A Hybrid Approach based on Winter s Model and Weighted Fuzzy Time Series for Forecasting Trend and Seasonal Data

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1 Joural of Mahemacs ad Sascs 7 (3): , 2011 ISSN Scece Publcaos Hybrd pproach based o Wer s Model ad Weghed Fuzzy me Seres for Forecasg red ad Seasoal Daa 1 Suharoo ad 2 Muhammad Hsyam Lee 1 Deparme of Sascs, Faculy of Mahemacs ad Naural Sceces, Isue echology Sepuluh Nopember, Idoesa 2 Deparme of Mahemacs, Faculy of Sceces, Uversy echology Malaysa bsrac: Problem saeme: I he leraure, he mos suded of fuzzy me seres for he purpose of forecasg s he frs order fuzzy me seres model I hs model, oly he frs lagged varable s used he cosrucg he frs order fuzzy me seres model herefore, such approaches fal o aalyze accuraely red ad seasoal me seres hch s a mpora class me seres models pproach: I hs paper, a hybrd approach s proposed order o aalyze red ad seasoal fuzzy me seres he proposed hybrd approach s based o Wer s model ad eghed fuzzy me seres he Wer s model ad he WFS model are used oly, amg o capure dffere forms of paer he me seres daa he order of hs model s deermed by ulzg graphcal order fuzzy relaoshp real me seres abou ours arrvals daa s aalyzed h hs mehod o sho he effcecy of he proposed hybrd mehod Resuls: he resuls obaed from he proposed mehod are compared h he oher mehods, e, Decomposo, Wer s ad RIM models s a resul, s observed ha more accurae resuls are obaed from he proposed hybrd mehod Cocluso: he emprcal resuls h ours arrvals daa clearly sugges ha he hybrd model s able o ouperform each compoe model used solao he paer of me seres daa Moreover, hese emprcal evdeces sugges ha by usg dssmlar models or models ha dsagree each oher srogly, he hybrd model ll have loer geeralzao varace or error ddoally, because of he possble usable or chagg paers he daa, usg he hybrd mehod ca reduce he model uceray hch ypcally occurred sascal ferece ad me seres forecasg Key ords: Fuzzy me seres, hybrd, Wer s model, graphcal order, seasoaly, red INRODUCION he defos of fuzzy me seres ere frsly roduced by Sog ad Chssom (1993a; 1993b) ad hey developed he model by usg fuzzy relao equaos ad approxmae reasog Furhermore, Sog ad Chssom (1994) dvded he fuzzy me seres o o ypes, amely me-vara ad me-vara, hose dfferece reles o heher here exss he same relao beee me ad s pror me - (here = 1, 2,,m) If he relaos are all he same, s a me-vara fuzzy me seres; lese, f he relaos are o he same, he s me-vara Recely, Lu (2009) proposed a egraed fuzzy me seres forecasg sysem hch he forecased value ll be a rapezodal fuzzy umber sead of a sgle-po value ad effecvely deal h saoary, red ad seasoal me seres Laer, Egroglu e al (2009) proposed a e hybrd approach based o SRIM ad paral hgh order bvarae fuzzy me seres for forecasg seasoal daa Elaal e al (2010) roduced fuzzy cluserg o selec membershp fucos fuzzy me seres model ddoally, Lee ad Suharoo (2010) also proposed a e eghed fuzzy me seres for forecasg me seres h seasoal paer I hs paper, a e hybrd model based o he Wer s model ad eghed fuzzy me seres s proposed o mprove he forecas accuracy red ad seasoal daa hs approach follos he dea from Zhag (2003) ho proposed a hybrd model based o RIM ad Neural Neor model I hs e hybrd model, a lear chroologcal egh from Yu (2005) s expaded o a uform ad/or expoeal chroologcal egh as Lee ad Suharoo (2010) for forecasg he error seres from Wer s model hs sudy shos ha Correspodg uhor: Suharoo, Deparme of Sascs, Faculy of Mahemacs ad Naural Sceces, Isue echology Sepuluh Nopember, Surabaya 60111, Idoesa 177

2 J Mah & Sa, 7 (3): , 2011 he graphcal order fuzzy relaoshp could be used effecvely o selec a approprae order of fuzzy me seres ddoally, hs sudy also shos ha by usg a seres of mohly ours arrvals o Bal, Idoesa, he hybrd approach h a expoeal chroologcal egh (Lee ad Suharoo, 2010) ouperforms he hybrd fuzzy me seres proposed by Che (1996); Yu (2005) ad Cheg e al (2008) ad some classcal mehods, e, Decomposo, Wer s ad RIM models Daa sources: real mohly daases abou he umber of ours arrvals o Bal, Idoesa, from 1989 o 1997, s used as case sudy hs seres as obaed from he Idoesa Ceral Bureau of Sascs (see bpsgod) Bal s he ma desao of he eraoal ourss ho vs Idoesa ad hese daa also have red ad seasoal paer Ismal e al (2009) aalyzed hese oursm daa usg erveo aalyss ad recely Suharoo (2011) also used hese daa for evaluag he effec of addve or mulplcave order SRIM model For hs daases, he las 12 observaos are reserved as he es for forecasg evaluao ad comparso (ou-sample daase or esg daa) MERILS ND MEHODS Che (1996) mproved he approach proposed by Sog ad Chssom (1993a; 1993b) Che s mehod uses a smple operao, sead of complex marx operaos, he esablshme sep of fuzzy relaoshps he algorhm of Che s mehod ca be gve as follos: Sep 1: Defe he uverse of dscourse ad ervals for rules absraco Based o he ssue doma, he uverse of dscourse ca be defed as: U = [sarg, edg] s he legh of erval s deermed U ca be paroed o several equally legh ervals Sep 2: Defe fuzzy ses based o he uverse of dscourse ad fuzzfy he hsorcal daa Sep 3: Fuzzfy observed rules Sep 4: Esablsh Fuzzy Logcal Relaoshps (FLRs) ad group (FLRG) hem based o he curre saes of he daa of he fuzzy logcal relaoshps Case 2: here s oly oe fuzzy logcal relaoshp he fuzzy logcal relaoshp sequece If, he F(), forecas value, s equal o Case 3: If value, s equal o,,, 2,,, 2, he F(), forecas Sep 6: Defuzzfy If he forecas of F() s,,,, he defuzzfed resul s equal o he 2 arhmec average of he mdpos of,,, 2 Yu s mehod: Yu (2005) proposed eghed models o acle o ssues fuzzy me seres forecasg, amely, recurrece ad eghg he mehod proposed by Yu apples a lear chroologcally eghs ad produces more accurae forecass ha Che s frs order fuzzy me seres mehod he seps of he algorhm of he eghed mehod proposed by Yu (2005) ca be gve belo Sep 1: Defe he dscourse of uverse ad subervals Based o m ad max values he daa se, D m ad D max varables are defed he choose o arbrary posve umbers hch are D 1 ad D 2 order o dvde he erval evely, U = [D D, D D ] m 1 max 2 Sep 2: Defe fuzzy ses based o he uverse of dscourse ad fuzzfy he hsorcal daa Sep 3: Fuzzfy observed rules Sep 4: Esablsh fuzzy logcal relaoshps (revsed Che s mehod) he recurre FLRs are ae o accou by revsg Sep 4 Che s mehod For example, here are 5 FLRs h he same LHS, 1 2,1 1, 1 1,1 3,1 1 hese FLRs are used o esablsh fuzzy logcal relaoshp group as: 1 2, 1, 1, 3,1 Sep 5: Forecas Use he same rule as Che s Sep 6: Defuzzfy Suppose he forecas of F() s,,, 2 marx of he mdpos of he defuzzfed marx s equal o a,,, 2 : Sep 5: Forecas Le F( 1) = Case 1: If he fuzzy logcal relaoshp of s empy;, he F(), forecas value, s equal o M() = [m,m,,m 2 ] here, M() represes he defuzzfed forecas of F() 178

3 J Mah & Sa, 7 (3): , 2011 Sep 7: ssgg eghs Suppose he forecas of F() s,,, he correspodg eghs for 2 2,,,, say 1, 2,, are specfed as: = h=1 h here, 1=1, = for 2 We he oba he egh marx as: W() = [,,, ] =,,, here, h s he correspodg egh for h Sep 8: Calculag he fal forecas values I he eghed model, he fal forecas s equal o he produc of he defuzzfed marx ad he raspose of he egh marx: ˆF() = M() W() = [m,m,,m 2 ] 1 2,,, = Marx produc operaor M() = 1 marx W() = 1 marx, respecvely Cheg s mehod: Cheg e al (2008) proposed fuzzy me seres based o adapve expecao model for oba forecass he mehod proposed by Cheg e al produces more accurae forecass ha Che s ad Yu s mehod o o real daa, amely IEX ad he erollmes of he Uversy of labama he seps of he algorhm of he mehod proposed by Cheg e al (2008) are gve belo Sep 1: Defe he dscourse of uverse ad subervals as Yu s Sep 3: Fuzzfy observed rules Sep 4: Esablsh fuzzy logcal relaoshps (revsed Che s mehod) he FLRs h he same LHSs ca be grouped o form of FLR Group For example, here are 5 FLRs h he same LHS, 1 2,1 1, 1 1,1 3,1 1 hese FLRs are used o esablsh fuzzy logcal relaoshp group as: 1 2, 1, 1, 3,1 ll FLRs ll cosruc a flucuao-ype marx Hece, he flucuao-ype marx s: [ ] W() = [,,, ] = 1, 1, 2, 1, 3 Sep 5: ssgg eghs he marx from Sep 4 s furher sadardzed o W ad mulpled by he L df deffuzfed marx,, o produce he forecas value hese eghs should sadardzed o oba he egh marx, e W () = [ 1, 2,, ] hs egh should be ormalzed by applyg he sadardze egh marx equao as follos: 1 2 W () =,,, Sep 6: Calculae forecas value From Sep 5, e ca oba he sadardzed egh marx, o ge he forecas value by usg: F() = L ( 1) W ( 1) df here, L df ( 1) s he deffuzfed marx ad W ( 1) s he egh marx Sep 7: Employ he adapve forecasg equao o produce a coclusve forecas Lee s mehod: Lee ad Suharoo (2010) proposed a uform ad expoeal chroologcally eghs o acle o ssues fuzzy me seres forecasg, amely, recurrece ad eghg, as exeso of Yu s mehod hs mehod produces more accurae forecass ha Che s, Yu s ad Cheg s mehods he seps of he algorhm of he eghed mehod proposed by Lee ad Suharoo (2010) are gve as follos Sep 2: Defe fuzzy ses based o he uverse of Sep 1: Defe he uverse of dscourse ad paro dscourse ad fuzzfy he hsorcal daa o ervals as Yu s mehod 179

4 J Mah & Sa, 7 (3): , 2011 Sep 2: Esablsh a relaed fuzzy se (lgusc value) for each observao he rag daase Sep 3: Esablsh fuzzy relaoshp Sep 4: Esablsh fuzzy relaoshps groups for all FLRs Sep 5: Selec he bes order of FLRs he graphcal orders for FLRs ad flucuao-ype marxes are used o defy he bes order of FLRs Sep 6: Forecas Sep 7: Defuzzfy Use he same rule as Yu (2005) acheved successes her o lear or olear domas, parcularly for forecasg red ad seasoal me seres Hoever, oe of hem s a uversal model ha s suable for all crcumsaces, parcularly WFS oly ored ell for saoary or seasoal me seres h o red Zhag (2003) saed ha sce s dffcul o compleely o he characerscs of he daa a real problem, hybrd mehodology ha has boh lear ad olear modelg capables ca be a good sraegy for praccal use By combg dffere models, dffere aspecs of he uderlyg paers may be capured s proposed by Zhag (2003), may be reasoable o cosder a me seres o be composed of a lear srucure ad a olear compoe ha s: Sep 8: ssgg eghs Suppose he forecas of F() s,,, he correspodg eghs for 2 2,,,, say 1, 2,, are: c c c W() =,,,, here, -1 1=1, = c for c 1 he correspodg egh for, 2 ad h s h hs proposed eghs become a expoeal eghs he c > 1 ad ed o gve he rece FLRs as more mpora ha he older oes ad geerally hgher values ha Yu s egh ddoally, hese proposed eghs also sho ha he c = 1 he he eghs ll have uformly chroologcal paer hch mply he same mpora me of chroologcal relaoshp Sep 9: Calculae he fal forecas values he fal forecas s equal o he produc of he defuzzfed marx ad he raspose of he egh marx: ˆF() = M() W() = [m,m,,m 2 ] -1 1 c c,,, = Marx produc operaor, M() = 1 marx Y = L + N (1) L = he lear compoe N = he olear compoe hese o compoes have o be esmaed from he daa I hs paper, frs, e le Wer s model o model he lear compoe parcularly red ad seasoal compoes, he he resduals from Wer s model ll become saoary seres ad may coa oly he olear relaoshp hus, e propose o cosder he forecas of me seres o be composed of o compoes, Ŷ 1, ad Ŷ 2,, as follos: Y ˆ = Y ˆ ˆ 1, + Y 2, (2) Ŷ 1, = model Forecas value for me from he Wer s Ŷ 2, = Forecas value from he WFS for he resdual a me from he Wer s model he four equaos used mulplcave Wer s model are as follos (Hae ad Wcher 2009): he expoeally smoohed seres: W() = 1 marx Y = α( ) + (1 α)( ) (3) S-L Proposed Model ad lgorhm: Boh Wer s ad Weghed Fuzzy me Seres (WFS) models have he red esmae: 180

5 J Mah & Sa, 7 (3): , 2011 = β( -1) + (1 β)-1 (4) he seasoaly esmae: Y S = γ( ) + (1 γ)s-l (5) Forecas p perods o he fuure: Ŷ +p = ( + p ) S -L+p (6) Le e deoe he resdual a me from he Wer s model, he: e = Y Y ˆ (7) 1, here Ŷ 1, s he forecas value for me from he esmaed Wer s model Resduals are mpora dagoss of he suffcecy of lear models lear model s o suffce f here are sll lear correlao srucures lef he resduals Hoever, resdual aalyss s o able o deec ay olear paers he daa I fac, here s currely o geeral dagosc sascs for olear auocorrelao relaoshps herefore, eve f a model has passed dagosc checg, he model may sll o be adequae ha olear relaoshps have o bee appropraely modeled By modelg resduals usg WFS, olear relaoshps ca be dscovered I summary, he proposed mehodology of he hybrd sysem cosss of o seps I he frs sep, a Wer s model s used o aalyze he red ad seasoal par of he problem I he secod sep, a WFS model s developed o model he resduals from he Wer s model I hs secod sep, e apply four WFS models proposed by Che (1996); Yu (2005); Cheg e al (2008) ad Lee ad Suharoo (2010) he resuls from he WFS ca be used as predcos of he error erms for he Wer s model he hybrd model explos he uque feaure ad sregh of Wer s model as ell as WFS model deermg dffere paers hus, could be advaageous o model red, seasoal ad olear paers separaely by usg dffere models ad he combe he forecass o mprove he overall modelg ad forecasg performace o valdae he mehodology of hybrd model for forecasg red ad seasoal me seres daa, a e algorhm s proposed as follos Sep 2: pply WFS mehod o model he resduals from he Wer s model ad ge he secod forecas compoe, Ŷ 2, I hs sep, four WFS mehods proposed by Che (1996); Yu (2005); Cheg e al (2008) ad Lee ad Suharoo (2010) are appled o fd he bes forecased values Sep 3: Calculae he fal forecas values by addg he forecas values a he frs ad secod seps as Eq 2 RESULS o demosrae he effecveess of hs hybrd mehod, e use daa abou he umber of ours arrvals o Bal, Idoesa, va Ngurah Ra arpor from Jauary 1989 ul December 1997 as a case sudy he me seres plo a Fg 1 llusraes ha he daa have boh red ad seasoal paer o assess he forecasg performace of dffere models, each daa se s dvded o o samples of rag ad esg he rag daa se ha coas 96 records (Jauary 1989 ul December 1996) s used exclusvely for model developme ad he he las 12 records (Jauary 1997 ul December 1997) as es sample s used o evaluae he esablshed model I hs sudy, all hybrd modelg s mplemeed va o pacage programs, e, MINIB for Wer s model a he frs sep ad MLB for WFS model a he secod sep he resuls are compared h hree classcal me seres models, amely Decomposo mehod, RIM ad Wer s models Oly he -sepahead forecasg s cosdered he Roo Mea Squared Error (RMSE) s seleced o be he forecasg accuracy measures Sep 1: pply Wer s model a Eq 3-6 o ge he frs forecas compoe, Ŷ 1, ad he resduals, e 181 Fg 1: Mohly daa abou he umber of ours arrvals o Bal

6 J Mah & Sa, 7 (3): , 2011 able 1: Comparso of RMSEs boh rag ad esg daases rag esg Mehod RMSE Rao RMSE Rao Decomposo Wer's α,β, γ= rma (0,1,1),(0,1,1) Hybrd model: Wer's+WFS models he 1 s order Che's Yu's (2005) Cheg's (2008), α= Lee's (2010), c= he 12 h order Che's (1996) (seasoal) Yu's (2005) Cheg's (2008), α= Lee's (2010), c= he resuls of RMSEs obaed usg he hybrd models ad hree classcal me seres models, boh rag ad esg daa, are lsed able 1 Colum rao llusraes he rao beee each mehod o he resul of RIM model he value s less ha 1 sho ha he resul s beer ha RIM DISCUSSION he resuls a able 1 geeral sho ha he overall forecasg errors ca be sgfcaly reduced by usg he hybrd models (by combg o models ogeher), boh rag ad esg daases I erms of RMSE, he performace evaluao rag daa shos ha hybrd model, e, a combao beee Wer s ad Cheg s WFS mehods a he 12h order FLR yelds he mos accurae forecased values ha oher models ddoally, hese resuls also sho ha mos of hybrd models yeld more accurae forecased values ha RIM ad o oher classcal me seres models Moreover, he hybrd model beee Wer s ad Lee s WFS mehods a he frs order FLR yelds he bes forecased values ha oher models a esg daase ddoally, he resuls esg daa also sho ha all he proposed hybrd mehods he frs order FLR yeld more accurae forecas ha oher hybrd mehods ad o classcal me seres models, e Decomposo ad RIM models he resuls also sho ha Wer s model could recosruc ell he red ad seasoal compoe of he seres ad he WFS could f ell he resdual from Wer s model o mprove he forecas accuracy CONCLUSION me seres aalyss ad forecasg s a acve research area over he las fe decades he accuracy of be eased 182 me seres forecasg s fudameal o may decso processes ad hece he research for mprovg he effecveess of forecasg models has ever sopped Wh he effors of Box ad Jes (1976), he RIM model has become oe of he mos popular mehods he forecasg research ad pracce More recely, WFS have sho her promse me seres forecasg applcaos h her olear modelg capably I hs sudy, e propose o ae a hybrd approach based o Wer s ad WFS models ad apply for forecasg red ad seasoal daa, e, ours arrvals daa he Wer s model ad he WFS model are used oly, amg o capure dffere forms of paer he me seres daa he emprcal resuls h ours arrvals daa clearly sugges ha he hybrd model s able o ouperform each compoe model used solao he paer of me seres daa Varous combg mehods have bee proposed he leraure Hoever, mos of hem are desged o combe he smlar mehods Zhag (2003) saed ha heorecal as ell emprcal evdeces he leraure sugges ha by usg dssmlar models or models ha dsagree each oher srogly, he hybrd model ll have loer geeralzao varace or error ddoally, because of he possble usable or chagg paers he daa, usg he hybrd mehod ca reduce he model uceray hch ypcally occurred sascal ferece ad me seres forecasg Furhermore, by fg he Wer s model frs o he red ad seasoal daa, he fg problem h hgher order relaed o fuzzy me seres model ca

7 J Mah & Sa, 7 (3): , 2011 REFERENCES Box, GEP ad GM Jes, 1976 me Seres alyss: Forecasg ad Corol 1s Ed, Sa Frassco, Holde-Day, ISBN: , pp: 1-24 Che, SM, 1996 Forecasg erollmes based o fuzzy me seres Fuzzy Ses Sys, 81: DOI: / (95) Cheg, CH, L Che, HJ eoh ad CH Chag, 2008 Fuzzy me seres based o adapve expecao model for IEX forecasg Exp Sys ppl, 34: DOI: /esa Egroglu, E, ladag, CH, Yolcu, U, Basara, M ad Uslu, VR 2009 e hybrd approach based o SRIM ad paral hgh order bvarae fuzzy me seres forecasg model Exp Sys ppl, 36: DOI: /esa Elaal, K, H Hefy ad H Elahab, 2010 Cosrucg fuzzy me seres model based o fuzzy cluserg for a forecasg J Compu Sc, 6: DOI: /cssp Hae, JE ad D Wcher, 2009 Busess Forecasg Prece Hall, Ne Jersey, ISBN- 10: , pp: Ismal, Z, Suharoo, Yahaya ad R Efed, 2009 Ierveo model for aalyzg he mpac of errorsm o oursm dusry J Mah Sa, 5: DOI: /mssp Lee, MH ad Suharoo 2010 ovel eghed fuzzy me seres model for forecasg seasoal daa Proceedg he 2d Ieraoal Coferece o Mahemacal Sceces, Nov 30-Dec 3, Kuala Lumpur, Malaysa, pp: ISBN: Lu, H, 2009 egraed fuzzy me seres forecasg sysem Exp Sys ppl, 36: DOI: /esa Sog, Q ad BS Chssom, 1993a Fuzzy forecasg erollmes h fuzzy me seres-par 1 Fuzzy Ses Sys, 54: 1-9 DOI: / (93)90355-L Sog, Q ad BS Chssom, 1993b Fuzzy me seres ad s models Fuzzy Ses Sys, 54: DOI: / (93)90372-O Sog, Q ad BS Chssom, 1994 Fuzzy forecasg erollmes h fuzzy me seres-par 2 Fuzzy Ses Sys, 62: 1-8 DOI: / (94) Suharoo, 2011 me seres forecasg by usg seasoal auoregressve egraed movg average: subse, mulplcave or addve model J Mah Sa, 7: DOI: /mssp Yu, HK 2005 Weghed fuzzy me-seres models for IEX forecasg Physc : Sa Mech ppl, 349: DOI: /JPHYS Zhag, GP, 2003 me seres forecasg usg a hybrd RIM ad eural eor model Neurocompug, 50: DOI: /S (01)