References for the Holonomic Gradient Method (HGM) and
the Holonomic Gradient Descent Method (HGD)
Papers and Tutorials
Early papers related to HGM.
- Yoshihito Tachibana, Yoshiaki Goto, Tamio Koyama, Nobuki Takayama,
Holonomic Gradient Method for Two Way Contingency Tables,
- F.H.Danufane, K.Ohara, N.Takayama,
Holonomic Gradient Method for the Distribution Function of the Largest Root of Complex Non-central Wishart Matrices,
An integral formula for the powered sum of the independent, identically and normally distributed random variables,
- H.Hashiguchi, N.Takayama, A.Takemura,
Distribution of Ratio of two Wishart Matrices and Evaluation of Cumulative Probability
by Holonomic Gradient Method,
- R.Vidunas, A.Takemura,
Differential relations for the largest root distribution
of complex non-central Wishart matrices,
The A-hypergeometric System Associated with the Rational Normal Curve and
System of Partial Differential Equations for the Hypergeometric Function 1F1 of a Matrix Argument on Diagonal Regions,
- Y.Goto, K.Matsumoto,
Pfaffian equations and contiguity relations of the hypergeometric function of type (k+1,k+n+2) and their applications,
Holonomic gradient method for the probability content of a simplex
with a multivariate normal distribution,
- N.Takayama, S.Kuriki, A.Takemura,
A-Hpergeometric Distributions and Newton Polytopes,
- G.Weyenberg, R.Yoshida, D.Howe,
Normalizing Kernels in the Billera-Holmes-Vogtmann Treespace,
- C.Siriteanu, A.Takemura, C.Koutschan, S.Kuriki, D.St.P.Richards, H.Sin,
Exact ZF Analysis and Computer-Algebra-Aided Evaluation
in Rank-1 LoS Rician Fading,
- K.Ohara, N.Takayama,
Pfaffian Systems of A-Hypergeometric Systems II ---
Holonomic Gradient Method,
The Annihilating Ideal of the Fisher Integral,
- T.Koyama, A.Takemura,
Holonomic gradient method for distribution function of a weighted sum
of noncentral chi-square random variables,
Contiguity relations of Lauricella's F_D revisited,
T.Koyama, H.Nakayama, K.Ohara, T.Sei, N.Takayama,
Software Packages for Holonomic Gradient Method,
Mathematial Software --- ICMS 2014,
4th International Conference, Proceedings.
Edited by Hoon Hong and Chee Yap,
Springer lecture notes in computer science 8592,
- N.Marumo, T.Oaku, A.Takemura,
Properties of powers of functions satisfying second-order linear differential equations with applications to statistics,
- J.Hayakawa, A.Takemura,
Estimation of exponential-polynomial distribution by holonomic gradient descent
- C.Siriteanu, A.Takemura, S.Kuriki,
MIMO Zero-Forcing Detection Performance Evaluation by Holonomic Gradient Method
Holonomic Modules Associated with Multivariate Normal Probabilities of Polyhedra,
- T.Hibi, K.Nishiyama, N.Takayama,
Pfaffian Systems of A-Hypergeometric Equations I,
Bases of Twisted Cohomology Groups,
(major revision v2 of arxiv:1212.6103).
Accepted version is at
T.Hibi et al, Groebner Bases : Statistics and Software Systems , Springer, 2013.
Introduction to the Holonomic Gradient Method (movie), 2013.
movie at youtube
- T.Sei, A.Kume,
Calculating the Normalising Constant of the Bingham Distribution on the Sphere using the Holonomic Gradient Method,
Statistics and Computing, 2013,
- T.Koyama, A.Takemura,
Calculation of Orthant Probabilities by the Holonomic Gradient Method,
- T. Koyama, H. Nakayama, K. Nishiyama, N. Takayama,
Holonomic Rank of the Fisher-Bingham System of Differential Equations,
Journal of Pure and Applied Algebra (online),
T. Koyama, H. Nakayama, K. Nishiyama, N. Takayama,
Holonomic Gradient Descent for the Fisher-Bingham Distribution on the d-dimensional Sphere,
Computational Statistics (2013)
- Hiroki Hashiguchi, Yasuhide Numata, Nobuki Takayama, Akimichi Takemura,
Holonomic gradient method for the distribution function of the largest root of a Wishart matrix,
Journal of Multivariate Analysis, 117, (2013) 296-312,
- 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,
- T.Koyama, A Holonomic Ideal which Annihilates the Fisher-Bingham Integral,
Funkcialaj Ekvacioj 56 (2013), 51--61.
Hiromasa Nakayama, Kenta Nishiyama, Masayuki Noro, Katsuyoshi Ohara,
Tomonari Sei, Nobuki Takayama, Akimichi Takemura ,
Holonomic Gradient Descent and its Application to Fisher-Bingham Integral,
Advances in Applied Mathematics 47 (2011), 639--658,
H.Dwinwoodie, L.Matusevich, E. Mosteig,
Transform methods for the hypergeometric distribution,
Statistics and Computing 14 (2004), 287--297.
Three Steps of HGM
- Finding a holonomic system satisfied by the normalizing constant.
We may use computational or theoretical methods to find it.
Groebner basis and related methods are used.
- Finding an initial value vector for the holonomic system.
This is equivalent to evaluating the normalizing constant and its derivatives
at a point.
This step is usually performed by a series expansion.
- Solving the holonomic system numerically. We use several methods
in numerical analysis such as the Runge-Kutta method of solving
ordinary differential equations and efficient solvers of systems of linear
Software Packages for HGM
CRAN package hgm (for R).
Some software packages are experimental and temporary documents are found in
"asir-contrib manual" (auto-autogenerated part), or
"Experimental Functions in Asir", or "miscellaneous and other documents"
or in this folder.
The nightly snapshot of the asir-contrib can be found in the asir page below,
or look up our
- Command line interfaces are in the folder OpenXM/src/hgm
in the OpenXM source tree. See
OpenXM distribution page .
- Experimental version of hgm package for R (hgm_*tar.gz, hgm-manual.pdf) for the step 3.
To install this package in R, type in
R CMD install hgm_*.tar.gz
- The following packages are
for the computer algebra system
They are in the asir-contrib collection.
- yang.rr (for Pfaffian systems) ,
nk_restriction.rr (for D-module integrations),
tk_jack.rr (for Jack polynomials),
ko_fb_pfaffian.rr (Pfaffian system for the Fisher-Bingham system),
are for the steps 1 or 2.
- nk_fb_gen_c.rr is a package to generate a C program to perform
maximal Likehood estimates for the Fisher-Bingham distribution by HGD (holonomic gradient descent).
- ot_hgm_ahg.rr (HGM for A-distributions, very experimental).
Programs to try examples of our papers
- d-dimensional Fisher-Bingham System
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