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

Papers and Tutorials

  1. Yoshihito Tachibana, Yoshiaki Goto, Tamio Koyama, Nobuki Takayama, Holonomic Gradient Method for Two Way Contingency Tables, arxiv:1803.04170
  2. F.H.Danufane, K.Ohara, N.Takayama, Holonomic Gradient Method for the Distribution Function of the Largest Root of Complex Non-central Wishart Matrices, arxiv:1707.02564
  3. T.Koyama, An integral formula for the powered sum of the independent, identically and normally distributed random variables, arxiv:1706.03989
  4. H.Hashiguchi, N.Takayama, A.Takemura, Distribution of Ratio of two Wishart Matrices and Evaluation of Cumulative Probability by Holonomic Gradient Method, arxiv:1610.09187
  5. R.Vidunas, A.Takemura, Differential relations for the largest root distribution of complex non-central Wishart matrices, arxiv:1609.01799
  6. S.Mano, The A-hypergeometric System Associated with the Rational Normal Curve and Exchangeable Structures, arxiv:1607.03569
  7. M.Noro, System of Partial Differential Equations for the Hypergeometric Function 1F1 of a Matrix Argument on Diagonal Regions, ACM DL
  8. Y.Goto, K.Matsumoto, Pfaffian equations and contiguity relations of the hypergeometric function of type (k+1,k+n+2) and their applications, arxiv:1602.01637
  9. T.Koyama, Holonomic gradient method for the probability content of a simplex region with a multivariate normal distribution, arxiv:1512.06564
  10. N.Takayama, S.Kuriki, A.Takemura, A-Hpergeometric Distributions and Newton Polytopes, arxiv:1510.02269
  11. G.Weyenberg, R.Yoshida, D.Howe, Normalizing Kernels in the Billera-Holmes-Vogtmann Treespace, arxiv:1506.00142
  12. 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, arxiv:1507.07056
  13. K.Ohara, N.Takayama, Pfaffian Systems of A-Hypergeometric Systems II --- Holonomic Gradient Method, arxiv:1505.02947
  14. T.Koyama, The Annihilating Ideal of the Fisher Integral, arxiv:1503.05261
  15. T.Koyama, A.Takemura, Holonomic gradient method for distribution function of a weighted sum of noncentral chi-square random variables, arxiv:1503.00378
  16. Y.Goto, Contiguity relations of Lauricella's F_D revisited, arxiv:1412.3256
  17. 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, 706--712. DOI
  18. N.Marumo, T.Oaku, A.Takemura, Properties of powers of functions satisfying second-order linear differential equations with applications to statistics, arxiv:1405.4451
  19. J.Hayakawa, A.Takemura, Estimation of exponential-polynomial distribution by holonomic gradient descent arxiv:1403.7852
  20. C.Siriteanu, A.Takemura, S.Kuriki, MIMO Zero-Forcing Detection Performance Evaluation by Holonomic Gradient Method arxiv:1403.3788
  21. T.Koyama, Holonomic Modules Associated with Multivariate Normal Probabilities of Polyhedra, arxiv:1311.6905
  22. 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). Accepted version is at DOI
  23. Intro T.Hibi et al, Groebner Bases : Statistics and Software Systems , Springer, 2013.
  24. Intro Introduction to the Holonomic Gradient Method (movie), 2013. movie at youtube
  25. 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
  26. T.Koyama, A.Takemura, Calculation of Orthant Probabilities by the Holonomic Gradient Method, arxiv:1211.6822
  27. 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), DOI
  28. 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
  29. 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
  30. 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
  31. T.Koyama, A Holonomic Ideal which Annihilates the Fisher-Bingham Integral, Funkcialaj Ekvacioj 56 (2013), 51--61. DOI
  32. 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
Early papers related to HGM.
  1. H.Dwinwoodie, L.Matusevich, E. Mosteig, Transform methods for the hypergeometric distribution, Statistics and Computing 14 (2004), 287--297.

Three Steps of HGM

  1. 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.
  2. 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.
  3. 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 equations.

Software Packages for HGM

Programs to try examples of our papers

  1. d-dimensional Fisher-Bingham System
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