# rCauchy

##### Simulate Neyman-Scott Point Process with Cauchy cluster kernel

Generate a random point pattern, a simulated realisation of the Neyman-Scott process with Cauchy cluster kernel.

##### Usage

`rCauchy(kappa, omega, mu, win = owin(), eps = 0.001)`

##### Arguments

- kappa
- Intensity of the Poisson process of cluster centres. A single positive number, a function, or a pixel image.
- omega
- Scale parameter for cluster kernel. Determines the size of clusters. A positive number, in the same units as the spatial coordinates.
- 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`

. - eps
- Threshold below which the values of the cluster kernel will be treated as zero for simulation purposes.

##### Details

This algorithm generates a realisation of the Neyman-Scott process
with Cauchy cluster kernel, 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
according to a Cauchy kernel.

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 model can be fitted to data by the method of minimum contrast,
using `cauchy.estK`

, `cauchy.estpcf`

or `kppm`

.
The algorithm can also generate spatially inhomogeneous versions of
the 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).

`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 `cauchy.estK`

or `cauchy.estpcf`

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

Ghorbani, M. (2012) Cauchy cluster process.
*Metrika*, to appear.

Jalilian, A., Guan, Y. and Waagepetersen, R. (2011)
Decomposition of variance for spatial Cox processes.
*Scandinavian Journal of Statistics* **40**, 119-137.

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

##### See Also

##### Examples

```
# homogeneous
X <- rCauchy(30, 0.01, 5)
# inhomogeneous
Z <- as.im(function(x,y){ exp(2 - 3 * x) }, W= owin())
Y <- rCauchy(50, 0.01, Z)
```

*Documentation reproduced from package spatstat, version 1.34-1, License: GPL (>= 2)*