Package ‘hgmFB’March 27, 2015
Type Package
Depends R (>= 2.6.0)
Title Fisher-Bingham Distribution by HGM
Version 1.0
Date 2015-03-26
Author Nobuki Takayama, Tamio Koyama, Tomonari Sei, Hiromasa Nakayama, Kenta Nishiyama
Maintainer Nobuki Takayama <takayama@math.kobe-u.ac.jp>
Description This package evaluates the normalizing constant for the Fisher-Bingham distributions and solves MLE problems by utilizing the holonomic gradient method (HGM)
License GPL-2
LazyLoad yes
URL http://www.openxm.org

R topics documented:
hgmFB-package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
hgm.z.mleFBByOptim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Index
hgmFB-package

1
2
4

HGM

Description
This package evaluates the normalizing constant for the Fisher-Bingham distributions and solves
MLE problems by utilizing the holonomic gradient method.
Details1

2

hgm.z.mleFBByOptim
Package:Type:
License:LazyLoad:

hgm_fbPackage
GPL-2
yes

This package evaluates the normalizing constant for the Fisher-Bingham distributions and solves
MLE problems by utilizing the holonomic gradient method. The HGM and HGD are proposedin the paper below. This method based on the fact that a broad class of normalizing constantsof unnormalized probability distributions belongs to the class of holonomic functions, which aresolutions of holonomic systems of linear partial differential equations.
References
N3OST2 Hiromasa Nakayama, Kenta Nishiyama, Masayuki Noro, Katsuyoshi Ohara, Tomonari Sei,Nobuki Takayama, Akimichi Takemura, Holonomic Gradient Descent and its Application toFisher-Bingham Integral, Advances in Applied Mathematics 47 (2011), 639–658, http://dx.doi.org/10.1016/j.aam.2011.03.001
dojo Edited by T.Hibi, Groebner Bases: Statistics and Software Systems, Springer, 2013, http://dx.doi.org/10.1007/978-4-431-54574-3• http://www.openxm.org
See Also
hgm.z.mleFBByOptim,hgm.z.mleFBByOptim

MLE of Fisher-Bingham distribution by optim and HGM.

Description
It makes the maximal likelihood estimate (MLE) for the Fisher-Bingham distribution on S^d.
Usage
hgm.z.mleFBByOptim(d=0,ss=NULL,data=NULL,start=NULL,lb=NULL,ub=NULL,bigValue=10000)
Arguments
dThe dimension of the sphere

ssSufficient statistics

dataThe argument data is a set of data on the d-dimensional sphere. Its format is an
n by (d+1) matrix where n is the number of data. When data is given, ss must be
NULL and ss is calculated from data by hgm.ssFB(data).

start

Starting point for the function optim. The default value is a random vector.

hgm.z.mleFBByOptim

3lb

An array of length sslen = (d+1)*(d+2)/2 + (d+1), each of which is the lower
bound of the parameter. The default value is -100.

ub

An array of length sslen = (d+1)*(d+2)/2 + (d+1), each of which is the upperbound of the parameter. The default value is 100.

bigValue

It is used as a value wall to avoid that the evaluation point is out of the searchdomain defined by lb and ub.

Details
It solves the MLE for the Fisher-Bingham distribution. The normalizing constant is evaluated byhgm_ko_ncfb (external program, which should in the path). The function
optim is used for the optimization. The output is used as a starting point for the holonomic gradientmethod. See nk_fb_gen_c.rr of http://www.math.kobe-u.ac.jp/Asir. This function generates
temporary work files whose names start with tmp. data <- read.table(fileName,sep=",")
can be used to read CSV data from a file.
ValueThe output format is that of the function optim().
Author(s)T.Koyama, H.Nakayama, K.Nishiyama, N.Takayama.
References
T. Koyama, H. Nakayama, K. Nishiyama, N. Takayama, Holonomic Gradient Descent for theFisher-Bingham Distribution on the d-dimensional Sphere, Computational Statistics (2013) http://dx.doi.org/10.1007/s00180-013-0456-z
See Also
optim
Examples## =====================================================## Example 1. Asteroid data in [N3OST2]## =====================================================## Not run:d <- 2
ss <- c(0.3119,0.0292,0.0707,0.3605,0.0462,0.3276,0.0063,0.0054,0.0762)
start <- c(0.1,0.1,1,1,1,-1,-1,-1,-1)
hgm.z.mleFBByOptim(d=d,ss=ss,start=start)## End(Not run)Index∗Topic Fisher-Bingham distribution

on S^dhgm.z.mleFBByOptim, 2∗Topic Fisher-Bingham distribution
hgmFB-package, 1∗Topic HGD
hgmFB-package, 1∗Topic HGM
hgm.z.mleFBByOptim, 2
hgmFB-package, 1∗Topic Holonomic gradient method
hgm.z.mleFBByOptim, 2∗Topic MLE
hgm.z.mleFBByOptim, 2∗Topic holonomic gradient descent
hgmFB-package, 1∗Topic holonomic gradient method
hgmFB-package, 1∗Topic package
hgmFB-package, 1
hgm.z.mleFBByOptim, 2, 2
HGM_FB (hgmFB-package), 1
hgm_fb (hgmFB-package), 1
hgm_fb-package (hgmFB-package), 1
hgmFB-package, 1
optim, 34