rFisher: Generate Fisher and Watson distributions on the sphere

In baddstats/spherstat: Spatial Point Pattern Analysis, Geometry and Trigonometry for the Sphere

Generate Fisher and Watson distributions on the sphere

Description

Generates variates of the Fisher and Watson distributions on the sphere.

Usage

rFisher(n, mode, kappa, win=sphwin(type="sphere"), nsim=1, 
drop=TRUE, as.sp=TRUE, ndim="2")
rWatson(n, mode, kappa, win=sphwin(type="sphere"), squeeze=FALSE, nsim=1, 
drop=TRUE, as.sp=TRUE, ndim="2")
rWatson.bipolar(n, mode, kappa, win=sphwin(type="sphere"), squeeze=FALSE, nsim=1,
drop=TRUE, as.sp=TRUE, ndim="2")
rWatson.girdle(n, mode, kappa, win=sphwin(type="sphere"), squeeze=FALSE, nsim=1,
drop=TRUE, as.sp=TRUE, ndim="2")

Arguments

n	The number of points to be simulated. An integer.
mode	A length 2 numeric giving the mode of the distribution
kappa	The concentration parameter. Must be positive for the Fisher and Watson bipolar distributions. Must be negative for the Watson girdle distribution.
win	The window, an object of type sphwin.
squeeze	Logical. For Watson distributions; if TRUE then applies a squeeze test which can speed up the simulation if kappa is large.
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.
as.sp	Logical. If TRUE, returns an object of class as defined by sp.dim. Otherwise, returns a matrix. See Value.
ndim	A string, taking value "2" or "3". Specifies whether the object should contain the locations of the points in spherical coordinates (ndim="2") or Cartesian coordinates (ndim="3").

Details

rFisher generates a realisation of n points of the Fisher distribution inside the window win. The Fisher distribution is the analogue on the sphere of the normal distribution; the maximum of the density occurs at mode and the minimum at the point diametrically opposite mode (i.e. the antimode). The density of this distribution on any small or great circle that is orthogonal to the axis containing the mode and antimode, is uniform. The algorithm was given in Fisher et al. (1981) and in Fisher et al. (1993).

rWatson.bipolar generates a realisation of of n points of the Watson bipolar distribution inside the window n. This distribution is the bipolar analogue of the Fisher distribution, with maximum density at two diametrically opposite modes (one of which is specified as mode in the function) and minimum density at the great circle halfway between the modes. The algorithm was given in Best and Fisher (1986) and in Fisher et al. (1993).

rWatson.girdle generates a realisation of of n points of the Watson bipolar distribution inside the window n. This distribution is the ‘opposite’ of a the Watson bipolar in that the minimum density is at two diametrically opposite antimodes (one of which is specified as mode in the function) and maximum density at the great circle halfway between the antimodes. The algorithm was given in Best and Fisher (1986) and in Fisher et al. (1993).

rWatson simulates either a Watson bipolar distribution (using rWatson.bipolar) or a Watson girdle distribution (using rWatson.girdle) depending on what value of kappa is provided.

The algorithsm used in these simulations are found in Section 3.6.2. of N. I. Fisher et al.

Value

If nsim=1 and drop=FALSE then a single item as described below; otherwise a list containing nsim items.

An item is determined by the values of as.sp and ndim: If as.sp=FALSE and ndim="2", a two column matrix giving the locations of the simulated points. If as.sp=FALSE and ndim="3", a three column matrix giving the locations of the simulated points. If as.sp=TRUE and ndim="2", an object of class sp2 giving the locations of the simulated points. If as.sp=TRUE and ndim="3", an object of class sp3 giving the locations of the simulated points.

Author(s)

Tom Lawrence <email:[e-mail]>

References

Best, D. J. and Fisher, N. I. (1986) ‘Goodness-of-fit and discordancy tests for samples from the Watson distribution on the sphere’, Australasian Journal of Statistics, 28, 13–31.

Fisher, N. I., Embleton, B. J. J. and Lewis, T. (1987) ‘Statistical analysis of spherical data’, 1st paperback edition. Cambridge University Press.

Fisher, N. I., Lewis, T. and Willcox, M. E. (1981) ‘Tests of discordancy for samples from Fisher’s distribution on the sphere' Journal of the Royal Statistical Society Series C, 30, 230–237.

Lawrence, T.J. (2017) Master's Thesis, University of Western Australia.

See Also

runif.sphwin

Examples

rFis <- rFisher(n=100, mode=c(0,0), kappa=50, win=sphwin())
rFis

rFis2 <- rFisher(n=100, mode=c(pi/3, pi/3), kappa=0.1, 
win=sphwin(type="band", param=c(0, pi/3), ref=c(pi/3, pi/3)))
rFis2

rWB <- rWatson.bipolar(n=100, mode=c(0,0), kappa=50, win=sphwin())
rWB

rWB2 <- rWatson.bipolar(n=100, mode=c(pi/3, pi/3), kappa=0.1, 
win=sphwin(type="band", param=c(0, pi/3), ref=c(pi/3, pi/3)))
rWB2

rWG <- rWatson.girdle(n=100, mode=c(0,0), kappa=-50, win=sphwin())
rWG

rWG2 <- rWatson.girdle(n=100, mode=c(pi/3, pi/3), kappa=-0.1, 
win=sphwin(type="band", param=c(0, pi/3), ref=c(pi/3, pi/3)))
rWG2

rWat <- rWatson(n=100, mode=c(0,0), kappa=50, win=sphwin())
rWat

rWat2 <- rWatson(n=100, mode=c(pi/3, pi/3), kappa=-0.1, 
win=sphwin(type="band", param=c(0, pi/3), ref=c(pi/3, pi/3)))
rWat2

rWatFish <- rWatson(n=100, mode=c(0,0), kappa=0, win=sphwin())
rWatFish

baddstats/spherstat documentation built on Feb. 6, 2023, 1:45 a.m.