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)
Intensity of the Poisson process of cluster centres. A single positive number, a function, or a pixel image.
Shape parameter for the cluster kernel. A number greater than -1.
Scale parameter for cluster kernel. Determines the size of clusters. A positive number in the same units as the spatial coordinates.
Mean number of points per cluster (a single positive number) or reference intensity for the cluster points (a function or a pixel image).
Window in which to simulate the pattern.
An object of class
or something acceptable to
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.
Number of simulated realisations to be generated.
drop=TRUE (the default), the
result will be a point pattern, rather than a list
containing a point pattern.
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.
Numeric. Size of window expansion for generation of parent
points. By default determined by calling
clusterradius with the numeric threshold value given
Numerical threshold below which the model will be treated as a Poisson process. See Details.
Logical value indicating whether to save the locations of the parent points as an attribute.
A point pattern (an object of class
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
This algorithm generates a realisation of the Neyman-Scott process
with Variance Gamma (Bessel) cluster kernel, inside the window
The process is constructed by first
generating a Poisson point process of ``parent'' points
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 (
and the points being placed independently and uniformly
according to a Variance Gamma kernel.
The shape of the kernel is determined by the dimensionless
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
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
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
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
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
mu is a
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
kppm, or using
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
This avoids computations that would otherwise require huge amounts
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.