dspheremix: Density functions for mixtures of spherical von Mises-Fisher

Description

Density functions for nine finite mixtures of spherical von Mises-Fisher allowing different numbers of modes.

Usage

dspheremix(x, model = NULL)

Value

A numeric vector of density evaluated on x.

Arguments

x: A matrix whose rows represent points on the unit sphere in Cartesian coordinates. If a row norm is different from one, a message appears indicating that they must be standardized.
model: Number between 1 and 9, corresponding to a density model defined in Saavedra-Nieves and Crujeiras (2021). See Details.

Author

Paula Saavedra-Nieves and Rosa M. Crujeiras.

Details

These nine spherical models are obtained as mixtures of von Mises distributions where the density $f$ is given by: $$ f=\sum_{i=1}^I w_i K_{vM}(x;m_i;k_i), w_i\geq 0;\sum_{i=1}^I w_i=1 $$ with $K_vM$ denoting the von Mises-Fisher kernel density; $m_i$, $k_i$ and $w_i$ the mean, concentration and weight corresponding to each component. More details can be found in Hornik and Grun (2014) and Wood (1994). The combination of means, concentration parameters and the weights of spherical models from Saavedra-Nieves and Crujeiras (2021) are specified below:

S1: (0, 0, 1) ($m$); 10 ($k$); 1 ($w$).

S2: (0, 0, 1), (0, 0, -1) ($m$); 1, 1 ($k$); 1/2, 1/2 ($w$).

S3: (0, 0, 1), (0, 0, -1) ($m$); 10, 1 ($k$); 1/2, 1/2 ($w$).

S4: (0, 0, 1); (0, 1/$\sqrt2$, 1/$\sqrt2$) ($m$); 10, 10 ($k$); 1/2, 1/2 ($w$).

S5: (0, 0, 1); (0, 1/$\sqrt2$, 1/$\sqrt2$) ($m$); 10, 10 ($k$); 2/5, 3/5 ($w$).

S6: (0, 0, 1); (0, 1/$\sqrt2$, 1/$\sqrt2 $) ($m$); 10, 5 ($k$); 1/5, 4/5 ($w$).

S7: (0, 0, 1), (0, 1, 0), (1, 0, 0) ($m$); 5, 5, 5 ($k$); 1/3, 1/3, 1/3 ($w$).

S8: (0, 0, 1), (0, 1, 0), (1, 0, 0) ($m$); 5, 5, 5 ($k$); 2/3, 1/6, 1/6 ($w$).

S9: (0, 0, 1); (0, 1/$\sqrt 2$, 1/$\sqrt 2$), (0, 1, 0) ($m$); 10, 10, 10 ($k$); 1/3, 1/3, 1/3 ($w$).

References

Hornik, K. and Grun, B. (2014). movMF: an R package for fitting mixtures of von Mises-Fisher distributions. Journal of Statistical Software, 58(10), 1-31.
Saavedra-Nieves, P. and Crujeiras, R. M. (2021). Nonparametric estimation of directional highest density regions. Advances in Data Analysis and Classification, 1-36.
Wood, A. T. (1994). Simulation of the von Mises Fisher distribution. Communications in Statistics-Simulation and Computation, 23(1), 157-164.

Examples

Run this code

# Density function evaluation from model S1
data <- rbind(c(1,0,0),c(0,1,0),c(0,0,1))
dspheremix(data, model=1)

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