
`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 1.31-3, License: GPL (>= 2)*