Directional (version 7.0)

Simulation of random values from rotationally symmetric distributions: Simulation of random values from rotationally symmetric distributions

Description

Simulation of random values from rotationally symmetric distributions. The data can be spherical or hyper-spherical.

Usage

rvmf(n, mu, k)
riag(n, mu)
rspcauchy(n, mu, rho)
rpkbd(n, mu, rho)

Value

A matrix with the simulated data.

Arguments

n

The sample size.

mu

A unit vector showing the mean direction for the von Mises-Fisher or the spherical Cauchy distribution. The mean vector of the Independent Angular Gaussian distribution does not have to be a unit vector.

k

The concentration parameter (\(\kappa\)) of the von Mises-Fisher distribution. If \(\kappa=0\), random values from the spherical uniform will be drwan.

rho

The \(\rho\) parameter of the spherical Cauchy or the Poisson kernel-based distribution.

Author

Michail Tsagris.

R implementation and documentation: Michail Tsagris and Giorgos Athineou <[e-mail]>.

Details

The von Mises-Fisher uses the rejection smapling suggested by Wood (1994). For the Independent Angular Gaussian, values are generated from a multivariate normal distribution with the given mean vector and the identity matrix as the covariance matrix. Then each vector becomes a unit vector. For the spherical Cauchy distribution the algortihm is described in Kato and McCullagh (2020) and for the Poisson kernel-based distribution, it is described in Sablica, Hornik and Leydold (2023).

References

Wood A.T.A. (1994). Simulation of the von Mises Fisher distribution. Communications in Statistics-Simulation and Computation, 23(1): 157--164.

Dhillon I. S. and Sra S. (2003). Modeling data using directional distributions. Technical Report TR-03-06, Department of Computer Sciences, The University of Texas at Austin. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.75.4122&rep=rep1&type=pdf

Kato S. and McCullagh P. (2020). Some properties of a Cauchy family on the sphere derived from the Mobius transformations. Bernoulli, 26(4): 3224--3248. https://arxiv.org/pdf/1510.07679.pdf

Sablica L., Hornik K. and Leydold J. (2023). Efficient sampling from the PKBD distribution. Electronic Journal of Statistics, 17(2): 2180--2209.

See Also

vmf.mle, iag.mle rfb, racg, rvonmises, rmixvmf

Examples

m <- rnorm(4)
m <- m/sqrt(sum(m^2))
x <- rvmf(100, m, 25)
m
vmf.mle(x)