# 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
`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.

Neyman-Scott models where `kappa`

is a single number
and `rcluster = list(mu,f)`

can be fitted to data
using the function `kppm`

.

##### 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.

##### Inhomogeneous Neyman-Scott Processes

There are several different ways of specifying a spatially inhomogeneous Neyman-Scott process:

- The point process of parent points can be 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 according to which the parent points are generated. - The number of points in a typical cluster can
be spatially varying.
If the argument
`rcluster`

is a list of two elements`mu, f`

and the first entry`mu`

is a`function(x,y)`

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

), then`mu`

is interpreted as the reference intensity 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(x, y) * g(x-x0, y-y0)`

where`g`

is the probability density of the offspring displacements generated by the function`f`

.Equivalently, clusters are first generated with a constant expected number of points per cluster: the constant is

`mumax`

, the maximum of`mu`

. Then the offspring are randomly*thinned*(see`rthin`

) with spatially-varying retention probabilities given by`mu/mumax`

. - The entire mechanism for generating a cluster can
be dependent on the location of the parent point.
If the argument
`rcluster`

is a function, then the cluster associated with a parent point at location`(x0,y0)`

will be generated by calling`rcluster(x0, y0, ...)`

. The behaviour of this function could depend on the location`(x0,y0)`

in any fashion.

Note that if `kappa`

is an
image, the spatial domain covered by this image must be large
enough to include the *expanded* window in which the parent
points are to be generated. This expanded window consists of
`as.rectangle(win)`

extended by the amount `rmax`

in each direction. This requirement means that `win`

must
be small enough so that the expansion of `as.rectangle(win)`

is contained in the spatial domain of `kappa`

. As a result,
one may wind up having to simulate the process in a window smaller
than what is really desired.

In the first two cases, the intensity of the Neyman-Scott process
is equal to `kappa * mu`

if at least one of `kappa`

or
`mu`

is a single number, and is otherwise equal to an
integral involving `kappa`

, `mu`

and `f`

.

##### 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 1.34-1, License: GPL (>= 2)*