This site provides programs and machine readable data for the paper "Properties and applications of Fisher distribution on the rotation group" by Tomonari Sei(清), Hiroki Shibata(柴田), Akimichi Takemura(竹村), Katsuyoshi Ohara(小原), Nobuki Takayama(高山).

Abstract

We study properties of Fisher distribution (von Mises-Fisher distribution, matrix Langevin distribution) on the rotation group SO(3). In particular we apply the holonomic gradient descent method, introduced in Holonomic Gradient Descent and its Application to the Fisher-Bingham Integral and a method of series expansion for evaluating the normalizing constant of the distribution and for computing the maximum likelihood estimate. The rotation group can be identified with the Stiefel manifold of two orthonormal vectors. Therefore from the viewpoint of statistical modeling, it is of interest to compare Fisher distributions on these manifolds. We illustrate the difference with an example of near-earth objects data. Programs and machine readable data are for the R (a system for statistics) and the Risa/Asir (a computer algebra system).

Programs and Data for V_2(R^3) (Stiefel manifold)

Folder for V_2(R^3)

Risa/Asir program to get the Pfaffian system (sf.rr)

   start Risa/Asir with the asir-contrib library. 
   load("sf.rr");
   bynd();
   myprune();
   A=mypf()$    /* Pfaffian system is saved in the file tmp-pf-sf32.ab  
                   It is a list of 6 matrices A[0], ..., A[5],
                   which are coefficients of the Pfaffian system.
                   V=dsf(3,2)[1] a list of variables.
                   Let F be a solution vector. Then, dF/dV[i] = A[i]F holds.
                */

   The function dsf(3,2) returns the system of differential equations for c(\Theta).

Risa/Asir program for the holonomic gradient descent (HGD) (sf2i.rr) See 2011-09-16-comets.txt how to use it.

R program for evaluation of c(\Theta) by a seires (etrcalc.r)

  Start R.
  source("etrcalc.r");
  SO.init(Ts.Xm)   # Initial value for HGD for commets data. The output is embedded in sf2.rr 
  SO.init(TsAst,XmAst) # Initial value for HGD for asteroids data.
  SO.int2(Ts) # value of c(\Theta) for Ts.

Programs and Data for SO(3)

Risa/Asir program to get a partial Grobner basis for SO(3) It takes a very long time to get the basis on a 64 bit machine with 48G memory. The result is in tmp-dso3.ab (right click for the download. It is huge size).
```
 load("sdo.rr");
 bynd()$
```

Folder for HGD-MLE for SO(3)

A note (old) for the files and programs in this folder (in Japanese)

Pfaffian data for SO(3) (pf.ab) (right click for saving)

  Start Risa/Asir
  Pfs=bload("pf.ab")$
 /* 
    Pfs[0] is a list of 9 matrices, which are the coefficients of the Pfaffian system.
    A hashed data structure is used to save polynomials.
  */
The data is generated by  mydso9.rr 
  MLE by the HGD for SO(3), commets data(old). 
See the files Log.txt and Log16.txt to see how to run it
(A fast machine will be recommended to run it). 
    Start Risa/Asir
    load("hgd-test.rr");
    load("hgd-test16.rr");

A note on HGD-MLE (SO(3)) for the commets data