rCauchy: Simulate Neyman-Scott Point Process with Cauchy cluster...

In spatstat.random: Random Generation Functionality for the 'spatstat' Family

Simulate Neyman-Scott Point Process with Cauchy cluster kernel

Description

Generate a random point pattern, a simulated realisation of the Neyman-Scott process with Cauchy cluster kernel.

Usage

 rCauchy(kappa, scale, mu, win = square(1),
            nsim=1, drop=TRUE,
            ...,
            algorithm=c("BKBC", "naive"),
            nonempty=TRUE, 
            thresh = 0.001, poisthresh=1e-6,
            expand = NULL,
            saveparents=FALSE, saveLambda=FALSE,
            kappamax=NULL, mumax=NULL)

Arguments

kappa	Intensity of the Poisson process of cluster centres. A single positive number, a function, or a pixel image.
scale	Scale parameter for cluster kernel. Determines the size of clusters. A single 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.
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.
...	Passed to clusterfield to control the image resolution when saveLambda=TRUE, and to clusterradius when expand is missing or NULL.
algorithm	String (partially matched) specifying the simulation algorithm. See Details.
nonempty	Logical. If TRUE (the default), a more efficient algorithm is used, in which parents are generated conditionally on having at least one offspring point. If FALSE, parents are generated even if they have no offspring. Both choices are valid; the default is recommended unless you need to simulate all the parent points for some other purpose.
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.
poisthresh	Numerical threshold below which the model will be treated as a Poisson process. See Details.
expand	Window expansion distance. A single number. The distance by which the original window will be expanded in order to generate parent points. Has a sensible default, determined by calling clusterradius with the numeric threshold value given in thresh.
saveparents	Logical value indicating whether to save the locations of the parent points as an attribute.
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.
kappamax	Optional. Numerical value which is an upper bound for the values of kappa, when kappa is a pixel image or a function.
mumax	Optional. Numerical value which is an upper bound for the values of mu, when mu is a pixel image or a function.

Details

This algorithm generates a realisation of the Neyman-Scott process with Cauchy 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 Cauchy kernel.

Note that, for correct simulation of the model, the parent points are not restricted to lie inside the window win; the parent process is effectively the uniform Poisson process on the infinite plane.

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.

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.

Value

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.

Simulation Algorithm

Two simulation algorithms are implemented.

• The naive algorithm generates the cluster process by directly following the description given above. First the window win is expanded by a distance equal to expand. Then the parent points are generated in the expanded window according to a Poisson process with intensity kappa. Then each parent point is replaced by a finite cluster of offspring points as described above. The naive algorithm is used if algorithm="naive" or if nonempty=FALSE.

• The BKBC algorithm, proposed by Baddeley and Chang (2023), is a modification of the algorithm of Brix and Kendall (2002). Parents are generated in the infinite plane, subject to the condition that they have at least one offspring point inside the window win. The BKBC algorithm is used when algorithm="BKBC" (the default) and nonempty=TRUE (the default).

The naive algorithm becomes very slow when scale is large, while the BKBC algorithm is uniformly fast (Baddeley and Chang, 2023).

If saveparents=TRUE, then the simulated point pattern will have an attribute "parents" containing the coordinates of the parent points, and an attribute "parentid" mapping each offspring point to its parent.

If nonempty=TRUE (the default), then parents are generated subject to the condition that they have at least one offspring point in the window win. nonempty=FALSE, then parents without offspring will be included; this option is not available in the BKBC algorithm.

Note that if kappa is a pixel image, its domain must be larger than the window win. This is because an offspring point inside win could have its parent point lying outside win. In order to allow this, the naive simulation algorithm first expands the original window win by a distance equal to expand and generates the Poisson process of parent points on this larger window. If kappa is a pixel image, its domain must contain this larger window.

If the pair correlation function of the model is very close to that of a Poisson process, with maximum deviation less than poisthresh, then the model is approximately a Poisson process. This is detected by the naive algorithm which then simulates a Poisson process with intensity kappa * mu, using rpoispp. This avoids computations that would otherwise require huge amounts of memory.

Fitting cluster models to data

The Cauchy cluster model with homogeneous parents (i.e. where kappa is a single number) where the offspring are either homogeneous or inhomogeneous (mu is a single number, a function or pixel image) can be fitted to point pattern data using kppm, or fitted to the inhomogeneous K function using cauchy.estK or cauchy.estpcf.

Currently spatstat does not support fitting the Cauchy cluster process model with inhomogeneous parents.

A Cauchy cluster process model fitted by kppm can be simulated automatically using simulate.kppm (which invokes rCauchy to perform the simulation).

Author(s)

Original algorithm by Abdollah Jalilian and Rasmus Waagepetersen. Adapted for spatstat by \adrian. Brix-Kendall-Baddeley-Chang algorithm implemented by \adrian and \yamei.

References

\baddchangclustersim

Brix, A. and Kendall, W.S. (2002) Simulation of cluster point processes without edge effects. Advances in Applied Probability 34, 267–280.

Ghorbani, M. (2013) Cauchy cluster process. Metrika 76, 697-706.

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, rMatClust, rThomas, rVarGamma, rNeymanScott, rGaussPoisson.

For fitting the model, see kppm, clusterfit.

Examples

 # homogeneous
 X <- rCauchy(30, 0.01, 5)
 # inhomogeneous
 ff <- function(x,y){ exp(2 - 3 * abs(x)) }
 Z <- as.im(ff, W= owin())
 Y <- rCauchy(50, 0.01, Z)
 YY <- rCauchy(ff, 0.01, 5)

spatstat.random documentation built on Sept. 30, 2024, 9:46 a.m.