A Hidden Markov Model-based Forecasting Model for Fuzzy Time Series

Transcription

1 A Hdde Markov Model-based Forecasg Model for e Seres SHENG-UN LI YI-CHUNG CHENG Isue of Iforao Maagee Naoal Cheg Kug Uversy awa R.O.C. Depare of Idusral ad Iforao Maagee Naoal Cheg Kug Uversy awa R.O.C. Depare of Ieraoal rade aa Woa s College of Ars & echology awa R.O.C. No. a-hsueh Road aa 0 awa R.O.C. Absrac Vague ad coplee daa represeed as lgusc values assvely exss dverse real-word applcaos. he ask of forecasg fuzzy e seres uder ucera crcusaces s hus of grea pora bu dffcul. he here uceray volvg e evoluo usually akes he raso of saes a syse probablsc. I hs paper we proposed a ew forecasg odel based o Hdde Markov Model for fuzzy e seres o realze he probablsc sae raso. We coduc experes of forecasg a real-world eperaure applcao o valdae he beer accuracy of he proposed odel acheved over radoal fuzzy e seres odels. Key-Word:- e Seres ses relaos Hdde Markov Model e seres forecasg Iroduco he forecasg proble of e seres plays a pora role varous probles. However he rado odel of e seres cao hadle he vagueess ad copleeess daa. I 99 Sog ad Chsso roduced ha fuzzy logc o he proble ad proposed a ew paradg of forecasg e seres fuzzy e seres whch s capable of dealg wh vague ad coplee daa represeed as lgusc values uder ucera crcusaces [ ]. he odel was aly coposed of four seps: ( deerg ad parog her uverse of dscourse ( defg fuzzy ses fro he uverse of dscourse ad fuzzfyg he e seres ( dervg fuzzy logcal relaoshps exsg he fuzzfed e seres ad ( forecasg ad defuzzfyg he forecasg oupus. Sog ad Chsso valdaed her odel by usg he erolle of he Uversy of Alabaa. Sce Sog ad Chsso s work a uber of researches have bee repored abou forecasg fuzzy e seres. All he work revewed prarly focused o provg seps ( ad ( Sog ad Chsso's fraework o prove he forecasg accuracy or reduce he copuao overhead. Frsly o allevae he overhead of copuao e dervg he fuzzy relaoshp Sog ad Chsso s odel Sullva ad Woodall proposed he Markov-based odel [] by usg coveoal arx ulplcao. Sullva ad Woodall se he forecasg fuzzy e seres odel wh a e vara Markov usg lgusc labels wh probably dsrbuos ad oralzg all values of rows of ebershp arx o uy. Uforuaely wo pora properes ha he effec of freuecy of rasos o he esao of relaoshp arx s o ake o cosderao ad s oly suable for oe varable. However accordg o he work of Che ad Hwag [] who proposed a wo varables e-vara fuzzy e seres odel o predc he eperaure hey foud ha he forecased eperaure usually s relaed o he eperaure of he rece days []. herefore he probably of he sae caused by earler saes should be cosdered whe dealg wh he forecasg. For accoodag such reuree he propery of Hdde Markov Model (HMM [0] s us rgh acklg hs proble. I a HMM a uderlyg ad uobserved seuece of saes follow a Markov cha wh a fe sae space ad he probably dsrbuo of he observao a ay e s deered oly he curre sae of ha Markov cha.

2 Oher relaed work o fuzzy e seres ca be foud he leraure. Che preseed a effce forecas procedure for erolles of Uversy of Alabaa usg splfed arhec operaos ad proved he forecasg accuracy []. Che laer exeded he prevous work ad proposed a hgh-order fuzzy e seres odel order o reduce he forecasg error []. he work of Hwag Che ad Lee [] showed ha he varao of erolles for ex year s relaed o he erolle red of pas years. Huarg proposed heursc odels by egrag proble-specfc heursc kowledge wh Che s odel o prove forecasg []. Sog ad Chsso appled frs-order e-vara odels forecasg he erolle ad dscussed he dfferece bewee e-vara ad e-vara odels []. Recely saur Yag ad Wag appled he cocep of eropy o easure he degrees of fuzzess whe a e-vara relao arx s derved [6]. Oher ssue abou he oher sep of Sog ad Chsso's fraework o fuzzy e seres ca be foud [ 8 9 ]. I hs paper we focus o he ehacee of seps ( ad ( Sog ad Chsso's fraework. We devoe ourselves o acklg he ssues of provg forecasg accuracy by cobed he propery of HMM ad Che s work. We proposed a ew forecasg odel based o HMM whch he possbly probably of a sae depeds o he pervous saes. We coduc experes forecasg he daly average eperaure ad cloud desy fro Jue 996 o Sepeber 996 ape awa. he resul s ue ecouragg because s accuracy s beer ha oes he leraure. here are fve secos hs paper. I Seco we brefly roduce he basc cocep of fuzzy e seres beforehad prepare he fuzzfed hsorcal daa. I Seco he ew forecasg odel o HMM s proposed by llusrag he exaple of forecasg he daly average eperaure ad cloud desy fro Jue 996 o Sepeber 996 ape awa. I Seco he perforace evaluao ad coparso ers of ea suared error s gve ad dscussed. he las seco s cocluso ad fuure work. e seres I hs seco we brefly descrbe he cocep of fuzzy e seres ad s uderlyg heory. Defo : Le { Y ( R = K p} s a e seres U s he uverse of dscourse whch s a subse of U R. { u = U = K } u s a ordered = paro se of he uverse of dscourse U. A fuzzy e seres s defed as F( { f = K p} = ad f ca be expressed as follows: f = µ Y / A + µ ( Y / A + L + ( Y / A ( µ ( Y R = K } A where µ s he ebershp degree of e seres { Y ( p belogs o. he sybol + eas he operao of uo sead of he operao of suao ad he sybol / dcaes he separaor raher ha he cooly used algebrac sybol of dvso. ses A = K are defed o a ordered paro se of he uverse of dscourse U ad are assocaed wh approprae lgusc values. Le µ Y µ ( Y L µ ( Y µ =[ ] ( be he ebershp degree vecor of f defed o U. For a radoal crsp e seres oe eeds a fuzzfyg procedure o oba he correspodg fuzzy e seres. Followg s he rules for deerg he degree of he ebershp of he hsorcal daa Y belogg o fuzzy se A. he geeraly ragular ebershp fuco s skeched ou follow. Rule : If he hsorcal daa s less ha where s he dpo of he frs ordered paro erval u he he ebershp degree s of A oherwse zero. Rule : If he hsorcal daa Y s greaer ha where Y s he dpo of he las ordered paro erval u he he ebershp degree s of A oherwse zero. Rule : If he hsorcal daa + s wh he rage of ad = K he he ebershp degree s + for A ad + + u u + Y Y Y + for A + where ad are he dpos of he ordered paro erval ad respecvely. Defo : Le F( be a fuzzy e seres. F( s caused by F(- f here exss a fuzzy relaoshp R(- such ha F( = F( R( where represes a fuzzy operaor R( s called he fuzzy relaoshp arx. Forecasg Model Based o HMM

3 Sullva ad Woodall have proposed a sasc forecasg fuzzy e seres odel based o Markov based for oe varable []. However here s very led use for ulvarable. I s proper o apply hdde Markov process o solve such proble ha a Markov process ay o be powerful eough. We odel a forecasg fuzzy e seres usg a Hdde Markov odel where here s a uderlyg hdde Markov process. he hdde Markov process has wo assupos oe s assued ha he ex sae s depede oly upo he curre sae ad he oher assupo s each raso probably does o vary e ha s s a e vara odel. Hdde Markov Model (HMM s a sascal odel o deal wh sybols or sgals fro a syse whch s assued o be a Markov process. I a HMM coa wo ses of saes ad hree ses of probables. For a weaher exaple he sae of he cloud s probablscally relaed o he sae of he eperaure varao ha s he eperaure varao ad cloud saes are closely lks. Soeoe ay perhaps o access o drec eperaure observao bu does have a varao of eperaure by he cloud. I he case whch he daly average eperaure ad cloud desy fro Jue 996 o Sepeber 996 ape awa as dscussed [] as show able. We uoe Sog ad Chsso s fraework o deal wh he forecasg proble four seps as he saed above. Frs o be far copare wh Che s odel we copue he varao of he eperaure seres bewee wo couous daa show as able ad paro o seve eve legh ervals u u K u u [ ] ( ] where =..6 u =.6.0 u = (.0 0.] u = ( 0. 0.] u = ( ] u = ( 0.8.] u = (..0] 6 for hs case he lgusc fuzzy ses are s = (very bg decrease s = (bg decrease = (decrease s = (o chage s = (crease = (bg crease s = (very bg crease respecve. he cloud daa s paroed o seve eve legh ervals v v where v 90 v K v = [ ] = [ v [ v = [ 60 = [ 0 v [ 0 v = [ 0 = 60 6 = s 6 s v he lgusc fuzzy ses are o = (very very heavy o = (very heavy o = (heavy o = (oral o = (h o = (very h o = (very very h respecve. Nex defg fuzzy ses fro he uverse of dscourse ad fuzzfyg he e seres. As we 6 eoed ha he fuzzfyg rules he prevous seco. For sace he cloud desy Jue s belog o he erval he ebershp degree vecor s v 6 µ = [ ] 0. he axu ebershp degree s a se s o 6 he he fuzzy. All he resul of fuzzfg ad he ebershp degrees of he cloud desy Jue 996 are show able. Furherore derve fuzzy logcal relaoshps exsg he fuzzfed e seres. We have wo ses of saes he observable saes (he sae of he cloud o o K o ad he hdde saes (he sae of he varao of he eperaure sae probably π = π π K π = ( o 6 s s s K ( ( Pr( s ( s. he al Pr s = = K Pr = s he al sae probably ha s he probably of each sae occurs as he frs eve. π = Pr ( s = ad ( s π =. π = Pr( s = = N s he oal = N uber of hsory daa s he uber of he sae s ( s hsory daa Jue 9 8 π =. he sae raso A = s he sae raso probably a arx ( a fro sae s o sae.e. s Pr( s s a = where arx a 0 ad = a = b A s a. he cofuso arx ( B = s he probably of observg a observable sae ha he hdde odel s hdde sae b ( o s b = Pr where B s a b 0 ad b =. I Jue = o s gve.e. arx

4 B = ( o s b = Pr ( o s =. ( s herefore gve a Hdde Markov odel (HMM s a rple λ = ( π A B we choose he k-h colu of cofuso arx B B(: [ k] s he fuzzy relaoshp arx for he observao sae he forecasg resul s µ ( k = π * B(:[ k] Pr( k k K o k [ o s Pr( o s Pr( o s ] = k he he sae raso arx s ecessary excep he forecasg order large or eual ha wo. I hs case for frs order he cloud desy s o 6 Jue 996 he 6-h colu of cofuso arx B B(: [6] = µ ( k = π * B(:[ k] = = o oralze he probably 0.* 0 Nµ (k = 0 ha s o say he probably of he cloud desy s very h ( ad he varao of eperaure s bg o 6 s s s s6 decrease ( s / ad s / ad s s / ad s / ad s / ad s s / he oher s 0. he las sep defuzzfyg he forecasg oupus order o copare he perforace. he axu probably s a ad he dpo of erval u s s6 6. he forecasg eperaure s 6.+.=. Jue. Day able. he hsory daa of he average eperaure fro Jue 996 o Sepeber 996 ape awa eperaure Jue July Augus Sepeber he varao of eperaure se eperaure he varao of eperaure se eperaure he varao of eperaure se eperaure he varao of eperaure se 6. 0 s s. 0. s. -0. s.6. s s s s 9.0. s s s s 0.. s s s.. s s s s s s s s 8..6 s s s s 9..0 s s s s s s 8.. s.0 -. s 0.. s s s 8.. s s s s s s s 9.6. s s s s s s s. -. s 9.. s s s 9..0 s. -. s s s s s s s s s.9 0. s s s s s s s s s s s s s s s. -0. s s s s. -0. s s. -0. s

5 s. 0 s s.0-0. s. -0. s. 0 s s.0.0 s. 0. s s s.8 -. s s s s s 8.. s s s s s s. -. s. -. s s s s s s s s. 0. s s.. s able. he hsory daa of he cloud desy Jue 996 ape awa Day Cloud he degree of ebershp se desy µ µ µ µ µ µ 6 µ 6 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o Perforace evaluaos ad coparso I hs seco we wll evaluae he forecasg perforace of he proposed fuzzy e seres odel he eperaure ad copare wh he prevous odels. he forecasg accuracy s easured ers of he ea suare error (MSE = ( Forecasg _ Value Acual Value _ MSE = he perforace coparso s furher valdaed by forecasg errors perceage ad average forecasg errors perceage as follows: E FEP N = N = E FEP E AFEP whch are defed Forecasg _ value Acual _ Value 00% Acual _ Value able he forecasg accuracy s easured ers of he ea suare error (MSE Algorh-A[] Algorh-B[] Algorh-B*[] he proposed odel Jue.(w=.(w=.0(w= 0.668

6 July.(w=.(w=.(w= 0.96 Augus.6(w=.6(w=.0(w= 0.9 Sepeber.(w=.(w=.(w= 0.89 *he w wll be he lowes MSE [ ]. able he forecasg accuracy s easured ers of he average forecasg errors perceage ( E AFEP Algorh-A[] Algorh-B[] Algorh-B*[] he proposed odel Jue.0%(w=.90%(w=.88%(w=.% July.09%(w=.88%(w=.0%(w=.% Augus.9%(w=.9%(w=.%(w=.6% Sepeber.%(w=.%(w=.0%(w=.% *he w wll be he lowes average forecasg errors perceage [ ] he uber of MSE ad he average forecasg errors perceage are able ad. able lss he MSE of repored dffere odels by [ ] ad he proposed odel. he MSE perfored he proposed ehod are ad 0.89 Jue July Augus ad Sepeber respecve. I s beer ha oher odels. he copare of he average forecasg errors perceage s show able. he average forecasg errors perceage are.%.%.6% ad.% Jue July Augus ad Sepeber respecve. able ad hghlghs dffereces bewee he Algorh-A [] Algorh-B [] Algorh-B* [] ad he HMM odels. As far as oday s sudes we ca say ha he resul s forecased he bes by he proposed ehod. Coclusos ad Fuure Work he fuzzy e-seres has recely receved ore aeo because of s capably of dealg wh vague ad coplee daa. here have bee a varey of odels developed o eher prove forecasg accuracy or reduce copuao overhead. I hs paper we proposed a ew forecasg odel based o Hdde Markov Mode o deal wh he wo varables (wo saes forecasg proble of fuzzy e seres. he proposed odel s esablshed o he bass ha he probably dsrbuo of he observao a ay e s always depede o he prevous saes. he perforace evaluao o forecasg he eperaure awa cofrs ha he proposed odel ouperfors all prevously proposed odels ers of ea suared error. he fuure work o exedg he proposed odel o hadle he forecasg proble of ul-desoal fuzzy e seres s udergog. [] S.-M. Che Forecasg erolles based o hgh-order fuzzy e seres Cyberecs ad Syses: A Ieraoal Joural Vol. 00 pp. -6. [] Y.-Y. Hsu S.-M. se ad B. Wu A ew approach of bvarae fuzzy e seres aalyss o he forecasg of a sock dex Ieraoal Joural of Uceray ess ad Kowledge-Based Syses Vol. 00 pp [] K. Huarg Heursc odels of fuzzy e seres for forecasg Ses ad Syses Vol. 00 pp [6] K. Huarg Effecve leghs of ervals o prove forecasg fuzzy e seres Ses ad Syses Vol. 00 pp [] J. R. Hwag S.M. Che ad C. H. Lee Hadlg forecasg probles usg fuzzy e seres Ses ad Syse Vol pp. -8. [8].-S. Lee C.-C. Chu F.-C. L Predco of he ueploye rae usg fuzzy e seres wh Box-Jeks ehodology Ieraoal Joural of Syses Vol. 00 pp. -8. [9] S.-. L Y.-P. Che Naural paro-based forecasg odel for fuzzy e seres IEEE Ieraoal Coferece o Syses Budapes Hugary 00 pp. -9. [0] L.R. Raber A uoral o hdde Markov odels ad seleced applcaos speech recogo. Proceedgs of he IEEE Vol. 989 pp [] Q. Sog B. S. Chsso Forecasg erolles wh fuzzy e seres-par I Ses ad Syses Vol. 99 pp. -9. [] Q. Sog B. S. Chsso e seres ad s odels Ses ad Syses Vol. 99 pp [] Q. Sog B. S. Chsso Forecasg erolles wh fuzzy e seres-par Ⅱ Ses ad Syses Vol pp. -8. [] J. Sullva W. H. Woodall A coparso of fuzzy forecasg ad Markov odelg Ses ad Syses Vol pp [] C.-C. sa S.-J. Wu A sudy for secod-order odelg of fuzzy e seres Proceedgs of he IEEE I'l Cof. o Syses 999 pp. 9-. [6] R.-C. saur J.-C. Yag H.-F. Wag relao aalyss fuzzy e seres odel Copuers ad Maheacs wh Applcaos Vol.9 00 pp Refereces: [] S.-M. Che Forecasg erolles based o fuzzy e seres Ses ad Syses Vol pp. -9. [] S.-M. Che J.-R. Hwag eperaure predco usg fuzzy e seres IEEE ras. o Syses Ma ad Cyberecs-Par B: Cyberecs Vol pp. 6-.