Recursive segmentation procedure based on the Akaike information criterion test

Transcription

1 ecuve egmenaon pocedue baed on he Aae nfomaon ceon e A-Ho SAO Depamen of Appled Mahemac and Phyc Gaduae School of Infomac Kyoo Unvey a@.yoo-u.ac.jp JAPAN

2 Oulne Bacgound and Movaon Segmenaon pocedue baed on Aae Infomaon eon Numecal Analy Empcal analy and dcuon Summay A-Ho Sao ecuve Segmenaon Pocedue Baed on he Aae Infomaon eon e 37 h Annual ompue Sofwae and Applcaon onfeence OMPSA A-Ho Sao A ompehenve Analy of me See Segmenaon on he Japanee Soc Pce o appea n Poceda of ompue Scence axv:05.033

3 Movaon apalm ha no pupoe h ju a mechanm o develop a yem by ung pof whch wa ganed fom he yem. Fm n moden capalm mae a bune cycle. he boom-bu cycle obeved n fnancal mae.

4 ze of yem Bune cycle conucon deucon developmen Sauaon deucon deucon auaon auaon auaon developmen developmen developmen conucon conucon conucon me

5 og eun of foegn echange ae EU/JPY Jan 000 o Dec 009 he nonaonay of fnancal me ee one of he mpoan popee

6 How do we deal wh nonaonay me ee How do we ea he nonaonay me ee obeved n fnancal mae? Boh mple and elable mehod egadng acal gnfcance level of e ac hould be deed. he me ee egmenaon can be ued fo h pupoe.

7 A mehod o egmen non-aonay me ee no aonay egmen Goldfeld and Quand 973 A ecuve enopc cheme 0 [4] S.M. Goldfeld and.e. Quand: A Maov model fo wchng egeon Jounal of Economec Vol. 973 pp. 36 [5] S.A. heong.p. Fona G.H. ee J.. Ko W.S. Ym D.Y. Xu and Y. Zhang: he Japanee Economy n e: A me See Segmenaon Sudy Economc E-jounal on hp:// ep ep ep ep ep b a y b a y y h

8 One-dmenonal cae A-Ho Sao A ompehenve Analy of me See Segmenaon on he Japanee Soc Pce axv:05.033

9 Nonaonay me ee Nonaonay me ee ae aumed o con of eveal locally aonay me ee wh dffeen ac he me ee conng of 4 egmen. Each egmen ampled fom a zeo-mean nomal dbuon wh dffeen vaance. he vaance e a 3 and 4 fom he lef egmen. he lengh of each egmen e a 500.

10 Segmenaon Pocedue e g be a Gauan deny funcon paameezed by and : Aume wo ype of lelhood funcon fo obevaon = Defne he -lelhood ao beween and ep g g g g Null hypohe: Alenave hypohe:

11 Appomaon of elhood-ao g g g H g H n g H g nh g g g n n d whee 0 n n

12 Segmenaon pocedue ag ma * one ge ˆ ˆ ˆ emao he mamum lelhood a and eplacng n n n n n n n n n n ˆ ˆ ˆ

13 ecuve egmenaon pocedue he egmenaon pocedue appled o each egmen ecuvely. he emnaon condon gven by c. If ma le han c hen we do no apply he egmenaon pocedue any moe.

14 Wl heoem he hehold value c elaed o he gnfcance level of accepance. Accodng o Wl heom he pobably deny of appomaely ampled fom a chquaed deny wh a degee of feedom equal o he dffeence beween he numbe of fee paamee. p e

15 Infomaon ceon e fo M- dmenonal mulple me ee he Aae nfomaon ceon of a model wh K model paamee fo obevaon AI ˆ K ˆ whee he lelhood value of he model wh he mamum lelhood emao ˆ.

16 Sgnfcance level P 0 / dy e y AI AI y AI AI a: egulazed ncomplee gamma funcon

17 Afcal me ee p ep 0. = 0. = =3.0 e aume he me ee conng of 3 egmen. he f 00 pon ae ampled fom a nomal dbuon wh mean 0. and andad devaon.0. he econd 50 pon ae ampled fom a nomal dbuon wh mean -0. and andad devaon.5 and he hd 00 pon ae ampled fom a nomal dbuon wh zeo-mean and andad devaon 3.0.

18 Mean = 0. Va =.0 =00 Mean = -0. Va =.5 =00 Numecal Sudy Mean = 0.0 Va = 3.0 =50 Fom he me ee how we can deemne egmen bounday? he egmenaon pocedue appled o h afcal me ee wh c =0. he egmenaon pocedue can epaae he me ee no 3 elemen.

19 Empcal Analy of Japanee ecue he daa on daly pce of compane led on he f econ of he oyo Soc Echange fo he peod fom 4 h Januay 000 o h Decembe bune day. 43 compane whch la moe han 0 yea ae eleced Majo Japanee compane uch a oyoa Hach Panaonc and o on ae ncluded.

20 Daa and paamee og-eun of oc ae defned a E O M whee E and O ae endng and openng pce a day. N he numbe of oc: M=43 he numbe of obevaon: n=675 c fed a 0.

21 he daly numbe of egmen boundae I 003 o 007 II he end of 007 III Sepembe 008 IV Mach 990 V Apl 0 a June 000 b Apl 004 c Febuay 006 d 007 o 009 e Mach 0

22 Qunle ac lafy egmen no 5 caegoe fom he malle vaance: aegoy %-0% aegoy %-40% aegoy 3 4%-60% aegoy 4 6%-80% aegoy 5 8%-00%

23 he numbe of oc n each caegoy I 003 o 007 II he end of 007 III Sepembe 008 IV Mach 990 V Apl 0 eul fom me ee fo he peod fom 000 o 0

24 eul Fom he end of 00 he numbe of he fouh and ffh qunle deceaed ecovey phae Fom he begnnng of 004 he numbe of he f qunle nceaed ecovey phae Fom 004 o 007 he numbe of he f and econd qunle oo hgh value boom phae Fom he end of 007 he numbe of he f qunle haply dopped ah phae A he end of 008 he numbe of he ffh qunle haply nceaed phae Fom he end of 009 he numbe of he f qunle nceaed and he numbe of he ffh qunle deceaed ecovey phae

25 Sably nde Sably nde = #+#-#4-#5 Ea Japan Eahquae ehman hoc

26 Sably nde = #+#-#4-#5 OPIX OPIX 00 OPIX compoe ehman hoc Ea Japan Eahquae

27 Mulvaae cae A-Ho Sao ecuve Segmenaon Pocedue Baed on he Aae Infomaon eon e 37 h Annual ompue Sofwae and Applcaon onfeence OMPSA

28 Segmenaon pocedue fo muldmenonal me ee e = M = be he M- dmenonal mulple -eun me ee defned a = + whee = + he echange ae of -h cuency pa a me. e u aume ha he mulvaae me ee con of n egmen ampled fom n dffeen mulvaae nomal dbuon = n. M M p μ ep M j j / j / j

29 elhood-ao o deemne he n aonay egmen fom he gven mulple me ee = M we employ he ecuve egmenaon pocedue. p p p μ μ μ he null model: he alenave model:

30 elhood-ao μ μ μ p p p m - m

31 elhood-ao j j j j j j j j j ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

32 Geneal cae By ung M-dmenonal pobably deny p paameezed by we can we a whee H[p] Shannon enopy defned a ] [ ] [ ] [ θ θ θ θ θ θ p H p H p H p p p θ θ θ d d ] [ p p p H M

33 elhood-ao ˆ] [ ] ˆ [ ] ˆ [ ] [ ] [ ] [ θ θ θ θ θ θ p H p H p H p H p H p H Δ Empcally he -lelhood-ao compued fom mamum lelhood emao. p p p ag ma ˆ ag ma ˆ ag ma ˆ θ θ θ θ θ θ θ θ θ

34 AI fo he..d. M-dmenonal Gauan dbuon wh obevaon 3 ˆ ˆ 3 ˆ M M M M AI M M M M AI 3 3 ˆ ˆ ˆ M M M M AI AI AI

35 ecuve egmenaon pocedue he pecum of ha a mnmum a ome me ha we denoe by * * ag mn Δ AI A he emnaon condon a acal gnfcance level compued fom he booap dbuon.

36 Sacal gnfcance level e 0 < < be a ao o deemne he Jacnfe equence = +[ * ] whee andomly eleced wh he ame pobably fo 0 < < []+ = +[-*] whee andomly eleced wh he ame pobably fo * < < [ * ]+. Fom Jacnfe equence he Jacnfe e ac hough Jacnfe vaance-covaance mace.

37 he Jacnfe e ac he Jacnfe e ac compued fom whee 3 ~ ~ * ~ * * M M AI ' ' ' ' ' ' * ' ' * * * * * * * * * * * ~ ~ ~ ~ ~ ~ j j j j j j j j j j j

38 he emaon eo of he e ac he acal gnfcance level emaed fom he hocal pobably deny ~ K * 0 P * 0 AI AI K ~ whee K * 0 he numbe of AI ~ even whee * 0 afed. AI K : he numbe of booap al.

39 ecuve egmenaon pocedue * ag mn Δ * ag mn Δ

40 Sample eo he ample empcal vaance-covaance ma noe-deed becaue of fnene of me ee. Such ample eo can be evaluaed fom egenvalue dbuon. he egenvalue dbuon of an empcal vaance-covaance ma alo depend on he ao beween he lengh of he daa e and he dmenon of mulvaae me ee M.

41 Egenvalue deny epeenaon 3 d d d M M M M M M M M M M AI whee and epeen -h evenvalu of he coepondng vaance-covaance mace: and epecvely. and epecvely epeen peca of hee mace.

42 Sample eo In he cae of M-dmenonal mulvaae Gauan uncoelaed dencally dbued andom vaable M/< he deny of he egenvalu of he ample vaance-covaance ma appomaed by he Mačeno-Pau deny: M 0 ohewe whee and M / a cale faco elaed o he vaance of ndvdual degee of feedom.

43 Sample eo A M = he egenvalue deny can be appomaed a whee + =4. In h cae he negand n become ngula a =0 and n effec ll-defned. Fom acal unceany when M appoache and hu h mae paccally mpoble o emae * popely.

44 Sample eo he uaon even woe fo M/> nce hen he deny ha a pea a =0. he negand of well defned only f > M. ypcally o dnguh wo egenvalue of he covaance ma one need M > a whee he coeffcen a nveely popoonal o he quae of he dffeence of he egenvalue n equaon. houghou h analy we e a = 3. he lengh of each egmen hould be geae han 3M.

45 Numecal Analy e daa he vaance-covaance ma a egmen ξ A ξ A ξ A ξ A A A EA A A ξ ξ A dawn fom..d. andad nomal dbuon: lm 0 lm j

46 Numecal mulaon M=0 = a h =0.0 K=000

47 Empcal analy 30 cuency pa conng of AUD B AD HF EU GBP JPY MXN NZD SGD USD and ZA I analyzed daly -eun me ee of AUD/JPY B/JPY AD/JPY HF/JPY EU/AUD EU/SGD EU/USD EU/ZA GBP/JPY MXN/JPY NZD/JPY SGD/JPY USD/AUD USD/B USD/AD USD/HF USD/GBP USD/JPY USD/MNX USD/NZD USD/SGD USD/AD USD/HF USD/GBP USD/JPY USD/MNX USD/NZD USD/SGD USD/ZA and ZA/JPY. Duaon: Januay 3 00 o Decembe hee ae 760 daa pon n h mulple me ee.

48 Segmen Sa dae End dae M=30 K=000 a h = egmen Paba hoc ehman hoc Euo hoc US deb celng c

49 AUD/EU

50 AUD/EU Sa dae End dae mean vaance

51 oncluon An nfomaon ceon AI e wa popoed fo a mue of mulvaae Gauan dbuon. he popoed mehod wa confmed o deec he egmened bounday wh 6% elave eo. he popoed mehod wa appled fo a eun me ee conng of 30 cuency pa and egmen wee obaned. Some egmen wee confmed o coepond o ccal even uch a he Paba hoc n 007 he ehman hoc n 008 he Euopean Soveegn deb c n 00 and he US debcelng c n 0.

52 han you fo you nd aenon A-Ho Sao Depamen of Appled Mahemac and Phyc Gaduae School of Infomac Kyoo Unvey A-Ho Sao ecuve Segmenaon Pocedue Baed on he Aae Infomaon eon e 37 h Annual ompue Sofwae and Applcaon onfeence OMPSA