Generate a random point pattern, a simulated realisation of the Neyman-Scott process with Variance Gamma (Bessel) cluster kernel.

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

kappa

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

nu

Shape parameter for the cluster kernel. A number greater than -1.

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.

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`

.

This algorithm generates a realisation of the Neyman-Scott process
with Variance Gamma (Bessel) 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 Variance Gamma kernel.

The shape of the kernel is determined by the dimensionless
index `nu`

. This is the parameter
\(\nu^\prime = \alpha/2-1\) appearing in
equation (12) on page 126 of Jalilian et al (2013).

The scale of the kernel is determined by the argument `scale`

,
which is the parameter
\(\eta\) appearing in equations (12) and (13) of
Jalilian et al (2013).
It is expressed in units of length (the same as the unit of length for
the window `win`

).

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`

, or using `vargamma.estK`

or `vargamma.estpcf`

applied to the inhomogeneous \(K\) function.

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.

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.

# NOT RUN { # homogeneous X <- rVarGamma(30, 2, 0.02, 5) # inhomogeneous ff <- function(x,y){ exp(2 - 3 * abs(x)) } Z <- as.im(ff, W= owin()) Y <- rVarGamma(30, 2, 0.02, Z) YY <- rVarGamma(ff, 2, 0.02, 3) # }