![Overview
1
? оǤϡBArgȥԥåǥ(cDTM)һ㻯ФȥԥåȅgZ
ɥѥ`Ή֤ؓvԤ]Ǥ롢BArgե饯ʥ?ȥԥå
ǥ(cFTM)롣
? cFTMBArgǥʤΤǡĕΥॹפBA?ǷpäƤ
ܡġ(Ug?ڥƥȤҊ)ȥԥåֲӅgZֲL
Ԥդ]뤳ȤǤ롣
? ѥ`ƶϹŵĥȥԥåǥͬcFTMՓ|ӥ˥`
ǩ`ägYˤäơַȥԥåLrgԤޤϥդ
뤳ȤǤ뤳Ȥ_J롣](https://image.slidesharecdn.com/fractionaltopicv2-230317145407-70d44ae3/85/NLP2023-2-320.jpg)
![4
ȥԥåǥ(LDA)
Лr
Лr
ȥԥåֲ
20%
40%
40%
o
ӑ
ӻ
ĕ(BoW)
٥ȥԥåֲ
ȥԥåxk
ڥȥԥåȤ΅gZֲ
gZxk
ӑ
עҕ
o
90%
10%
5%
gZֲ
? LDAĕDZڵĤʥȥԥåֲ(Ԓ}γF_)ɤǥ롣
ĕڤθgZϤȥԥåǤ΅gZֲ(gZγF_)ˏäƳF
[Blei et.al.2003]](https://image.slidesharecdn.com/fractionaltopicv2-230317145407-70d44ae3/85/NLP2023-5-320.jpg)
![5
Лr
Лr
ȥԥåֲ
20%
40%
40%
Лr
Лr
Лr
o
ӑ
ӻ
ĕ
٥ȥԥåֲ
ȥԥåxk
(헷ֲ)
ӑ
עҕ
o
90%
10%
5%
gZֲ
?
??
? ?
?
??
? ?
ȥԥåǥ(LDA)
ڥȥԥåȤ΅gZֲ
gZxk
(헷ֲ)
ǥꥯֲ
Υѥ`
ǥꥯֲ
Υѥ`
?
ĕϿ]ʤ
? LDAˤƤĕϽQܤǤꡢrgϤʤ
ǥꥯֲ
ǥꥯֲ
ǥꥯֲ???헷ֲηֲ
헷ֲ???}xɢȤֲ](https://image.slidesharecdn.com/fractionaltopicv2-230317145407-70d44ae3/85/NLP2023-6-320.jpg)
![6
ʥߥå?ȥԥåǥ(DTM)
???1
???1
???1
???1
?
???1
??
??
??
??
?
??
???1
?
??
?
r? ? 1
ĕ
r?
ĕ
? ĕһˤϕrϵИ֤ġ
? rϵИ]Ǥȥԥåǥʥߥå?ȥԥåǥ(DTM)
?ȥԥåФ]??
ӑ
עҕ
o
90%
10%
5%
gZֲ
٥Υߥ
ʸ
90%
10%
5%
gZֲ
?ȥԥåڤ~ʹΉ仯]??
e.g., Փġ˥`
?
Лr
Лr
ȥԥåֲ
Υѥ`
ȥԥåֲ
(헷ֲ)
ȥԥå
gZֲ
Υѥ`
gZֲ
(헷ֲ)
softmax
softmax
[Blei and Lafferty 2006]](https://image.slidesharecdn.com/fractionaltopicv2-230317145407-70d44ae3/85/NLP2023-7-320.jpg)
![7
BArgʥߥå?ȥԥåǥ(cDTM)
? DTMǤĕΥॹפxɢĤĵgǤҪ롣
? FgĕϲgǰkФ졢ޤǷp륱`⤢ꡢˌI
ȥԥåֲȅgZֲΥѥ`????Brown\Ӥǥǥ
BArgʥߥå?ȥԥåǥ(cDTM)
g_^̤Brown\? = ?? ?0 ǤȤϡ
(1)?0 = 0 ?. ?.
(2)??BAǤ롣
(3)??϶Ԥ֤ġ
0 ?1 < ?2 < ? < ??, ??2
? ??1
, , ???
? ????1
Ϥ줾
(4)??ҎԤ֤ġ
? > ? 0, ?? ? ?
?~?(0, ? ? ?)
[Wang et.al. 2008]](https://image.slidesharecdn.com/fractionaltopicv2-230317145407-70d44ae3/85/NLP2023-8-320.jpg)
![9
? cDTMǤϡBrown\Ӥ|(1)ȥԥåФ??(2)ȥԥåڤ~ʹ
??Ή֤ˉ仯rgĤvS]Ǥʤ
ȥԥåֲȅgZֲΥѥ`????Fractional Brown\(fBm)ǥǥ
BArgե饯ʥ?ȥԥåǥ(cFTM)
g_^̤? 0,1 HurstָFractional Brown\
?? = ??
?
?0 ǤȤϡ
(1)?0 = 0 ?. ?.
(2)??BAǤ롣
(3)??ҎԤ֤ġ
? > ? 0, ?? ? ?
?~?(0, ? ? ? 2?)
BArgե饯ʥȥԥåǥ(cFTM)
[Mandelbrot and Ness 1968]](https://image.slidesharecdn.com/fractionaltopicv2-230317145407-70d44ae3/85/NLP2023-10-320.jpg)
![12
cFTMՓ|
1:cFTMΥȥԥåֲӅgZֲLԤޤϥդ֤
}: 2Ĥδ_^??, ?? = ?(??)ϥϩ`ָ?ǜyäͬLԤޤϥդ֤ġ
?(?)һAvнʤʤ餫vȤ롣
^: [Hayashi & Nakagawa TBA]
}:եȥޥåv?(?)(ֿܤ)ץåBAǤ
^: [Gao & Pavel 2017] Proposition 4
}: (ֿܤ)ץåBAvнһAv֤ġ
^: [ɼ 1980]
2:ɥեȤʤcFTMΥȥԥåֲȅgZֲɥѥ` (L)ȶ
mϹŵĤ(rgkչΤʤ)ȥԥåǥΥѥ`ƶˎŤǤ
Գ:r?ˤ????·ֲҎֲäʾĤ˱ȤǤ
Գ:إ`BAԤvΥ()һ¤뤳Ȥʾ(ƥ`չ_)](https://image.slidesharecdn.com/fractionaltopicv2-230317145407-70d44ae3/85/NLP2023-13-320.jpg)
![21
ޤȤ
LDA DTM cDTM cFTM
ĕΕrg]
ॹפǷp
g]
ȥԥåֲȅgZֲ
LԤȥդ]
ARǥ Brown\ Fractional Brown\
lӤǥ
? ΰkչȤơȥԥåֲ뤤υgZֲֲ΄ʵĤӋ㷽줹롣ˤä
꼚ȤǤΥȥԥåΉӤǥ뻯Ǥ롣
ޤоǤϿ]Ƥʤɥե헤ǾΤʥ˥`ͥåväѧ뤳Ȥ롣
? оȤƥȥԥåޤυgZֲLӛԤդ٬FĿĤΤɥե헤]
ʤäޤfBmһˤ϶Ԥ֤ʤᡢKalman FilterȤ˥ȥԥåֲ뤤υgZ
ֲֲʵĤӋ㤹뤳ȤǤʤ
?BArgե饯ʥ?ȥԥåǥ(cFTM)Фä](https://image.slidesharecdn.com/fractionaltopicv2-230317145407-70d44ae3/85/NLP2023-22-320.jpg)
BArgե饯ʥ?ȥԥåǥ(NLP2023 ?UgɥᥤΤZI)
- 1. The views expressed here are our own and do not necessarily reflect the views of Nomura Asset management. Any errors and inadequacies are our own. ZIѧ(NLP2023) Ʃ`ޥå 1 : ?UgɥᥤΤZI(1) 2023/3/15(ˮ) 17:30-19:00 BArgե饯ʥ?ȥԥåǥ a.Ұ奢åȥޥͥʽ b.|ѧѧԺѧо д ۡ ƽٱ a a b Yb\ȶ˼gо ꥵ`ե`
- 2. Overview 1 ? оǤϡBArgȥԥåǥ(cDTM)һ㻯ФȥԥåȅgZ ɥѥ`Ή֤ؓvԤ]Ǥ롢BArgե饯ʥ?ȥԥå ǥ(cFTM)롣 ? cFTMBArgǥʤΤǡĕΥॹפBA?ǷpäƤ ܡġ(Ug?ڥƥȤҊ)ȥԥåֲӅgZֲL Ԥդ]뤳ȤǤ롣 ? ѥ`ƶϹŵĥȥԥåǥͬcFTMՓ|ӥ˥` ǩ`ägYˤäơַȥԥåLrgԤޤϥդ 뤳ȤǤ뤳Ȥ_J롣
- 6. 5 Лr Лr ȥԥåֲ 20% 40% 40% Лr Лr Лr o ӑ ӻ ĕ ٥ȥԥåֲ ȥԥåxk (헷ֲ) ӑ עҕ o 90% 10% 5% gZֲ ? ?? ? ? ? ?? ? ? ȥԥåǥ(LDA) ڥȥԥåȤ΅gZֲ gZxk (헷ֲ) ǥꥯֲ Υѥ` ǥꥯֲ Υѥ` ? ĕϿ]ʤ ? LDAˤƤĕϽQܤǤꡢrgϤʤ ǥꥯֲ ǥꥯֲ ǥꥯֲ???헷ֲηֲ 헷ֲ???}xɢȤֲ
- 7. 6 ʥߥå?ȥԥåǥ(DTM) ???1 ???1 ???1 ???1 ? ???1 ?? ?? ?? ?? ? ?? ???1 ? ?? ? r? ? 1 ĕ r? ĕ ? ĕһˤϕrϵИ֤ġ ? rϵИ]Ǥȥԥåǥʥߥå?ȥԥåǥ(DTM) ?ȥԥåФ]?? ӑ עҕ o 90% 10% 5% gZֲ ٥Υߥ ʸ 90% 10% 5% gZֲ ?ȥԥåڤ~ʹΉ仯]?? e.g., Փġ˥` ? Лr Лr ȥԥåֲ Υѥ` ȥԥåֲ (헷ֲ) ȥԥå gZֲ Υѥ` gZֲ (헷ֲ) softmax softmax [Blei and Lafferty 2006]
- 8. 7 BArgʥߥå?ȥԥåǥ(cDTM) ? DTMǤĕΥॹפxɢĤĵgǤҪ롣 ? FgĕϲgǰkФ졢ޤǷp륱`⤢ꡢˌI ȥԥåֲȅgZֲΥѥ`????Brown\Ӥǥǥ BArgʥߥå?ȥԥåǥ(cDTM) g_^̤Brown\? = ?? ?0 ǤȤϡ (1)?0 = 0 ?. ?. (2)??BAǤ롣 (3)??϶Ԥ֤ġ 0 ?1 < ?2 < ? < ??, ??2 ? ??1 , , ??? ? ????1 Ϥ줾 (4)??ҎԤ֤ġ ? > ? 0, ?? ? ? ?~?(0, ? ? ?) [Wang et.al. 2008]
- 10. 9 ? cDTMǤϡBrown\Ӥ|(1)ȥԥåФ??(2)ȥԥåڤ~ʹ ??Ή֤ˉ仯rgĤvS]Ǥʤ ȥԥåֲȅgZֲΥѥ`????Fractional Brown\(fBm)ǥǥ BArgե饯ʥ?ȥԥåǥ(cFTM) g_^̤? 0,1 HurstָFractional Brown\ ?? = ?? ? ?0 ǤȤϡ (1)?0 = 0 ?. ?. (2)??BAǤ롣 (3)??ҎԤ֤ġ ? > ? 0, ?? ? ? ?~?(0, ? ? ? 2?) BArgե饯ʥȥԥåǥ(cFTM) [Mandelbrot and Ness 1968]
- 11. 10 ? = {??}?0 ?? = {??,?}?0 ??? = ??? ?? ?? + ????? ? ???,? = ???,? ??,? ?? + ??,???? ? ?? (?) = exp(?) exp(?) ?? (??) = exp(??) exp(??) ?~categorical(?? ) ?~categorical(?? ) BArgե饯ʥȥԥåǥ(cFTM) ? cFTMΥեǥ ? cFTMɥǥ
- 12. 11 cFTMՓ| ? cFTMcDTMһ㻯ǤBArgǥǤ롣 fBm|BAԤҎԣ餫 ? = 0.1 ? = 0.25 ? = 0. 5 ? = 0. 75 ? = 0. 9 ? = 0.5ΤȤfBmBrown\Ӥһ¤롣 ? > 1/2 ΤȤfBmΉ֤Lӛ(Pv)֤ġ ? < 1/2ΤȤfBmΉ֤ϥդ (Pؓv)֤ġ _^{??}?0ˌΉ??,? = ?? ? ??Lӛ(resp. դ)֤ĤȤϴΤɤĤȣ ? ?? Cov ?0,?, ? ??1 ?,?? = (< ), ?? > 0. Lӛ դ
- 13. 12 cFTMՓ| 1:cFTMΥȥԥåֲӅgZֲLԤޤϥդ֤ }: 2Ĥδ_^??, ?? = ?(??)ϥϩ`ָ?ǜyäͬLԤޤϥդ֤ġ ?(?)һAvнʤʤ餫vȤ롣 ^: [Hayashi & Nakagawa TBA] }:եȥޥåv?(?)(ֿܤ)ץåBAǤ ^: [Gao & Pavel 2017] Proposition 4 }: (ֿܤ)ץåBAvнһAv֤ġ ^: [ɼ 1980] 2:ɥեȤʤcFTMΥȥԥåֲȅgZֲɥѥ` (L)ȶ mϹŵĤ(rgkչΤʤ)ȥԥåǥΥѥ`ƶˎŤǤ Գ:r?ˤ????·ֲҎֲäʾĤ˱ȤǤ Գ:إ`BAԤvΥ()һ¤뤳Ȥʾ(ƥ`չ_)
- 15. 14 g^ ? ǥˤrϵv΄u뤿ᡢش٥(|ձġBrexit)ǰǤ ȥԥåƤˤĤƶԵĤuФ ? ǩ`å ˥`ӛ¥ǩ`äơUgȥԥåˤĤƱ^Фä ǩ`åȤϥ`˥`(https://jp.reuters.com/ )¤gΥǩ`ȡä (a)|ձĤǰ5g(2011/3/83/12) Τg˒i˥`ӛ178ʹä (b)BrexitͶƱǰ1(2016/6/12016/6/30) Τg˒i˥`ӛ316ʹä ? ӛ¤Ҋ~Ӌ㤷BoWä ~Hˡ4ӛ¤ˤޤƤʤgZ50%Ϥӛ¤˺ޤƤgZ ⤷
- 16. 15 ǥO äǥOˤĤƤ¤ͨꡣ ? ϩ`ָ0.1 ȥԥå˥դ , 0.5, 0.9(ȥԥåLԤ) ˤĤƱ^ ? ȥԥå٤5O ? ǥO gYΌg϶ڤǤ뤿ȥԥåֲgЉ仯һȥԥåڤ΅gZֲΉ仯 ĤСȤ붨롣 äƱgYǤϡȥԥåֲɥѥ`?ΤߕrgkչȤ ? ǥѧ MCMCץ(Stan)äƥǥѧФä ͻץwarmupƶϤ줾ǥեȤ2000,1000,1000,4 O
- 17. 16 20110309ˮ ʽЈ?Ĥ긶䡢ԭ?ݥȥR 11:57 `Ӣ`륹ȥʥI 11:34 ݥȥؤ꼱ϕNȜgҪȤΑ֤ 11:34 Lgסլ`ՈָϕNøƤ3¤֤ˮ 11:32 ӥɤΑ܇ꠡΎڈ 11:31 ΣЈ?P`ɥϕNݥȥ֧Ԯ 11:28 ȤwнֹOСӥϑ餦ե 11:14 ʽЈޥ`9գ 10:56 й˰v֤쥢`ݔΰkzߤե` 10:54 ̣ɣ£ϣңӢyЅf 10:34 ʽЈޥ`9գ 9:53 |ϥʽޤޤǥե㑱I 9:46 Lgסլ`ՈָϕNøƤ3¤֤ˮ 9:32 g^(˥`) ॹפ BA
- 18. 17 ǩ`å (|ձǰ) kǰӛ kӛ kǰϡvBĤ˷ڤݤԼƤ롣 kϡ?ԭkvB˥`ˉӤƤ롣 𡹤γFl ԭkγFl 2011/3/11 15:11 mDZh7ε𡢖|Ǥ5 2011/3/11 16:16 LΖ|ʽЈӰ푴_Jޤؤ չ_ 2011/3/12 13:02 u1ԭk©οԡͨ 侀Ubʡ 2011/3/8 1:23 йեƤeOؔ@A 2011/3/9 23:31 ӥɤΑ܇ꠡΎڈ 2011/3/10 23:45 ʽЈ?Ĥ긶ڥ¤ ʧTнyӋ
- 19. 18 ? = 0.1 ? = 0. 5 ? = 0. 9 5ĤΥȥԥåϤ줾졢ȥԥå1(ྀ)Ϲڥ˥`ȥԥå2()ϹЛrȥԥå3(v)Ϻ ˥`ȥԥå4(ྀ)ϺЛrȥԥå5(Ͼ)йvBȽዤǤ롣 ٥ǰϥӥݲй?UgȤä˥`ӺЛrΥȥԥåफä һǡ٥Ĥvڥ˥`ӹЛrΥȥԥåϕN ٥ǰǥȥԥåƤ 仯뤳ȤϤʤ ԥåvԤƤ롣 ٥˥ȥԥåΰk_ʤ ʉ(դ)ߤ Iߤgλø ?Hurstָ?뤳Ȥǥ뤬L/դ֤ĥȥԥå΄EǤ롣 g^ (|ձǰǤΥȥԥå仯)
- 20. 19 ǩ`å (Brexitǰ) xgʩKˤޤǤӛ xgʩӛ BrexitǷǤvͶƱǰϡBrexitvBԼͶƱŤBƉӃAä һǡͶƱKᤫ顢BrexitvBԼϼ˜p٤Ƥ ͶƱγFl xѡγFl 2016/6/28 7:56 ީ`åȥ飺Ӣݥ134롢 Ӣ¤ΤݥɉӤT 2016/6/29 2:41 6Mmָ10 θˮ 2016/6/30 15:37 սUƽС4վA졢Iһ Ѳȃҏߤǰ 2016/6/13 11:46 ǰսUƽϴA䡢ӢΣ xѤ 2016/6/21 5:23 ʽЈϕNӢţxё ͤǰĸ 2016/6/24 13:09 ӢͶƱţxɤأ £
- 21. 20 g^ (BrexitǰǤΥȥԥå仯) ? = 0.1 ? = 0. 5 ? = 0. 9 H=0.1ǤΥȥԥå1,2H=0.5,0.9ǤΥȥԥå1xѤƤιͶƱvBΥ˥`ʾƤ ͶƱޤǴεڤvB˥`ӤxћQᡢһܞƹͶƱvBΥ˥`Ϝp٤ xћQᡢH=0.1Ǥ϶ڵĤӛ¤Μp٤˷ӳƴȥԥåδ_ʤ¤ һHˏLڥȥɤȤƤӛ¤ΉӤӳơȥԥå_ʤS֤Ƥ롣 ȥԥåLԤӳ ͶƱvBΥ˥`xK LڵĤˤωӃAƶ ȥԥåΥդӳơ ͶƱvBΥ˥`x KᡢһܞƜp٤ƶ ?BrexitǰǤHurstָ?뤳Ȥǥȥԥå仯Ǥ뤳Ȥ_JǤ Iߤgλø
- 22. 21 ޤȤ LDA DTM cDTM cFTM ĕΕrg] ॹפǷp g] ȥԥåֲȅgZֲ LԤȥդ] ARǥ Brown\ Fractional Brown\ lӤǥ ? ΰkչȤơȥԥåֲ뤤υgZֲֲ΄ʵĤӋ㷽줹롣ˤä 꼚ȤǤΥȥԥåΉӤǥ뻯Ǥ롣 ޤоǤϿ]Ƥʤɥե헤ǾΤʥ˥`ͥåväѧ뤳Ȥ롣 ? оȤƥȥԥåޤυgZֲLӛԤդ٬FĿĤΤɥե헤] ʤäޤfBmһˤ϶Ԥ֤ʤᡢKalman FilterȤ˥ȥԥåֲ뤤υgZ ֲֲʵĤӋ㤹뤳ȤǤʤ ?BArgե饯ʥ?ȥԥåǥ(cFTM)Фä
- 23. 22 ο Gao, B., & Pavel, L. (2017). On the properties of the softmax function with application in game theory and reinforcement learning. arXiv preprint arXiv:1704.00805. Blei, D. M., Ng, A. Y., & Jordan, M. I. (2003). Latent dirichlet allocation. Journal of machine Learning research, 3(Jan), 993-1022. Blei, D. M., & Lafferty, J. D. (2006, June). Dynamic topic models. In Proceedings of the 23rd international conference on Machine learning (pp. 113-120). Wang, C., Blei, D., & Heckerman, D. (2008, July). Continuous time dynamic topic models. In Proceedings of the Twenty-Fourth Conference on Uncertainty in Artificial Intelligence (pp. 579-586). Mandelbrot, B. B., & Van Ness, J. W. (1968). Fractional Brownian motions, fractional noises and applications. SIAM review, 10(4), 422-437. ɼֹ. (1980) Aѧ 2 T 1 (Vol. 2). |ѧ. Kohei Hayashi & Kei Nakagawa (2023). Generating Fractional Time-Series with Latent Fractional-Net To be appear