References for the Holonomic Gradient Method (HGM) and the Holonomic Gradient Descent Method (HGD)

Papers and Tutorials

  1. T.Koyama, Holonomic Modules Associated with Multivariate Normal Probabilities of Polyhedra, arxiv:1311.6905
  2. T.Hibi, K.Nishiyama, N.Takayama, Pfaffian Systems of A-Hypergeometric Equations I, Bases of Twisted Cohomology Groups, arxiv:1212.6103 (major revision v2 of arxiv:1212.6103)
  3. Intro T.Hibi et al, Groebner Bases : Statistics and Software Systems , Springer, 2013.
  4. Intro Introduction to the Holonomic Gradient Method (movie), 2013. movie at youtube
  5. T.Sei, A.Kume, Calculating the Normalising Constant of the Bingham Distribution on the Sphere using the Holonomic Gradient Method, Statistics and Computing, 2013, DOI
  6. T.Koyama, A.Takemura, Calculation of Orthant Probabilities by the Holonomic Gradient Method, arxiv:1211.6822
  7. T. Koyama, H. Nakayama, K. Nishiyama, N. Takayama, Holonomic Rank of the Fisher-Bingham System of Differential Equations, to appear in Journal of Pure and Applied Algebra
  8. T. Koyama, H. Nakayama, K. Nishiyama, N. Takayama, Holonomic Gradient Descent for the Fisher-Bingham Distribution on the d-dimensional Sphere, Computational Statistics (2013) DOI
  9. 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, DOI
  10. 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
  11. T.Koyama, A Holonomic Ideal which Annihilates the Fisher-Bingham Integral, Funkcialaj Ekvacioj 56 (2013), 51--61. jstage
  12. 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, DOI

Three Steps of HGM

  1. Find 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.
  2. Find 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.
  3. Solve 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 equations.

Software Packages for HGM

  1. hgm package for R for the step 3.
  2. yang (for Pfaffian systems) , nk_restriction (for D-module integrations), tk_jack (for Jack polynomials) are for the steps 1 or 2 and in the asir-contrib

Programs to try examples of our papers

  1. d-dimensional Fisher-Bingham System 
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