# rVarGamma

0th

Percentile

##### Simulate Neyman-Scott Point Process with Variance Gamma cluster kernel

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

Keywords
spatial, datagen
##### Usage
rVarGamma(kappa, nu.ker, omega, mu, win = owin(),
eps = 0.001, nu.pcf=NULL)
##### Arguments
kappa
Intensity of the Poisson process of cluster centres. A single positive number, a function, or a pixel image.
nu.ker
Shape parameter for the cluster kernel. A number greater than -1.
omega
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.
eps
Threshold below which the values of the cluster kernel will be treated as zero for simulation purposes.
nu.pcf
Alternative specifier of the shape parameter. See Details.
##### Details

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.ker. This is the parameter $\nu^\prime = \alpha/2-1$ appearing in equation (12) on page 126 of Jalilian et al (2013). Instead of specifying nu.ker the user can specify nu.pcf which is the parameter $\nu=\alpha-1$ appearing in equation (13), page 127 of Jalilian et al (2013). These are related by nu.pcf = 2 * nu.ker + 1 and nu.ker = (nu.pcf - 1)/2. Exactly one of nu.ker or nu.pcf must be specified.

The scale of the kernel is determined by the argument omega, 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, using cauchy.estK, cauchy.estpcf or kppm. It can also be fitted by maximum composite likelihood using kppm. The algorithm can also generate spatially inhomogeneous versions of the cluster process:

• The parent points can be spatially inhomogeneous. If the argumentkappais afunction(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 argumentmuis afunction(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 cauchy.estK or cauchy.estpcf applied to the inhomogeneous $K$ function.

##### 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 rNeymanScott.

##### References

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.