APPLY MARKOV CHAINS MODEL AND FUZZY TIME SERIES FOR FORECASTING

Transcription

1 MINISRY OF EDUCAION AND RAINING VIENAM ACADEMY OF SCIENCE AND ECHNOLOGY GRADUAE UNIVERSIY OF SCIENCE AND ECHNOLOGY DAO XUAN KY APPLY MARKOV CHAINS MODEL AND FUZZY IME SERIES FOR FORECASING Major: Mah Fudameals for Iformacs Code: SUMMARY OF MAHEMAICS DOCORAL DISSERAION Ha No, 207

2 hs wor s compleed a: Graduae Uversy of Scece ad echology Veam Academy of Scece ad echology Supervsor : Assoc. Prof. Dr. Doa Va Ba Supervsor 2: Dr. Nguye Va Hug Revewer : Revewer 2: Revewer 3: hs Dsserao wll be offcally preseed fro of he Docoral Dsserao Gradg Commee, meeg a: Graduae Uversy of Scece ad echology Veam Academy of Scece ad echology A. hrs. day. moh. year. hs Dsserao s avalable a:. Lbrary of Graduae Uversy of Scece ad echology 2. Naoal Lbrary of Veam

3 LIS OF PUBLISHED WORKS [] Dao Xua Ky ad Luc r uye. A marov-fuzzy combao model for soc mare forecasg. Ieraoal Joural of Appled ahemacs ad SascsM, 55(3):09 2, 206. [2] Đào Xuâ Kỳ, Lục rí uye, va Phạm Quốc Vươg. A combao of hgher order marov model ad fuzzy me seres for soc mare forecasg. I Hộ hảo lầ hứ 9: Mộ số vấ đề chọ lọc của Côg ghệ hôg và ruyề hôg, Hà Nộ, pages 6, 206. [3] Đào Xuâ Kỳ, Lục rí uye, Phạm Quốc Vươg, va hạch hị Nh. Mô hh marov-chuỗ hờ ga mờ rog dự báo chứg hoá. I Hộ hảo lầ hứ 8: Mộ số vấ đề chọ lọc của Côg ghệ hôg và ruyề hôg, P HCM, pages 9 24, 205. [4] Lục rí uye, Nguyễ Vă Hug, hạch hị Nh, Phạm Quốc Vươg, Nguyễ Mh Đức, va Đào Xuâ Kỳ. A ormal-hdde marov model model forecasg soc dex. Joural of Compuer Scece ad Cyberecs, 28(3):206 26, 202. [5] Dao Xua Ky ad Luc r uye. A Hgher order Marov model for me seres forecasg. Ieraoal Joural of Appled ahemacs ad SascsM, vol 57(3):-8, 208.

4 Iroduco he me seres forcasg wh predve varable objec X chagg over me order o acheve predcve accuracy s always a challege o scess, o oly Veam bu also globally. Because s o easy o fd a suable probably dsrbuo for hs predcve varable objec a he po was bor. Hsorcal daa eed o be colleced ad aalyzed, order o fd a perfec f. I s, however, a dsrbuo ca oly f wh sascs a parcular me me seres aalyss, ad vares a oher cera po of me. herefore, he use of a fxed dsrbuo for he predcable objec s o applcable for hs aalyss. For he above meoed reaso, he buldg of predcable me seres forcasg model requres coeco ad sycogo bewee hsorcal ad fuure sascs, order o se up a depede model bewee daa obaed a prese ad he pas -, -2. If he coeco X X 2X2 px p qq s se up, we ca geerae a auoregressve egraed movg average (ARIMA) [5] model. hs model s applcaable wdely for s praccal heory ad ergraed o almos curre sascal sofware such as Evews, SPSS, malab, R, ad ec. I s, however, may real me sequecg shows ha hey do o chage learly. herefore, model such as ARIMA does o su. R. Parrell poed ou [28] ha here s a o-lerable coeco ecoomc or facal me seres varace dcaors. he geeralzed auoregressve codoal heerosedascy (GARCH) [25,28] s he mos popular o-lerable me seres forecasg aalyss o meo. he lmao of hs model les he assumpo ha sascs vary a fxed dsrbuo (ormally sadard dsrbuo), whle acual sascs shows ha dsrbuo s sascally sgfca [39] (whle sadard dsrbuo has a balaced varao). Aoher me seres forecasg s Arfcal Neural Newor (ANN whch was developed recely. ANN models do o based o deermsc dsrbuo of sascs; sead fucos le huma bra ryg o fd rules ad pahes o rag daa, expermeal esg, ad resul summarzg. ANN model s usually used for sascs classfcao purpose [23]. More recely, a ew heory of sascal mache learg called Suppor Vecor Mache (SVM) servg as aswer o forcas ad classfcao whch caugh aeo of sceess [36,,3]. SVM s appled wdely may areas such as approxmae fuco, regresso aalyss ad forecas [,3]. he bgges lmao of SVM s ha wh huge rag fles, requres eomous calculao as awell as complexy of he lear regesso exercse. o address he lmaos ad promoe he sreghs of exsg models, a ew ad redy research mehod was roduced whch s called Combed Aayss (CA) e. a combao of of dffere mehods o crease he forecas accuracy. Numerrous sudes have bee coduced based o hs mehod, ad may combed models have bee publshed [43,5,6]. Some mehods uses he Marov cha (MC) as well as hdde Marov (HMM). Reful Hassa [9] developed a ued model by machg a HMM wh a ANN ad GA o geerae forecas a day - soc prce. hs model ams o defy smlar

5 paers from hsorcal sascs. he ANN ad GA models are used o erpolae he eghbor values ò he defed sascs model. Yag [4] combed he HMM model usg sychoroous cluserg echque o crease he accuracy of he forecasg model. he weghed Marov model was used by Peg [27] predcg ad aalyzg desease rasmsso rae Jagsu, Cha. hese combed models proved o brg praccal ad meagful resuls, as well as crease he accuracy predco compared o radoal oes [27,4,9]. he above meoed models, despe havg mproved sgfcaly erms of accuracy predco, sll face dffcules wh fuzzy sascs (here are ucera molecules). o deal wh fuzzy sascs, a ew research dreco was roduced recely, whch s called Fuzzy me Seres (FS). he frs resul from hs heory worh o meo s Sog ad Chssom [34]. hese sudes focused o mprovg he Fuzzy me Seres model ad fdg ways for he forecasg aalyss. Jla ad Na combed Heursc model wh Fuzzy me Seres model o mprove he model accuracy [24]. Che ad Hwag expaded he Fuzzy me seres model o Bary model [4] ad he Hwag ad Yu developed o N-scale model o forecas soc dcaors [2]. I a rece paper [35], BaQug Su has expaded he Fuzzy me Seres model o mul-order o forecas soc prce he fuure. Qse Ca [0] combed he Fuzzy me Seres model wh a opmzao ad regesso o oba a beer oucome. I Veam, he Fuzzy me Seres model was recely appled a umber of specfc areas, some o meo clude he sudy of Nguye Duy Heu ad Parers [2] semac aalyss. Addoally, he sudy of Nguye Cog Deu [3,4] combed he Fuzzy me Seres model wh echques o adjus some parameer mahs or specfc characors of sascs amg o he forecas accuracy. he sudy of Nguye Ca Ho [] used soographc algebra Fuzzy me Seres model whch showed he hgher forecas accuracy compared o several exsg modesl. Up o ow, spe of may ew models combg exsg oe amg o mprove he forecas accuracy, here s a fac ha hese models are complex ye accuracy o mprovg. herefore, here may arse some oher dreco amg o smplfy he model whle esure he forecas accuracy. he objecve of hs dsserao focuses o wo ey ssues. Frsly, o modelze me seres by saes whch each s a deermsc probably dsrbuo (sadard dsrbuo) ad o evaluae he suably of he model based o expermeal resuls. Secodly, combe Marov cha ad ew Fuzzy me seres models o mprove he forecas accuracy. I addo, o expad he advaced Marov cha model o accommodae seasoal sascs. he dsserao cosss of 3 chapers Chaper I preses overall sudy of Marov cha ad hdde Maro ad Fuzzy me Seres models; Chaper II preses me seres modellg o saes whch ) each sae s sadard dsrbuo vs. average, varace 2,,2,..., mwh m s he sae; 2) saes over me followed Marov cha. he model, he was esed o VN-Idex dcaor o evaluae effcecy of model forecas. Las chaper preses he aalyss of lmaos ad umaches bewee forecasg models ad

6 deermsc probably dsrbuo as a movao for he combed model proposed Chaper 3. Chaper III preses combed Marov cha ad Fuzzy me Seres models me seres forecasg. hs chaper also preses he expaded ad advaced Marov cha wh wo cha coceps whch are coveoal hgher order Marov (CMC) ad mproved hgher order Marov (IMC). hese models, he, were programmed he R laguage ad esed w daa ses ha correspoded exacly wh comparso model ses... Marov cha... Defos Chaper - Overvew & Proposal Cosder a ecoomc or maeral sysem S wh m possble saes, deoed by I : I,2,..., m. Sysem S evolves radomly dscree me ( 0,,2,...,,... ), called C ad se o a radom varable corespodg o he sae ò he sysem S a he me (C I). Defo... Radom varable sequese ( C, ) s a Marov cha f ad oly f all c0,c,...,c I : Pr( C c C0 c0, C c,..., C c ) Pr( C c C c ) (..) (wh a codo hs probably maes sese) Defo..2. Marov cha s cosdererd comprable f ad oly f he possbly (..) s o depede o ad o-comparable oher cases.. For he me beg, we cosder he comparable case, whch Pr( C c C c ) j, Ad marx Γ by defo: Γ j. o defe fully he developme of a Marov cha, s ecessary o fx a al dsrbuo for sae C 0, for example, a vecor: p ( p, p2,..., p m ), I hs chaper, we sop a cosderg comparable Marov cha whch s feaured by couple ( p, Γ ). Defo.2.3. A Marov marx Γ s cosdered formal f here exss a posve eger, such ha all elemes of he marx are acually posve...2. Marov cha classfcao ( ) ae Iad pu d () s he larges geeral dvsor of a se of gers such ha 0. Defo.2.4. If d ( ), sae s cosdered a revolvg cycle d. () If d ( ), he sae s o revolvg. Easy o see, f 0 he s o revolvg. However, he oppose s o prey rue. Defo.2.5. Marov cha of whch all s saes o revolvg s call rrevolvg Marov cha. Defo.2.6. A sae s called reachg sae j (wre j) f exs a eger such ha j 0. C j meas ca o reach j.

7 Defo.2.7. Sae ad j s called er-coeced f jad j, or f j. We wre j. Defo.2.8. Sae s called esseal f coecs wh every sae ha reaches; he oppose s call o-esseal. Relaoshp deermes a equvele relaoshp sae space I resuled a class dvso o I. he equvale class coas symbol deoed by Cl(). Defo.2.9. Marov cha s called o expadable f here s oly oe equvale class o. Defo.2.0. Subse E of sae space I s cosdered closed f: j, vớ mọ E. je Defo.2.. Sae Iof Marov cha ( C ) s cosdered regressed f exss sae j Iad such ha 0. Opposely, s called forwardg sae (movg)...3. Marov marx esmao j Cosder Marov cha ( C ),,2,... ad suppose o observe ad oher saes c, c2,..., c. Symbols c c, c2,..., c geeraed by radom varables C he he logcal fuco of forwardg probably marx s gve by: Pr( C c ) Pr( C c) Pr C c C c 2 Pr Pr C c C c Pr( C c) 2 c c 2 Defe umbers of rasfer j umber of mes ha sae forwards, follwed by sae j cha C, he lelhood loos le: j L( p) Pr( C c) j j We eed o fd he maxmum raoal fuco Lp ( ) wh he hddes are j. o solve hs exercse, frs we ae logar of Lp ( ) o mae a oal fuco amg o ae he dervave easly.. ( p) log L( p) log Pr( C c) j log m j, j Due o j, each, j, ae he dervave by parameer: j j2j j j Gve dervave equals o 0 obaed a j we have: j ˆ j ˆ herefore ˆ j j ˆ rue wh all j herefore j ˆ j m j.2. Hdde Marov Model j A HMM cludes wo bass compoes: cha X,,..., cosss observaos ad C,,..,, {,2,..., m} whch were geeraed from hose observaos. I deed, HMM model s a specal case of mxed depede model [6] ad C whch are mxed compoes.

8 .2.. Defo ad Symbols ( ) ( ) Symbols X và C dsplayhsorcal sascs from po of me o po of me, whch ca be summarzed as he smples HMM model as follows: ( ) Pr( C C ) Pr( C C ), 2,3,...,. ( ) ( ) Pr( X X, C ) Pr( X C ), Now we roduce some symbols whch are used he sudy. I case of dscree observao, by defo: p x Pr X x C. I he case of couy, p ( x ) s X s probably fuco rage, f Marov cha receves sae a po of me. We symbolze a comparave Marov cha s forwardg marx as Γ wh s compoes j defed by: j Pr( C j C ). From ow o, m dsrbues p ( x ) s called depede depedeces of he model Lelhood ad maxmum esmao of lelhood For dscree observao u Pr C vớ,2,...,, we have: m Pr( X x) Pr m ( C ) Pr( X x C ) u( ) p( x). X, defe (.2.) For coveece calculag, fomula (.2.) ca be re-wre he form of he followg marx: p ( x) 0 0 Pr(X =x)=(u (),...,u(m) ()) 0 0 u() P( x 0 0 pm( x) ). whch P (x) s dagoal marx wh he eleme o he dagoal le p ( x ). O he oher had, by aure of he pure Marov cha, u() u() Γ wh u () s a al dsrbuo of Marov cha, usually deoed wh sop dsrbuo as δ. hus, we have Pr( X x) u() Γ P( x). (.2.2) Now call L s he lelhood fuco of he model wh observe x, x2,..., x he ( ) ( ) L Pr( X x ). Derved from he smuaeous probably formula ( ) ( ) Pr( X, C ) Pr(C ) Pr( C C ) Pr( X C ), (.2.3) We sum o all possble saes of C, he usg he mehod as he fomula (.2.2), we have L P( x) ΓP( x2)... ΓP( x ). If al dsrbuo δ s he sop dsrbuo of Marov cha, he L ΓP( x ) ΓP( x2)... ΓP( x). o calculae lelhood easly by algorhm, reduce he umber of operaos ha he compuer eeds o perform, we defe vecor α where,..., by P( x ) ΓP( x2)... ΓP( x ) P( x ) ΓP ( xs), (.2.4) s2 he we have L, và ΓP ( x ), 2. (.2.5) I s easy o calculae L by regresso algorhm. o fd he parameer se sasfes L maxmal, we ca perform wo mehods:

9 Drec esmao of exreme values fuco L (MLE): Frsly,from equao (.2.5) we eed o calculae logar of L effecvely o advaageous o fd he maxmum based o he progressve probables α. For 0,,...,, we defe he vecor / w, where w (), ad B P ( x ). We have w0 0 ; 0 ; w w B ; L w( ) w. he L w ( w / w ). From equao (.4.3) we have w w B, he log L log w / w log B. EM Algorhm: hs algorhm s called Baum-Welch algorhm[9] for cosse Marov cha (No ecessarly Marov sop). he algorhm uses forward probables (FWP) ad bacward probables (BWP) o calculae L Forecasg dsrbuo For dscree observaos, forecasg dsrbuo ( ) ( ) Pr( X h x X x ) s a rao of L based o codoal probably ( ) ( ) ( ) ( ) Pr( X x, Xh x) Pr( Xh x X x ) ( ) ( ) h P( xpr ) ( X... x ) ( ) B2B3 BΓ P x h Γ P( x( x )... B 2B3 B. ( ) ( ) h By / $, we have Pr( X h x X x ) P ( x ). Forecasg dsrbuo ca be wre as probably of depedecy radom varables: m ( ) ( ) Pr( X h x X x ) ( h) p ( x). h where he wegh ( h) s he compoe of vecor Verb algorhm he objecve of Verb algorhm s o fd he bes of sae sequeces, 2,..., correspodg o he observao sequece x, x2,..., x whch maxmzes he fuco L. Pr( C, X x ) p ( x ), where 2,3,..., Se ( ) ( ) ( ) ( ) max c, c2,..., c Pr( C c, C, ). X x he we ca see ha probably j sasfes he followg recurso process for 2,3,..., ad,2,..., m: j max (, j ) pj ( x ). he bes sae sequece, 2,..., s deermed by regresso from argmax ad,.., m for, 2,...,, we have argmax(, ).,..., m.2.5. Saus forecasg For saus forecasg, we oly use he Bayes formula classcal. For,2,..., m, ( ) ( ) h Pr( C h X x ) α h (, ) / L Γ (, ) Noe ha, whe h, Γ h moves owards he sop dsrbuo of he Marov cha

10 .3. Fuzzy me seres.3.. Some coceps Suppose U be he dscourse doma. hs space deermes a se of objecs. If A s a crsp se of U he we ca deerme exacly a feaure fuco: ( ) { Defo.3.. [34]: Suppose U be he dscource doma ad U { u, u2,..., u }. A fuzzy se A U defed: A= f A( u)/ u + f A( u2)/ u f A( u)/ u f A s membershp fuco of fuzzy se A ad f A : U [0;], fa( u ) s a degree of membershp (he ra) of u o A. Defo.3.2. [34]: Le Y( )( 0,,2,...) be a me seres ha s values he dscource, whch s a subse of real umbers. O whch, he fuzzy ses f( )( 0,,2,...) s deermed o Y, ad Fs () a colleco he ses f( ), f2( ),..., he F () s called fuzzy me seres o Y. Defo.3.3.[34]: Suppose ha F () s oly ferred from F ( ), deoed as F( ) F( ), hs relaoshp ca be expressed as follows F( ) F( ) or(, ), whch F( ) F( ) or(, ) s called as rs-order model of F( ), R(, ), s he fuzzy relaoshp bewee F ( ) ad F (), ad "o" s a compoe operaor Max M. Defo.3.4. [34]: Le R(, ) be a frs-order model of F (). For ay, R(, ) R(, 2), he F () s sad o be a me-vara fuzzy me seres. Oherwse, F () sa me-vara fuzzy me seres.

11 Chaper 2. HIDDEN MARKOV MODEL IN IMES SERIES FORECASING 2.. Hdde Marov model he me seres forecasg Accordg o Chapper, HMM model cosss of wo basc compoes: he cha of observaos X,,..., ad mx compoes C,,..,, {,2,..., m}. o llusrae he HMM model me seres forecasg easly, le us cosder he above me seres ad deoed as X,,...,. he real problem for vesors s o predc he value of he fuure o ow how log he soc dex wll go from he boom o he op. From observg he fac ha he soc dex a a ew pea wll o be a ha value (or flucuae slghly aroud ha value) forever ha wll go dow afer some me, smlarly. wh oscllaos from he boom o he op. So we ca be specfed X max s he loges me ha he soc's value from he boom o he op. he, 0 X Xmax (see Fgure 2.2.). Ivesors wa o regulae he sae of affars wh X, such as "wa fas", "wa que fas", "wa log", "wa very log" bu do o ow how o defe. o solve hs problem, we cosder each of hese saes a Posso dsrbuo wh he mea (also he varace),, 2,3, 4 ad s "hdde" he cha X. Assumg ha hese saes follow a Marov cha, We have a hdde Marov model for he me seres forcasg problem. Fgure 2... he Defo of he me seres forecasg 2... HMM model wh Posso dsrbuo o apply he HMM model for me seres forecasg, he dsserao llusraes boh paramerc esmao mehods descrbed Seco.3.2 Chaper. For MLE esmao, he dsserao performs programmg o R for he HMM model wh he sae as he Posso dsrbuo. Posso dsrbuo has he parameer 0 boh mea ad varace. Parameer esmao by MLE mehod s as follows: Algorhm 2. Maxmum reasoable fuco Ipu: x,m, lambda0,gamma0 Oupu: m, lambda0, gamma0, BIC, AIC, mll : procedure POIS.HMM.MLE (x,m, lambda0,gamma0,... ) 2: parvec0 pos.hmm.p2pw(m, lambda0,gamma0) { Chage model o free parameer } 3: mod lm(pos.hmm.mll, parvec0,x = x,m = m) { Esmae he parameer as a reasoable maxmum fuco }

12 4: p pos.hmm.pw2p(m,mod$esmae) { Chage he free parameer o he model parameer pm } 5: mll mod$mmum { Ge he max value assged o mll } 6: p legh(parvec0) { Cou of model parameers } 7: AIC < 2 (mll+p) { Calculae he sadard AIC } 8: < sum(!s.a(x)) { Calculae he umber of observaos } 9: BIC < 2 mll+p log() { Calculae he sadard BIC } 0: reur (lambda, gamma, mll, AIC, BIC) HMM model wh ormal dsrbuo I he model wh ormal dsrbuo, he parameers of he Marov cha are sll he same, bu he parameer of he mx dsrbuo s he mea ad he varace, whle he umber of saes of he model s also he sop dsrbuo of he Marov cha. Calculaos of FWP ad BWP are performed by he ormalzao fuco HMM.lalphabea (logarhm of FWP ad BWP). I whch, lalpha, lbea s he log of FWP ad BWP respecvely. Algorhm 2.3 Calculae he forward ad bacward probables of L Ipu: x,m,mu, sgma,gamma,dela Oupu: lalpha, lb = lbea : procedure NORM.HMM.LALPHABEA(x,m,mu, sgma,gamma,dela ) 2: f (s.ull(dela)) he dela solve((dag(m) gamma+), rep(,m)) { I he ulely eve of he al dsrbuo of he Marov cha} 3: Calculae he probables of FWP (.2.6) for lalpha 4: Calculaes probables for BWP (.2.7) for lbea 5: reur ls(la = lalpha, lb = lbea) Here, accordg o he EM algorhm Seco.3.2 of Chaper, we ca mmedaely perform he parameer esmao by orm.hmm.em Algorhm 2.4. Algorhm EM for Normal-HMM Ipu: x,m,mu(), sgma(),gamma(),dela(),maxer, ol Oupu: mu, sgma, gamma, dela, mll, AIC, BIC : procedure NORM.HMM.EM(x,m,mu, sgma,gamma,dela,maxer, ol ) 2: mu.ex mu(); sgma sgma();dela dela() { Assg a parameer o he orgal value } 3: for er : maxer do 4: f b orm.hmm.lalphabea(x,m,mu, sgma,gamma,dela= dela) {Calculae FWP ad BWP} 5: ll reasoable fuco value 6: for j :m do 7: for :m do

13 8: Calculae gamma[ j,] 9: Calculae mu[j] 0: Calculae sgma [ j] : Calculae dela 2: cr sum(abs(mu[j] mu()[j])) + sum(abs(gamma[j] gamma()[j])) + sum(abs(dela[j] dela()[j]))+sum(abs(sgma[j] sgma()[j])) { he coverge crera } 3: f cr < ol he 4: AIC -2 (ll p) { he crera AIC} 5: BIC -2 ll+p log() {he crera BIC} 6: reur (mu, sgma, gamma, dela, mll, AIC, BIC) 7: else { If o coverged } mu0 mu; sgma0 sgma; gamma0 gamma; dela0 dela { Reassg he ew orgal parameer } 8: No covergg laer, maxer, loop 2.2. Expermeal resuls for HMM wh Posso dsrbuo Parameer esmao able Esmae parameers of model Posso-HMM for me.b.o. wh saes m=2,3,4, , , , , ,704 5, , , , , , , , , , , , ,5252 0, , , , , , , , , , ,8 0,2 0,5 0,49 0,46 0,47 0,07 0,33 0,47 0,02 0,2 0,8 0 0,4 0,46 0,07 0,07 0,53 0,29 0, ,5 0,49 0,9 0,56 0,25 0 0,38 0,4 0,5 0,07 0 0,5 0,36 0 0,4 0 0,3 0 0,33 0,9 0,35 0 0,53 0, ,33 0 0, able Mea ad varace compared wh sample 26,840 7,243 59,898 54,6275 M Mea Varace 20, , , , , , , ,72

14 5 20, ,4568 Sample 20, ,083 he resuls show ha he Posso-HMM model wh 4 saes has a varace close o he sample varace. However, here s o eough evdece o cofrm ha he 4-sae model s he bes. I order o have beer mehods of seleco, we eed o have sadards for selecg models more deal Model seleco Gve he observed x,..., x were bor by a real" uow f model ad we model by wo dffere approxmaos { gg} và { g2 G2}. Purpose of model seleco s o defy he bes model some aspec. Now, apply 2 sadards AIC ad BIC o Posso-HMM model for daa se me.b.o., he resuls are lsed able able AIC ad BIC Sadards m AIC 44, , , ,255 BIC 448, , , , Forecas Dsrbuo As meoed above, rag daa for he HMM model was obaed from 3 Jauary 2006 o 9 Jue 203. We wll ge he followg daa from 4/06/203 o 22/08/203 o compare wh forecas resuls of he model. Fgure 2..2 shows flucuao of closg VN-Idex durg hs perod. As we see, he umber of sessos ha VN-Idex flucuaes from he boom (26/06/203) o he pea (9/08/203) s 35 days. hus, hs value correspods o he sae 3 of he model (Posso dsrbuo wh he mea a ). We wll wa o see he resuls of he forecas model Fgure V-Idex flucuao from 4/06/203 o 22/08/203 ad wag me from boom o pea ( ) ( ) Pr( X ) Now, we eed o fd he formula for predcg he dsrbuo h x X x. Wh he marx formulas as show he prevous secos, hs dsrbuo ca be compued as follows:

15 P X x, X h x X x h x Γ x2 Γ x3 Γ x Γ x x P X h δp P P P X Px ' h δp x ΓP x Γ 2 ΓP x ' 3 ΓP x ' ' Gve α / α ', we have h P X h x X x Γ P x. hese dsrbuos are summarzed able able Dsrbuo forecas formao & ervals Forecas Mode Forecas mea 42, ,680 25, , ,4849 2,9300 Esmaed rage wh probably over 90% Forecas erval [ ] [ ] [ ] [ ] [ ] [ ] Probably 0, , , , , , Realy Forecas saes I he prevous seco we have foud he codoal dsrbuo of he sae C gve ( ) observao X. I hs way we oly cosder he prese sae ad pas saes. However, s also possble o calculae he codoal dsrbuo for he fuure sae C h, hs s called sae forecas. h ( ) ( ) αγ (, ) h Pr( C h X x ) Γ (, ) L wh / α α. We perform forecasg sae of he Posso-HMM model, 4 saes of daa me.b.o. 6 mes wh resuls as show able able mes forecasg sae of me.b.o Sae = 0, , , , , , , , , , , , , , , , , , , , , , , , Experemeal resul of HMM model wh ormal dsrbuo Parameers esmao Wh ay al dsrbuo 0,977 (e.g.: (/ 0, 4,/ ,/ 0, 4,/ )), 0, esmaed 0000 by EM we have: 0, ,806 0, , 063 0, , ,8624 0, , , , 088 0,982 (453,9839;484,680;505,9007;530,8300) (0, 6857; 7,523; 6, 428;3, 0746) Fgure 2.3. descrbes values of VNIdex wh bes sae rage usg Verb algorhm. he dashed les represe he four saes whle he dar dos represe he bes sae for he value a each me.

16 Fgure VN-Idex daa: bes sae rage Model seleco Accordg o he heory of HMM o he BIC ad AIC crera for he VNI dex, AIC ad BIC seleced 4 saus. he values of he sadard gve he able able VN-Idex daa: selec sae umber Model -logl AIC BIC 2-sae HM.597, , ,32 3-sae HM.50, , ,204 4-sae HM.439, , ,02 5-sae HM No covergece Forecas dsrbuo As descrbed par.3.3 Chaper, Fgure represes 0 forecas dsrbuos for VNIdex value. We see he forecass dsrbuo move oward sop dsrbuo very fas. Fgure VN-Idex daa: forecas dsrbuo of 0 followg days. hus, he HMM model wh dsrbuos s ceraly suable wh he predco some cases, especally wh he daa acually fs wh he dsrbuo seleced he model. However, wheher he me seres geeraed by a radom varable ha fs o he ormal

17 dsrbuo (or mx wh ormal dsrbuos) or ay oher dsrbuo seleced s a queso ha wll deerme he appropraeess ad accuracy level of he forecass Forecase sae able Maxmum ably (probably) forecas for each sae of 30 followg days sag from he las dae 3/05/20 Days [,] [,2] [,3] [,4] [,5] [,6] Sae=[,] 0,0975 0,695 0,226 0,2709 0,3065 0,3350 [2,] 0,8062 0,6622 0,557 0,4665 0,4005 0,3492 [3,] 0,0799 0,35 0,724 0,97 0,228 0,2223 [4,] 0,062 0,0330 0,0496 0,0653 0,0800 0,0933 [,7] [,8] [,9] [,0] [,] [,2] [,3] [,] 0,3579 0,3764 0,395 0,4039 0,44 0,4225 0,4296 [2,] 0,3092 0,2778 0,2530 0,2334 0,277 0,2052 0,95 [3,] 0,2274 0,2296 0,2298 0,2288 0,2270 0,2248 0,2224 [4,] 0,053 0,60 0,255 0,338 0,40 0,473 0,527 [,4] [,5] [,6] [,7] [,8] [,9] [,20] [,] 0,4355 0,4405 0,4448 0,4484 0,455 0,4542 0,4565 [2,] 0,870 0,803 0,749 0,705 0,669 0,639 0,64 [3,] 0,2200 0,276 0,254 0,233 0,23 0,2096 0,2080 [4,] 0,573 0,63 0,647 0,676 0,70 0,722 0,739 [,2] [,22] [,23] [,24] [,25] [,26] [,27] [,] 0,4586 0,4604 0,469 0,4633 0,4646 0,4657 0,4667 [2,] 0,593 0,576 0,56 0,549 0,539 0,530 0,523 [3,] 0,2066 0,2053 0,204 0,203 0,2022 0,204 0,2007 [4,] 0,754 0,766 0,776 0,784 0,79 0,797 0,80 [,28] [,29] [,30] [,] 0,4676 0,4684 0,4692 [2,] 0,57 0,52 0,507 [3,] 0,2000 0,995 0,990 [4,] 0,805 0,807 0,809 We see he hghes probably he frs 7 days falls sae 2 ad he ex day falls o he sae. herefore, he model s o effecve log erm bu good for shor erm. However, we ca forecas by couously updag he daa auomacally. he dsserao wll be updaed from 4/5/20 o 23/6/20 wh 30 closg prce of he soc o compare he forecas value wh he acual value of he daa. Fgure shows ha he value of hese 30 days s mosly sae. hs proves ha he forecas s correc.

18 Fgure VNIdex daa: Comparso of forecas sae ad acual sae Resul comparso hs dsserao preses forecas resuls of he HMM model wh a umber of models [9] o some of he daa as soc dexes. Due o he value characerscs of he growh me seres recevg real values, he HMM model wh ormal dsrbuo s chose. he proposed model hs dsserao ad he comparave model are performed o he same rag se ad o he same es se o esure he accuracy of he comparso. he accuracy measureme used s he average perce error (MAPE) calculaed by: a p MAPE *00% a able Mulple MAPE ru by HMM model o Apple daa,82,778,790,784,85,777,82,794,779,788,802,86,778,800,790,789 Mea:,795. Accuracy mea,795 ad average forecas value are llusraed Frgure Fgure HMM forecas for Apple share prce:acual-real prce; predc-forecased prce Smlar s Ryaar Arles soc daa from Jauary 6, 2003 o Jauary 7, 2005; IBM Corporao from Jauary 0, 2003 o Jauary 2, 2005, ad Dell Ic. from 0/0/2003 o 2/0/2005. A comparso of he MAPE accuracy measureme wh 400 rag observaos s show able able Comparso of he MAPE accuracy measureme wh oher models Daa ARIMA model ANN model HMM model Apple,80,80,795 Ryaar,504,504,306 IBM 0,660 0,660 0,660 Dell 0,972 0,972 0,863 Gve he resuls able we see he model HMM wh ormal dsrbuo provde hgher forecas accuracy compared wh oher classc models ARIMA ad ANN.

19 Chaper 3. EXENSION OF HIGHER ORDER MARKOV CHAIN MODEL AND FUZZY IME CHAIN IN FORECASING 3.. Hgher-order Marov cha Assume ha a each daa po C a gve classfed daa seres, ae value he se I,2,..., m ad m s lmed,.e. he value se has m ypes or saes. A Marov cha order s a radom varable se ha Pr( C c C c,..., C c ) Pr( C c C c,..., C c ) I [30], Rafery proposes a hgher-order Marov cha model (CMC). hs model ca be wre as follows: P( C c C c,..., C c ) qc c (3..) Whereas, ad Q [ q j ] s a shf marx wh oal colum umber s, he: 0 qc,, c c c I (3..2) 3... Improved hgher-order Marov model (IMC) I hs subseco, he dsserao preses he exeso of he Rafery [30] model o a more geeral hgher-order Marov cha model by allowg Q o chage as varous degrees of laecy. Here we assume ha he o-egave wegh ha sasfes: (3..3) 0 We have (3..) re-wre as: C QC (3..4) Where C s he probably dsrbuo of saes a me ( ). Use (3..3) ad Q a raso probably marx, we have each eleme C bewee 0 ad, ad sum of all elemes s. I Rafery model, o gve s o egave he codos (3..2) are added o esure ha C s probably dsrbuo of all saes. Rafery model (3..4) ca be geeralzed as below: C Q C (3..5) 2 he oal umber of depede parameers he ew model s ( m ) Parameer esmao I hs seco, he auhor preses effecve mehods for esmag parameers Q ad wh, 2,...,. o esmae Q, we ca cosder Q as a marx o rasfer sep classfcao daa sequece C. Gve he classfcao daa C, we ca cou raso () frequecy f jl he sequece from ( sae ) l o sae ( ) j afer seps. Moreover, we ca buld he f f m marx o rasfer seps for sequece ( ) C as below: ( ) () f2 fm2 F ( ) ( ) qˆ () qˆ () m Gve F, we receve esmao for ( Q [ ] () ) qlj as ( ) follow: ( ) ˆ qˆf ˆ 2 q m mfmm 2 Q () f m lj () Where f ( ) ( ) qˆ ˆ m q 0 m flj () () l qˆ mm lj flj l 0 oher

20 () We oe ha he complex calculaos of he cosruco of F s he calculao of 2 () O( L ), where L s legh of daa sequece. So he oal complexy of he cosruco F s 2 he calculao of O( L ). Where s umber of laeces. We ow prese he seps for esmag he parameers as follow [5] ha he dsserao wll use o embed he proposed combed model. Gve C C whe move o fy, he C ca be esmaed from sequece C by calculag he occurrece of each sae he sequece ad we se as C ˆ. ˆ ˆ QC C hs gves us a esmao of he parameers (,..., ) as below. We cosder he followg mmum problem: m ˆ ˆ QC C wh codo ha, và 0, Here. s he sadard Vecor. Specal case, f selec., we have he below mmum problem: m max [ ˆ ˆ l QC C] l wh codo ha, và 0, Here [.] l defy he l eleme of Vecor. he dffculy here s he opmzao o esure he exsece of sable dsrbuo C. Nex, we cosder he above mmum problem o be cosruced as a lear problem: m wh codo ha ˆ ˆ ˆ ˆ ˆ ˆ ˆ 2 C [ QC Q2C... QC] ˆ ˆ ˆ ˆ ˆ ˆ ˆ 2 C [ QC Q2C... QC] 0,, và 0, We ca solve he lear problem above ad oba he parameer. Isead of solvg a m-max problem, we ca also choose. ad buld he followg mmum problem: m m [ ˆ ˆ ˆ Q C C] l l wh codo ha, và 0, he correspodg lear problem s gve below: m m l l wh codo ha 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ 2 C [ QX Q2C... QC] 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ 2 C [ QC Q2C... QC] m 0,,, và 0, m \I cosrucg he above lear problems, he umber of varables s equal o ad he umber of codos s equal (2m ). he complexy of solvg lear problems s calculaed 3 O( L ), where s varable ad L s he umber of bary bs eeded o sore all he daa (codos ad arge fucos) [8] Selec fuzzy me seres he combed model Cosder me seres wh observaos X, X 2,, X, wh growh cha x, x2,, x, (defed mmedaely below). We wa o classfy growh o dffere saes such as "slow",

21 "ormal", "fas" or eve more. However, each x a a me s uclear a wha degree eve hough we defe levels clearly. hs meas, x ca vary from oe level o aoher ad dffere levels of membershp. herefore, he fuzzy me seres heory Seco.4 of Chaper ca do hs o classfy he subse of x (defed he followg seco) o saes ha x are members. Assumg ha hese saes follow a Marov cha, he Marov model gves us a predced fuure sae. From he fuure sae, he predced value x s calculaed from he prevous defo of he fuzzy perod Defe ad segme base sequece Gve he rag se { y } N, we ca defe he base sequece for growh sapce by U m {,..., N} y ;max {,..., N} y wh 0 a posve umber s chose so ha growh he fuure does o exceed max {,..., N } y. Wh each daa we ca selec dffere. However, selec ca sasfy all soc growhs. I order o mae sequece U fuzzy o growh labels such as "fas crease," "slow crease," "seady crease," or eve levels, he base sequece U s dvded o erval (he smples s dvded o cosecuve equal ervals) u, u2,..., u. For example, f he zog of VN-Idex (Veam soc dex) s: U [ , 0.050] [ 0.050,0.049] [0.049,0.0448] he VN-Idex resuls are coded as able 3.3. able Fuzzy growh cha Dae x Idex growh ( y ) Code 04// ,5-0,05997 NA NA 05// ,5-0, , // ,9-0, , // , 0, , // ,4 0, ,0863 // ,6 0, , Fuzzy rule of me seres We defy fuzzy A, each A s assged wh a growh ag ad defed o sprecfc paragraphs u, u2,..., u. he he fuzzy A ca be descrbed as: A A ( u) / u A ( u2) / u2... A ( u ) / u where A s a member fuco of each u j, j,..., A,,...,. Each fuzzy value of me seres y s re-calculaed based o fuzzy rule A. For example: A / u 0.5/ u2 0 / u / u A2 0.5 / u / u2 0.5/ u / u... A 0 / u 0 / u2 0 / u3... / u. where y A2 s a uclear value, he he clear value s re-calculaed based o fuzzy rule by: y , m m m 2 where m, m2, m 3 are mdpos of u, u2, u 3 respecvely.

22 For dffere fuzy rules, he versed rule s dffere Combed model of Marov cha ad fuzzy me seres 3.3. Combed model a frs-order Marov cha I hs seco, we descrbe deal he combao of he Marov model - fuzzy me seres. hs combao s llusraed Fgure Deals of each sep are show below: Compue he reurs of he rag se ad defe he dscouse Mae paro of he dscource Fuzzfy he reurs daa ra he hgher order Marov model for fuzzy seres Forecas he ecoded seres ad defuzzy o forecas prce Sep Sep 2 Sep 3 Sep 4 Sep 5 Fgure Model srucure Sep : Gve observao daa of a me seres { x, x2,..., x } growh cha of rag daa s calculaed as follows: x x y, x We have x ( y ). x Gve D max ad D m are he maxmum value ad mmum values of he growh cha afer removg he exraeous value, he he base sequece U [ Dm, Dmax ] where 0 ca be se as a hreshold for he crease of chages. Sep 2: A paro of uverse s geeraed smples way by devde [ Dmax, D m ] o 2 equal ervals. he he uverse U u u2... u where u [ Dm, Dm ] ad u [ Dmax, Dmax ]. Sep 3: I hs research, he lgusc erms A, A2, A3,..., A of he me seres represeg by fuzy ses, are also defed he smples way as follows: A / u 0.5/ u2 0 / u / u A2 0.5 / u / u2 0.5/ u / u... A 0 / u 0 / u2 0 / u3... / u Each A s ecoded by for {,2,..., }. herefore, f a daum of he me seres belogs o u, s ecoded by ( {,2,..., } ). We oba a ecoded me seres { c }, c {,2,..., }. For sace, f a paro of dscourse VN-Idex (Veam soc dex) as par 3.2. Sep 4: hs sep explas how Marov chas are appled ecoded me seres. Acordg o seco 3.2, we assume ha he ecoded me seres s a Marov cha as defed Defo.2.. o esmae he parameers of he Marov cha as Seco.2.3, s easy o esmae he raso probably marx Γ [ γj],, j{, 2,..., }, where: j Pr( c j c ) If exs he sae c s he absorpo sae (see.2.), o esure regulary of Γ defe Pr( c j c ) wh all,2,...,. j hs meas probably chage from sae o ay oher sae s he same.

23 Sep 5: Nex we geerae he oe-sep-ahead forecas for ecoded me seres ad defuzzy forecas fuzzy se o forecas value of he reurs. Gve c, colum Γ[, c ] s he probably dsrbuo of c j, j,2,..., 2 2. If M ( ( m 0.5 m2 ), (0.5m m2 0,5 m3 ),, ( m 0.5 m)) where m s he mddle value of he erval u he he forecas reurs a s calculaed as below: yˆ Γ [,c ]*M a jc mj j I hs sep, he vecor M ca be seleced dfferely accordg o he fuzzy mehod Sep 2. Fally, he x value s calculaed as follows: xˆ ˆ ( y )* x Exeso of hgher-order Marov cha Hgher-order Marov cha model combed wh fuzzy me seres s dffere from Marov cha frs-order model Sep 4 ad Sep 5. Sep 4: For he coveoal hgher order Marov model assocaed wh fuzzy me seres (called CMC-Fuz), by maxmzg he same he frs-order Marov cha model, s easy o esmae he raso probably marx l dmeso Γ [... ], {,2,..., } l l j. As defed hgh-order Marov cha, l l... s he observed probably c gve he ow c,..., c l :... (,..., ) l l Pr c l c l c l For he ew combed hgh-order Marov model (called IMC-Fuz), rasfer marx s l m m Q as seco Sep 5: Nex, we geerae a oe-sep forecas for he me seres ecoded by he raso probables marx ad he verse of he predco value of he me seres. Wh model CMC-Fuz, gve c,..., c l, colum [, c,..., c l ] s probably dsrbuo of c j for all ecoded value j,2,...,. Forecas growh value a he me s compued by: xˆ [, c,..., c l ]* M jc... c m l j j Wh model IMC-Fuz, Forecas growh value a he me s calculaed by: l xˆ Q [, c ] Fally, value X forecas s compued by: Xˆ ˆ ( x )* X Algorhm 3. Combed Marov - Fuzzy algorhm Ipu: Daa,, ra, Order, Saes Oupu: : predc, RMSE, MAPE, MAE Daa Daa y, 2,..., ra Daa y 2: ra <- Remove ale elemes of 3: Dvde erval [m( ra) ;max( ra) ] o Saes equal erval A 4: f x Ac he ecoded 6: f Model = CMC-Fuz he esmae raso marx of model CMC_Fuz. 7: for :Order do esmae marx Q

24 8: f Model = IMC-Fuz he 9: C cous ( ecoded ) / sum( cous) 0: Solve opmum problem m-max m max Q C C for Order 2: IMC.Fuz.Ma Q dsrbuo 3: for close 4: f esse do close A he ecoded Esmae raso marx of model IMC-Fuz based o sop ecode ew observao, > ra M vecor (2 / 3( md ( A ) 0.5 md ( A )),/ 2(0.5 md ( A ) md ( A ) 5: 2 2 0,5 md( A3)),...,2 / 3(0.5 md( A ) md( A))) reverse fuzzy rule wh ( of erval A A ) s mddle po predc ( raso. Mas[, ecoded, ecoded,..., ecoded ]%*% M )* Daa 7: errors (RMSE, MAPE, MAE) f ( predc acual ) calculae accuracy 6: 2 Order Where, ra s he observao umber he rag se; Order s order of hghorder Marov cha ad Saes s he umber of saes (he A ) he model. hus, models CMC-Fuz ad IMC-Fuz wh order Order cocdes wh he combed model of frs-order as em As a resul, he expermeal resuls for he frs order Marov cha model were performed smulaeously he hgh-order Marov cha model Expermeal resul Daa colleco I order o compare resuls [9, 20, 7, 26, 38, 33], we use smlar daa used [40, 29, 7, 37]. Moreover, oher dffere daa are also used o chec he accuracy of he model. Deals are gve able able Comparave daa ses Daa Name From o Frequecy Apple Compuer Ic. 0/0/2003 2/0/2005 Daly IBM Corporao 0/0/2003 2/0/2005 Daly Dell Ic. 0/0/2003 2/0/2005 Daly Ryaar Arles 06/0/2003 7/0/2005 Daly AIEX (awa exchage dex) 0/0/200 3/2/2009 Daly SSE(Shagha Soc Exchage) 2/06/2006 3/2/202 Daly DJIA( Dow Joes Idusral Average Idex) 04/08/2006 3/08/202 Daly S&P500 04/08/2006 3/08/202 Daly Uemployme rae 0/0/948 0/2/203 Mohly Ausrala elecrcy 0/0/956 0/08/995 Mohly Polad Elecrcy Load From 990 s 500 values Daly

25 hs sudy does o fxed ses of rag ad es daa. herefore readers ca mae approprae chages whe hey apply specfc daa. I may cases, he expermeal resuls show ha he rag daa s bewee 75% ad 85% for he bes predced resul. Resuls compared wh oher models he frs model o compare s he model [9]. rag ad es daa ses o Apple c., Dell comp., IBM cor., Ryaar Arles are used smlarly (ra = 400 ). Brsh Arles ad Dela Arles are o compared due o daawarehouse o hp://face.yahoo.com// are o complee ad correspode wh [9]. Soc able Compare MAPEs agas oher models HMM-based forecasg model Fuso HMM-ANN- GA wh weghed average (MAPE) Combao of HMM-fuzzy model(mape) CMC-Fuz model Saes =6 Order = IMC-Fuz model Saes =6 Ryaar Ar.,928,377,356,275,27 Apple 2,837,925,796,783,783 IBM,29 0,849 0,779 0,660 0,656 Dell Ic.,02 0,699 0,405 0,837 0,823 Order =2 able 3.3.3, wh Saes =6, we ca see model IMC-Fuz wh Order = s beer ha model CMC-Fuz wh Order =. Boh models are beer ha he models compared agas 4 daa [9]. Comparave resuls are show able Comparso resuls of he IMC-Fuz ad CMC-Fuz models are slghly beer ha oher models for SSE daa ad much beer for he DJIA ad S & P500 daa able Compare dffere models usg SSE, DJIA và S\&P500 daa ses Daa Measureme IMC-Fuz CMC- SSE DJIA S&P500 Fuz BPNN BPNN SNN SVM PCA- PCA- SNN MAE 20,549 20, , , , , ,0844 RMSE 27, ,439 30, , , , ,2975 MAPE 0,8750 0,877,0579 0,9865,290 0,969 0,9540 MAE 90,385 90, , , , ,963 92,769 RMSE 23,205 23, ,65 258, , , ,4365 MAPE 0,7304 0,7304 2,0348,893 2,2677,7404,583 MAE 0,4387 0, ,759 22,833 22,9334 6,838 5,58 RMSE 4,2092 4, ,23 25, ,996 20,5378 9,2467 MAPE 0,8074 0,8074,8607,6725,7722,282,872

26 I he rece publshed [33], he auhors proposeda ew fuzzy me predco model ad comparso wh he dffere mehods he AIEX forecas from 200 o Daa from Jauary o Ocober of each year used as rag daa ad he res from November o December for forecasg ad accuracy calculao. able shows ha our model wh Saes = 6 ad Order =.2 s beer ha all he models meoed. able Compare RMSEs of AIEX for he years from 200 o 2009 wh Saes = 6 Mehod Average Che 996[2] 04,25 9,33 68,06 73,64 60,7 64,32 7,62 30,52 92,75 8,36 ARIMA 97,43 2,23 7,23 70,23 58,32 64,43 69,33 306, 94,39 6,97 Yu 2005[42] 00,54 9,33 65,35 7,50 57,00 63,8 68,76 30,09 9,32 6,34 ES 96,80 9,43 68,0 72,33 54,70 63,72 65,04 303,39 95,60 5,45 Yu 2005 [42] 98,69 9,8 63,66 70,88 54,69 60,87 67,69 308,40 89,78 4,87 Huarg 2006[22] 97,86 6,85 6,32 70,22 52,36 58,37 67,69 306,07 87,45 3,3 Che 20[3] 96,39 4,08 6,38 66,75 52,8 55,83 65,48 304,35 85,06,28 ARFIMA 95,8 5,3 59,43 58,47 50,78 5,23 63,77 35,7 89,23 0,93 Javeda 204 [32] 94,80,70 59,00 64,0 49,80 55,30 63,0 30,70 84,80 09,37 Sadae206 [33] 89,47 04,37 49,67 59,43 37,80 47,30 54,43 294,37 78,80 0,74 Sadae206 [33] 86,67 0,62 45,04 55,80 34,9 45,4 52,88 293,96 74,98 99,00 IMC-Fuz Order= 7,73 68,44 55,96 56,58 55,97 5,87 59,36 06,9 7,5 82,7 Order=2 5,75 67,5 53,75 56,58 55,97 5,73 59,36 05,2 7,5 8,92 CMC-Fuz Order 6,52 68,45 55,97 56,58 55,97 5,87 59,37 06,9 7,5 82,57 Order 2 9,42 7,5 54,8 56,93 60,2 53,57 64,32 06,97 82,03 85,52 hs dsserao preses he Marov cha model (boh frs-order ad hgher-order) ad fuzzy me seres me seres forecasg. Frs, he fuzzy se of me seres proposed by he fuzzy ses becomes he saes of a Marov cha. Secod, he model exeso for he classc hgh-order Marov cha ad he mproved advaced Marov cha correspoded o he hghorder Marov cha parameerzao algorhms. hrd, performg expermes o he same rag se ad es se for rece predco models suggess ha he proposed model has sgfcaly hgher accuracy alhough he algorhm s smpler. Dsserao resuls CONCLUSION Wh he am o develop he forecasg model by egrag exsg models o ew models o mprove predcve accuracy, he dsserao has carred ou he research coes: A overvew of he Marov cha, he hgher-order Marov cha ad he Marov cha parameerzao mehods. Aalyze he poeal applcaos of he Marov cha me seres predco. he dsserao fds ha he fuzzy me seres model predcg me seres cosras s uclear erms of me seres, so some fuzzy me seres heory as well as some

27 predcve algorhms he use of fuzzy me seres s geeralzed. From he bass of he advaages ad lmaos of exsg forecasg models, he dsserao proposes ew combed predcve models o mprove forecas accuracy. Frs, apply he Hdde Marov Model (HMM) for he Posso dsrbuo ad he Normal dsrbuo for he predco model for a parcular me seres based o he aalyss of daa compably wh he model. (Seco 2.). A seres of algorhms are mplemeed ad ru o real daa ha shows he reasoableess of forecass for shor perods of me Secodly, o overcome he dsadvaages of he HMM model (based o he deermsc probably dsrbuo ha emprcal dsrbuo does o follow) ad overcome he (uclear) blurrg of me seres daa. Oupu model combes Marov cha ad fuzzy me seres me seres forecas. Algorhms ha combe he wo models have bee esablshed ad esed o a wde rage of daa compared o rece forecasg models ha show sgfca mproveme predced accuracy. I parcular, he hgher-order Marov model corporaes a poeally large fuzzy me seres appled o seasoal me seres forecasg. he corbuos of he dsserao are already salled expermeally o he R programmg laguage ad he fucos are gve he Appedx. Developme of dsserao opc Icorporae he Marov cha wh more complex fuzzy rules order o more accuraely deerme he role of each value he me seres for a fuzzy se. hs ca furher mprove he accuracy of he forecas. Expad he model for mulvarae me seres, whch he me seres compoes deped o each oher. he arge me sequece s relaed o he oher chas (cha of eves) accordg o he Marov saes defed o hese mpac sequeces. From may mpac sequeces, s possble o combe wh he ANN model o buld predcve models ag o accou exeral facors. hs s le wh realy. Paramerc opmzao problem s sll a ope dreco. Specfcally, he proposed dsserao model s mplemeed wh Order 2 ad Saes 7 suffce for comparso wh oher models. However, hey are o he bes parameer. herefore, he cosruco of a ferece facly ad he algorhm ha deermes he bes parameers for he model are also ssues ha ca be exeded.