Directional (version 6.8)

Density of some (hyper-)spherical distributions: Density of some (hyper-)spherical distributions

Description

Density of some (hyper-)spherical distributions.

Usage

dvmf(y, mu, k, logden = FALSE )
iagd(y, mu, logden = FALSE)
dpurka(y, theta, a, logden = FALSE)
dspcauchy(y, mu, rho, logden = FALSE)
dpkbd(y, mu, rho, logden = FALSE)

Value

A vector with the (log) density values of y.

Arguments

y

A matrix or a vector with the data expressed in Euclidean coordinates, i.e. unit vectors.

mu

The mean direction (unit vector) of the von Mises-Fisher, the IAG, the spherical Cauchy distribution, or of the Poisson kernel based distribution.

theta

The mean direction (unit vector) of the Purkayastha distribution.

k

The concentration parameter of the von Mises-Fisher distribution.

a

The concentration parameter of the Purkayastha distribution.

rho

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

logden

If you the logarithm of the density values set this to TRUE.

Author

Michail Tsagris and Zehao Yu.

R implementation and documentation: Michail Tsagris and Zehao Yu .

Details

The density of the von Mises-Fisher, of the IAG, of the Purkayastha, of the spherical Cauchy distribution, or of the Poisson kernel based distribution is computed.

References

Mardia K. V. and Jupp P. E. (2000). Directional statistics. Chicester: John Wiley & Sons.

Purkayastha S. (1991). A Rotationally Symmetric Directional Distribution: Obtained through Maximum Likelihood Characterization. The Indian Journal of Statistics, Series A, 53(1): 70--83

Cabrera J. and Watson G. S. (1990). On a spherical median related distribution. Communications in Statistics-Theory and Methods, 19(6): 1973--1986.

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

Golzy M. and Markatou M. (2020). Poisson kernel-based clustering on the sphere: convergence properties, identifiability, and a method of sampling. Journal of Computational and Graphical Statistics, 29(4): 758--770.

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

Zehao Yu and Xianzheng Huang (2024). A new parameterization for elliptically symmetric angular Gaussian distributions of arbitrary dimension. Electronic Journal of Statististics, 18(1): 301--334.

See Also

kent.mle, rkent, esag.mle

Examples

m <- colMeans( as.matrix( iris[,1:3] ) )
y <- rvmf(1000, m = m, k = 10)
dvmf(y, k=10, m)