Risa/Asir ,,. 2 Yoshihito Tachibana, Yoshiaki Goto, Tamio Koyama, Nobuki Takayama, Holonomic Gradient Method for Two Way Contingency Table

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1 Risa/Asir ,,. 2 Yoshihito Tachibana, Yoshiaki Goto, Tamio Koyama, Nobuki Takayama, Holonomic Gradient Method for Two Way Contingency Tables, arxiv: import("gtt ekn.rr");. 1 / 12

2 1 2 r 1 r 2 u. 2 p ij 0 (i, j). 3, β r 1 r 2, p u 1 u! Z(β; p), Z(β; p) =. β (i, j). u β E ij (p) = p ij log Z(β; p) p ij 5 2. p (, MLE). p u u! 2 / 12

3 \ ( ) T, β T = E(p) = (E ij (p)) p. 3 / 12

4 gtt ekn.rr gtt ekn, p = (p ij ) E ij (p) 1( ), 2(MLE). 1 1 : pu 1 u! Z. toric ideal (MCMC, 4 ), IPS,. 2 2 :. (i, j) exact method, p p ij =.. 1 Z(β; p) β (, HGM)? 2 Z Aomoto-Gel fand E(k, n) : Saito, Sturmfels, Takayama, Hypergeometric Polynomials and Integer Programming, Compositio Mathematica 115 (1999) / 12

5 , 1 E(k, n) : Y.Goto, K.Matsumoto, Pfaffian equations and contiguity relations of the hypergeometric function of type (k+1,k+n+2) and their applications, arxiv: modular method (,, (1989),, (1999), J.Faugerre (1999), linbox.org ) : Y.Tachibana,, 2016,. 3 2??? : N.Takayama, S.Kuriki, A.Takemura, A-hypergeometric distributions and Newton polytopes. arxiv: Risa/Asir gtt ekn.rr ( 2900, ) asir contrib update( update=1); 2 : p ij = 0 ( 0 ). cmle 3 5 / 12

6 E ij ( ) E ij, r i > 0, c j > 0,. p ij r i c j p ij p ij p kl p il p kj ( odds ), E odds. (p )/(r, c ) odds. 1 r i i. c j j. 2 odds i, k, j, l (?) 6 / 12

7 Theorem (,,, 2015) Newton New(Z) ( Z ξ p ) n d., ( odds log ). E[U] : R n /Im A T relint(new(z)) R n relint., f (ξ) = η ξ log Z(β, p(ξ)). 1 MLE.. 2, Newton. grad E. cmle 3. 3 MLE : (nuissance parameters r i, c j ) Sections 7 and / 12

8 , \ Before After CMLE: p = , CMLE , CMLE / 12

9 acetaminophen diclofenac sodium mefenamic acid death survival ( ) CMLE: ( ) odds : acetaminophen diclofenac sodium mefenamic acid death survival / 12

10 number_eval(gtt_ekn.expectation([[7,11],[12,4,2]], [[1,436481/55862,185809/66567],[1,1,1]]) 10 / 12

11 #!/bin/bash # $OpenXM: OpenXM/src/asir-port/cgi/cgi-cmle.sh,v /03/19 06:14:4 ## Replace $OpenXM_HOME by an absolute path of your cite. export OpenXM_HOME=/private/taka/OpenXM export HOME=/tmp source $OpenXM_HOME/rc/dot.bashrc export TMP=/tmp export CGI_ASIR_ALLOW="[(version) (asir_contrib_copyright) (gtt_ekn.cmle_3) (gtt_ekn.expectation) (number_eval)]" export CGI_ASIR_TLIMIT="3600" export OX_ASIR_LOAD_FILE=$OpenXM_HOME/src/asir-contrib/packages /src/gtt_ekn.rr export CGI_ASIR_MSG="<pre> Example. gtt_ekn.cmle_3([[3,3,1],[9,1,1]] by_g=1) </pre><br>" WFILE=/tmp/tmp-post-$$.txt tee /tmp/t.txt oxdecode_post >${WFILE} env CONTENT_LENGTH= wc -m ${WFILE} awk {print $1} sm1 -q -s "[(parse) (cgiasir.sm1) pushfile] extension oxnox cgiasir quit " <${WFILE} rm -f ${WFILE} exit 0 11 / 12

12 asir cgi apache24, httpd.conf, (...) LoadModule cgi_module modules/mod_cgi.so ScriptAlias /cgi-bin/ "/usr/local/www/apache24/cgi-bin/" (OS ) <Directory /usr/local/www/apache24/cgi-bin> (OS ) Options ExecCGI SetHandler cgi-script </Directory> /etc/crontab 59 * * * * root /private/taka/openxm/src/kan96xx/doc/httpd-asir2-kill.sh 59 * * * * root /usr/bin/killall sm1 CGI ASIR ALLOW asir.. CGI ASIR TLIMIT. OX ASIR LOAD FILE. oxdecode post POST GET. sm1 cgi ox asir. OpenXM/src/asir-port/cgi. 12 / 12

( ) (, ) arxiv: hgm OpenXM search. d n A = (a ij ). A i a i Z d, Z d. i a ij > 0. β N 0 A = N 0 a N 0 a n Z A (β; p) = Au=β,u N n 0 A

( ) (, ) arxiv: hgm OpenXM search. d n A = (a ij ). A i a i Z d, Z d. i a ij > 0. β N 0 A = N 0 a N 0 a n Z A (β; p) = Au=β,u N n 0 A ( ) (, ) arxiv: 1510.02269 hgm OpenXM search. d n A = (a ij ). A i a i Z d, Z d. i a ij > 0. β N 0 A = N 0 a 1 + + N 0 a n Z A (β; p) = Au=β,u N n 0 A-. u! = n i=1 u i!, p u = n i=1 pu i i. Z = Z A Au