| rVarGamma {spatstat} | R Documentation |
Generate a random point pattern, a simulated realisation of the Neyman-Scott process with Variance Gamma (Bessel) cluster kernel.
rVarGamma(kappa, nu.ker, omega, mu, win = owin(),
eps = 0.001, nu.pcf=NULL)
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 |
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. |
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' = 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 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 cauchy.estK
or cauchy.estpcf
applied to the inhomogeneous K function.
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.
Abdollah Jalilian and Rasmus Waagepetersen. Adapted for spatstat by Adrian Baddeley Adrian.Baddeley@uwa.edu.au http://www.maths.uwa.edu.au/~adrian/
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.
rpoispp,
rNeymanScott,
cauchy.estK,
cauchy.estpcf,
kppm.
# homogeneous
X <- rVarGamma(30, 2, 0.02, 5)
# inhomogeneous
Z <- as.im(function(x,y){ exp(2 - 3 * x) }, W= owin())
Y <- rVarGamma(30, 2, 0.02, Z)