# 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, scale, mu, win = owin(), thresh = 0.001,
nsim=1, drop=TRUE,
saveLambda=FALSE, expand = NULL, …,
poisthresh=1e-6, saveparents=TRUE)
```

##### Arguments

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

- scale
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`

.- thresh
Threshold relative to the cluster kernel value at the origin (parent location) determining when the cluster kernel will be treated as zero for simulation purposes. Will be overridden by argument

`expand`

if that is given.- nsim
Number of simulated realisations to be generated.

- drop
Logical. If

`nsim=1`

and`drop=TRUE`

(the default), the result will be a point pattern, rather than a list containing a point pattern.- saveLambda
Logical. If

`TRUE`

then the random intensity corresponding to the simulated parent points will also be calculated and saved, and returns as an attribute of the point pattern.- expand
Numeric. Size of window expansion for generation of parent points. By default determined by calling

`clusterradius`

with the numeric threshold value given in`thresh`

.- …
Passed to

`clusterfield`

to control the image resolution when`saveLambda=TRUE`

and to`clusterradius`

when`expand`

is missing or`NULL`

.- poisthresh
Numerical threshold below which the model will be treated as a Poisson process. See Details.

- saveparents
Logical value indicating whether to save the locations of the parent points as an attribute.

##### 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,
maximum composite likelihood or Palm likelihood using
`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).

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`

.

If the pair correlation function of the model is very close
to that of a Poisson process, deviating by less than
`poisthresh`

, then the model is approximately a Poisson process,
and will be simulated as a Poisson process with intensity
`kappa * mu`

, using `rpoispp`

.
This avoids computations that would otherwise require huge amounts
of memory.

##### Value

A point pattern (an object of class `"ppp"`

) if `nsim=1`

,
or a list of point patterns if `nsim > 1`

.

Additionally, some intermediate results of the simulation are returned
as attributes of this point pattern (see
`rNeymanScott`

). Furthermore, the simulated intensity
function is returned as an attribute `"Lambda"`

, if
`saveLambda=TRUE`

.

##### References

Ghorbani, M. (2013) Cauchy cluster process.
*Metrika* **76**, 697-706.

Jalilian, A., Guan, Y. and Waagepetersen, R. (2013)
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

`rpoispp`

,
`rMatClust`

,
`rThomas`

,
`rVarGamma`

,
`rNeymanScott`

,
`rGaussPoisson`

,
`kppm`

,
`clusterfit`

.

##### Examples

```
# NOT RUN {
# homogeneous
X <- rCauchy(30, 0.01, 5)
# inhomogeneous
ff <- function(x,y){ exp(2 - 3 * abs(x)) }
Z <- as.im(ff, W= owin())
Y <- rCauchy(50, 0.01, Z)
YY <- rCauchy(ff, 0.01, 5)
# }
```

*Documentation reproduced from package spatstat, version 1.55-1, License: GPL (>= 2)*