# rNeymanScott

##### Simulate Neyman-Scott Process

Generate a random point pattern, a realisation of the Neyman-Scott cluster process.

##### Usage

`rNeymanScott(kappa, rmax, rcluster, win = owin(c(0,1),c(0,1)), ..., lmax=NULL)`

##### Arguments

- kappa
- Intensity of the Poisson process of cluster centres. A single positive number, a function, or a pixel image.
- rmax
- Maximum radius of a random cluster.
- rcluster
- A function which generates random clusters, or other data specifying the random cluster mechanism. See Details.
- win
- Window in which to simulate the pattern.
An object of class
`"owin"`

or something acceptable to`as.owin`

. - ...
- Arguments passed to
`rcluster`

- lmax
- Optional. Upper bound on the values of
`kappa`

when`kappa`

is a function or pixel image.

##### Details

This algorithm generates a realisation of the
general Neyman-Scott process, with the cluster mechanism
given by the function `rcluster`

.
The clusters must have a finite maximum possible radius `rmax`

.

First, the algorithm
generates a Poisson point process of ``parent'' points
with intensity `kappa`

. Here `kappa`

may be a single
positive number, a function `kappa(x, y)`

, or a pixel image
object of class `"im"`

(see `im.object`

).
See `rpoispp`

for details.
Second, each parent point is
replaced by a random cluster of points.
These clusters are combined together to yield a single point pattern
which is then returned as the result of `rNeymanScott`

.

The argument `rcluster`

specifies the cluster mechanism.
It may be either:

- A
`function`

which will be called to generate each random cluster (the offspring points of each parent point). The function should expect to be called in the form`rcluster(x0,y0,...)`

for a parent point at a location`(x0,y0)`

. The return value of`rcluster`

should specify the coordinates of the points in the cluster; it may be a list containing elements`x,y`

, or a point pattern (object of class`"ppp"`

). If it is a marked point pattern then the result of`rNeymanScott`

will be a marked point pattern. - A
`list(mu, f)`

where`mu`

specifies the mean number of offspring points in each cluster, and`f`

generates the random displacements (vectors pointing from the parent to the offspring). In this case, the number of offspring in a cluster is assumed to have a Poisson distribution, implying that the Neyman-Scott process is also a Cox process. The first element`mu`

should be either a single nonnegative number (interpreted as the mean of the Poisson distribution of cluster size) or a pixel image or a`function(x,y)`

giving a spatially varying mean cluster size (interpreted in the sense of Waagepetersen, 2007). The second element`f`

should be a function that will be called once in the form`f(n)`

to generate`n`

independent and identically distributed displacement vectors (i.e. as if there were a cluster of size`n`

with a parent at the origin`(0,0)`

). The function should return a point pattern (object of class`"ppp"`

) or something acceptable to`xy.coords`

that specifies the coordinates of`n`

points.

If required, the intermediate stages of the simulation (the parents
and the individual clusters) can also be extracted from
the return value of `rNeymanScott`

through the attributes `"parents"`

and `"parentid"`

.
The attribute `"parents"`

is the point pattern of parent points.
The attribute `"parentid"`

is an integer vector specifying
the parent for each of the points in the simulated pattern.

##### 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 Details.

##### References

Neyman, J. and Scott, E.L. (1958)
A statistical approach to problems of cosmology.
*Journal of the Royal Statistical Society, Series B*
**20**, 1--43.

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

##### See Also

`rpoispp`

,
`rThomas`

,
`rGaussPoisson`

,
`rMatClust`

,
`rCauchy`

,
`rVarGamma`

##### Examples

```
# each cluster consist of 10 points in a disc of radius 0.2
nclust <- function(x0, y0, radius, n) {
return(runifdisc(n, radius, centre=c(x0, y0)))
}
plot(rNeymanScott(10, 0.2, nclust, radius=0.2, n=5))
# multitype Neyman-Scott process (each cluster is a multitype process)
nclust2 <- function(x0, y0, radius, n, types=c("a", "b")) {
X <- runifdisc(n, radius, centre=c(x0, y0))
M <- sample(types, n, replace=TRUE)
marks(X) <- M
return(X)
}
plot(rNeymanScott(15,0.1,nclust2, radius=0.1, n=5))
```

*Documentation reproduced from package spatstat, version 1.25-5, License: GPL (>= 2)*