rVarGamma

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.

Usage

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

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. 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.

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.

See Also

rpoispp, rNeymanScott, kppm.

vargamma.estK, vargamma.estpcf.

Aliases

• rVarGamma

Examples

# 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)
# }