hgm.so3nc: The function hgm.ncso3 evaluates the normalization constant...

In hgm: Holonomic Gradient Method and Gradient Descent

The function hgm.ncso3 evaluates the normalization constant for the Fisher distribution on SO(3).

Description

The function hgm.ncso3 evaluates the normalization constant for the Fisher distribution on SO(3).

Usage

hgm.ncso3(a,b,c,t0=0.0,q=1,deg=0,log=0)

Arguments

a	See the description of c.
b	See the description of c.
c	This function evaluates the normalization constant for the parameter Theta=diag(theta_ii) of the Fisher distribution on SO(3). The variables a,b,c stand for the parameters theta_11, theta_22, theta_33 respectively.
t0	It is the initial point to evaluate the series. If it is set to 0.0, a default value is used.
q	If it is 1, then the program works in a quiet mode.
deg	It gives the approximation degree of the power series approximation of the normalization constant near the origin. If it is 0, a default value is used.
log	If it is 1, then the function returns the log of the normalizing constant.

Details

The normalization constant c(Theta) of the Fisher distribution on SO(3) is defined by integral( exp(trace( transpose(Theta) X)) ) where X is the integration variable and runs over S0(3) and Theta is a 3 x 3 matrix parameter. A general HGM algorithm to evaluate the normalization constant is given in the reference below. We use the Corollary 1 and the series expansion in 3.2 for the evaluation.

Value

The output is an array of c(Theta) and its derivatives with respect to Theta_11,Theta_22,Theta_33. It is the vector C of the reference below. When log=1, the output is an array of log of them.

Author(s)

Nobuki Takayama

References

Tomonari Sei, Hiroki Shibata, Akimichi Takemura, Katsuyoshi Ohara, Nobuki Takayama, Properties and applications of Fisher distribution on the rotation group, Journal of Multivariate Analysis, 116 (2013), 440–455, doi: 10.1016/j.jmva.2013.01.010

Examples

## =====================================================
## Example 1. Computing normalization constant of the Fisher distribution on SO(3)
## =====================================================
hgm.ncso3(1,2,3)[1]

## =====================================================
## Example 2. Asteroid data in the paper
## =====================================================
hgm.ncso3(19.6,0.831,-0.671)[1]

hgm documentation built on Feb. 16, 2023, 7:44 p.m.