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9:
10: <h1> References for the Holonomic Gradient Method (HGM) and
11: the Holonomic Gradient Descent Method (HGD) </h1>
12:
13: <h2> Papers and Tutorials</h2>
14: <ol>
1.24 takayama 15: <li> Yoshihito Tachibana, Yoshiaki Goto, Tamio Koyama, Nobuki Takayama,
16: Holonomic Gradient Method for Two Way Contingency Tables,
1.25 ! takayama 17: <a href="https://arxiv.org/abs/1803.04170"> arxiv:1803.04170 </a>
! 18: <li> F.H.Danufane, K.Ohara, N.Takayama, C.Siriteanu,
! 19: Holonomic Gradient Method-Based CDF Evaluation for the Largest Eigenvalue of a Complex Noncentral Wishart Matrix
! 20: (Title of the version 1: Holonomic Gradient Method for the Distribution Function of the Largest Root of Complex Non-central Wishart Matrices),
1.23 takayama 21: <a href="https://arxiv.org/abs/1707.02564"> arxiv:1707.02564 </a>
22: <li> T.Koyama,
23: An integral formula for the powered sum of the independent, identically and normally distributed random variables,
24: <a href="https://arxiv.org/abs/1706.03989"> arxiv:1706.03989 </a>
1.21 takayama 25: <li> H.Hashiguchi, N.Takayama, A.Takemura,
26: Distribution of Ratio of two Wishart Matrices and Evaluation of Cumulative Probability
27: by Holonomic Gradient Method,
28: <a href="https://arxiv.org/abs/1610.09187"> arxiv:1610.09187 </a>
29:
1.18 takayama 30: <li> R.Vidunas, A.Takemura,
31: Differential relations for the largest root distribution
32: of complex non-central Wishart matrices,
33: <a href="http://arxiv.org/abs/1609.01799"> arxiv:1609.01799 </a>
34:
1.20 takayama 35: <li> S.Mano,
36: The A-hypergeometric System Associated with the Rational Normal Curve and
37: Exchangeable Structures,
38: <a href="http://arxiv.org/abs/1607.03569"> arxiv:1607.03569 </a>
39:
1.19 takayama 40: <li> M.Noro,
41: System of Partial Differential Equations for the Hypergeometric Function 1F1 of a Matrix Argument on Diagonal Regions,
42: <a href="http://dl.acm.org/citation.cfm?doid=2930889.2930905"> ACM DL </a>
43:
1.12 takayama 44: <li> Y.Goto, K.Matsumoto,
45: Pfaffian equations and contiguity relations of the hypergeometric function of type (k+1,k+n+2) and their applications,
1.13 takayama 46: <a href="http://arxiv.org/abs/1602.01637"> arxiv:1602.01637 </a>
47:
48: <li> T.Koyama,
49: Holonomic gradient method for the probability content of a simplex
50: region
51: with a multivariate normal distribution,
52: <a href="http://arxiv.org/abs/1512.06564"> arxiv:1512.06564 </a>
53:
54:
55: <li> N.Takayama, S.Kuriki, A.Takemura,
56: A-Hpergeometric Distributions and Newton Polytopes,
57: <a href="http://arxiv.org/abs/1510.02269"> arxiv:1510.02269 </a>
58:
59: <li> G.Weyenberg, R.Yoshida, D.Howe,
60: Normalizing Kernels in the Billera-Holmes-Vogtmann Treespace,
61: <a href="http://arxiv.org/abs/1506.00142"> arxiv:1506.00142 </a>
62:
1.17 takayama 63: <li> C.Siriteanu, A.Takemura, C.Koutschan, S.Kuriki, D.St.P.Richards, H.Sin,
64: Exact ZF Analysis and Computer-Algebra-Aided Evaluation
65: in Rank-1 LoS Rician Fading,
66: <a href="http://arxiv.org/abs/1507.07056"> arxiv:1507.07056 </a>
67:
1.13 takayama 68: <li> K.Ohara, N.Takayama,
69: Pfaffian Systems of A-Hypergeometric Systems II ---
70: Holonomic Gradient Method,
71: <a href="http://arxiv.org/abs/1505.02947"> arxiv:1505.02947 </a>
72:
73: <li> T.Koyama,
74: The Annihilating Ideal of the Fisher Integral,
75: <a href="http://arxiv.org/abs/1503.05261"> arxiv:1503.05261 </a>
76:
77: <li> T.Koyama, A.Takemura,
78: Holonomic gradient method for distribution function of a weighted sum
79: of noncentral chi-square random variables,
80: <a href="http://arxiv.org/abs/1503.00378"> arxiv:1503.00378 </a>
81:
82: <li> Y.Goto,
83: Contiguity relations of Lauricella's F_D revisited,
84: <a href="http://arxiv.org/abs/1412.3256"> arxiv:1412.3256 </a>
1.12 takayama 85:
1.15 takayama 86: <li>
87: T.Koyama, H.Nakayama, K.Ohara, T.Sei, N.Takayama,
88: Software Packages for Holonomic Gradient Method,
89: Mathematial Software --- ICMS 2014,
90: 4th International Conference, Proceedings.
91: Edited by Hoon Hong and Chee Yap,
92: Springer lecture notes in computer science 8592,
93: 706--712.
94: <a href="http://link.springer.com/chapter/10.1007%2F978-3-662-44199-2_105">
95: DOI
96: </a>
97:
1.11 takayama 98: <li>N.Marumo, T.Oaku, A.Takemura,
99: Properties of powers of functions satisfying second-order linear differential equations with applications to statistics,
100: <a href="http://arxiv.org/abs/1405.4451"> arxiv:1405.4451</a>
101:
1.8 takayama 102: <li> J.Hayakawa, A.Takemura,
103: Estimation of exponential-polynomial distribution by holonomic gradient descent
104: <a href="http://arxiv.org/abs/1403.7852"> arxiv:1403.7852</a>
105:
106: <li> C.Siriteanu, A.Takemura, S.Kuriki,
107: MIMO Zero-Forcing Detection Performance Evaluation by Holonomic Gradient Method
108: <a href="http://arxiv.org/abs/1403.3788"> arxiv:1403.3788</a>
109:
1.4 takayama 110: <li> T.Koyama,
1.1 takayama 111: Holonomic Modules Associated with Multivariate Normal Probabilities of Polyhedra,
112: <a href="http://arxiv.org/abs/1311.6905"> arxiv:1311.6905 </a>
113:
114: <li> T.Hibi, K.Nishiyama, N.Takayama,
115: Pfaffian Systems of A-Hypergeometric Equations I,
116: Bases of Twisted Cohomology Groups,
117: <a href="http://arxiv.org/abs/1212.6103"> arxiv:1212.6103 </a>
1.22 takayama 118: (major revision v2 of arxiv:1212.6103).
119: Accepted version is at
120: <a href="http://dx.doi.org/10.1016/j.aim.2016.10.021"> DOI </a>
1.1 takayama 121:
122: <li> <img src="./wakaba01.png" alt="Intro">
123: <a href="http://link.springer.com/book/10.1007/978-4-431-54574-3">
124: T.Hibi et al, Groebner Bases : Statistics and Software Systems </a>, Springer, 2013.
125:
126: <li> <img src="./wakaba01.png" alt="Intro">
127: Introduction to the Holonomic Gradient Method (movie), 2013.
128: <a href="http://www.youtube.com/watch?v=SgyDDLzWTyI"> movie at youtube </a>
129:
1.2 takayama 130:
1.1 takayama 131: <li> T.Sei, A.Kume,
1.2 takayama 132: Calculating the Normalising Constant of the Bingham Distribution on the Sphere using the Holonomic Gradient Method,
1.1 takayama 133: Statistics and Computing, 2013,
134: <a href="http://dx.doi.org/10.1007/s11222-013-9434-0">DOI</a>
135:
1.4 takayama 136: <li> T.Koyama, A.Takemura,
1.2 takayama 137: Calculation of Orthant Probabilities by the Holonomic Gradient Method,
138: <a href="http://arxiv.org/abs/1211.6822"> arxiv:1211.6822</a>
139:
1.1 takayama 140: <li>T. Koyama, H. Nakayama, K. Nishiyama, N. Takayama,
141: Holonomic Rank of the Fisher-Bingham System of Differential Equations,
142: <!-- <a href="http://arxiv.org/abs/1205.6144"> arxiv:1205.6144 </a>-->
1.11 takayama 143: Journal of Pure and Applied Algebra (online),
144: <a href="http://dx.doi.org/10.1016/j.jpaa.2014.03.004"> DOI </a>
1.1 takayama 145:
146: <li>
147: T. Koyama, H. Nakayama, K. Nishiyama, N. Takayama,
148: Holonomic Gradient Descent for the Fisher-Bingham Distribution on the d-dimensional Sphere,
149: <!-- <a href="http://arxiv.org/abs/1201.3239"> 1201.3239 </a> -->
150: Computational Statistics (2013)
151: <a href="http://dx.doi.org/10.1007/s00180-013-0456-z"> DOI </a>
152:
153: <li> Hiroki Hashiguchi, Yasuhide Numata, Nobuki Takayama, Akimichi Takemura,
154: Holonomic gradient method for the distribution function of the largest root of a Wishart matrix,
155: <!-- <a href="http://arxiv.org/abs/1201.0472"> 1201.0472 </a> -->
156: Journal of Multivariate Analysis, 117, (2013) 296-312,
157: <a href="http://dx.doi.org/10.1016/j.jmva.2013.03.011"> DOI </a>
158:
159: <li> Tomonari Sei, Hiroki Shibata, Akimichi Takemura, Katsuyoshi Ohara, Nobuki Takayama,
160: Properties and applications of Fisher distribution on the rotation group,
161: <!-- <a href="http://arxiv.org/abs/1110.0721"> 1110.0721 </a> -->
162: Journal of Multivariate Analysis, 116 (2013), 440--455,
163: <a href="http://dx.doi.org/10.1016/j.jmva.2013.01.010">DOI</a>
164:
165: <li>T.Koyama, A Holonomic Ideal which Annihilates the Fisher-Bingham Integral,
166: Funkcialaj Ekvacioj 56 (2013), 51--61.
1.11 takayama 167: <a href="http://dx.doi.org/10.1619/fesi.56.51">DOI</a>
168: <!-- <a href="https://www.jstage.jst.go.jp/article/fesi/56/1/56_51/_article">jstage</a> -->
1.1 takayama 169:
170: <li>
171: Hiromasa Nakayama, Kenta Nishiyama, Masayuki Noro, Katsuyoshi Ohara,
172: Tomonari Sei, Nobuki Takayama, Akimichi Takemura ,
173: Holonomic Gradient Descent and its Application to Fisher-Bingham Integral,
174: <!-- <a href="http://arxiv.org/abs//1005.5273"> arxiv:1005.5273 </a> -->
175: Advances in Applied Mathematics 47 (2011), 639--658,
176: <a href="http://dx.doi.org/10.1016/j.aam.2011.03.001"> DOI </a>
1.13 takayama 177:
1.1 takayama 178: </ol>
179:
1.13 takayama 180: Early papers related to HGM. <br>
181: <ol>
182: <li>
183: H.Dwinwoodie, L.Matusevich, E. Mosteig,
184: Transform methods for the hypergeometric distribution,
185: Statistics and Computing 14 (2004), 287--297.
186: </ol>
187:
188:
189:
1.2 takayama 190: <h2> Three Steps of HGM </h2>
191: <ol>
1.10 takayama 192: <li> Finding a holonomic system satisfied by the normalizing constant.
1.2 takayama 193: We may use computational or theoretical methods to find it.
194: Groebner basis and related methods are used.
1.10 takayama 195: <li> Finding an initial value vector for the holonomic system.
1.2 takayama 196: This is equivalent to evaluating the normalizing constant and its derivatives
197: at a point.
198: This step is usually performed by a series expansion.
1.10 takayama 199: <li> Solving the holonomic system numerically. We use several methods
1.2 takayama 200: in numerical analysis such as the Runge-Kutta method of solving
201: ordinary differential equations and efficient solvers of systems of linear
202: equations.
203: </ol>
204:
1.1 takayama 205: <h2> Software Packages for HGM</h2>
1.14 takayama 206:
1.15 takayama 207: <ul>
208: <li>
1.16 takayama 209: CRAN package <a href="https://cran.r-project.org/web/packages/hgm/index.html"> hgm </a> (for R).
1.14 takayama 210:
1.15 takayama 211: <li>
1.14 takayama 212: Some software packages are experimental and temporary documents are found in
1.6 takayama 213: "asir-contrib manual" (auto-autogenerated part), or
214: "Experimental Functions in Asir", or "miscellaneous and other documents"
215: of the
216: <a href="http://www.math.kobe-u.ac.jp/OpenXM/Current/doc/index-doc.html">
1.7 takayama 217: OpenXM documents</a>
1.8 takayama 218: or in <a href="./"> this folder</a>.
1.10 takayama 219: The nightly snapshot of the asir-contrib can be found in the asir page below,
1.6 takayama 220: or look up our <a href="http://www.math.sci.kobe-u.ac.jp/cgi/cvsweb.cgi/">
1.8 takayama 221: cvsweb page</a>.
1.1 takayama 222: <ol>
1.9 takayama 223: <li> Command line interfaces are in the folder OpenXM/src/hgm
224: in the OpenXM source tree. See <a href="http://www.math.kobe-u.ac.jp/OpenXM">
225: OpenXM distribution page </a>.
1.14 takayama 226: <li> Experimental version of <a href="http://www.math.kobe-u.ac.jp/OpenXM/Math/hgm/"> hgm package for R </a> (hgm_*tar.gz, hgm-manual.pdf) for the step 3.
1.11 takayama 227: To install this package in R, type in
228: <pre>
229: R CMD install hgm_*.tar.gz
230: </pre>
1.10 takayama 231: <li> The following packages are
232: for the computer algebra system
233: <a href="http://www.math.kobe-u.ac.jp/Asir"> Risa/Asir</a>.
234: They are in the asir-contrib collection.
235: <ul>
236: <li> yang.rr (for Pfaffian systems) ,
237: nk_restriction.rr (for D-module integrations),
238: tk_jack.rr (for Jack polynomials),
239: ko_fb_pfaffian.rr (Pfaffian system for the Fisher-Bingham system),
240: are for the steps 1 or 2.
241: <li> nk_fb_gen_c.rr is a package to generate a C program to perform
1.7 takayama 242: maximal Likehood estimates for the Fisher-Bingham distribution by HGD (holonomic gradient descent).
1.10 takayama 243: <li> ot_hgm_ahg.rr (HGM for A-distributions, very experimental).
244: </ul>
1.1 takayama 245: </ol>
246:
1.15 takayama 247: </ul>
248:
1.1 takayama 249: <h2> Programs to try examples of our papers </h2>
250: <ol>
251: <li> <a href="http://www.math.kobe-u.ac.jp/OpenXM/Math/Fisher-Bingham-2"> d-dimensional Fisher-Bingham System </a>
252: </ol>
253:
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