Performance evaluation of series and parallel strategies for financial time series forecasting

Khashei, Mehdi; Haji ahimi, Zah a A icle Pe o mance e alua ion o se ies and pa allel s a egies o inancial ime se ies o ecas ing Financial Inno a ion P o ided in Coope a ion wi h: Sp inge Na u e Sugges ed Ci a ion: Khashei, Mehdi; Haji ahimi, Zah a (2017) : Pe o mance e alua ion o se ies and pa allel s a egies o inancial ime se ies o ecas ing, Financial Inno a ion, ISSN 2199-4730, Sp inge , Heidelbe g, Vol. 3, Iss. 24, pp. 1-24, h ps://doi.o g/10.1186/s40854-017-0074-9 This Ve sion is a ailable a : h ps://hdl.handle.ne /10419/176464 S anda d-Nu zungsbedingungen: Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen Zwecken und zum P i a geb auch gespeiche und kopie we den. Sie dü en die Dokumen e nich ü ö en liche ode komme zielle Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich machen, e eiben ode ande wei ig nu zen. 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I he documen s ha e been made a ailable unde an Open Con en Licence (especially C ea i e Commons Licences), you may exe cise u he usage igh s as speci ied in he indica ed licence. h p://c ea i ecommons.o g/licenses/by/4.0/ RESEARCH Open Access Pe o mance e alua ion o se ies and pa allel s a egies o inancial ime se ies o ecas ing Mehdi Khashei and Zah a Haji ahimi * * Co espondence: [email p o ec ed] Depa men o indus ial and sys em enginee ing, Is ahan Uni e si y o Technology, Is ahan, I an Abs ac Backg ound: Imp o ing inancial ime se ies o ecas ing is one o he mos challenging and i al issues acing nume ous inancial analys s and decision make s. Gi en i s di ec impac on ela ed decisions, a ious a emp s ha e been made o achie e mo e accu a e and eliable o ecas ing esul s, o which he combining o indi idual models emains a widely applied app oach. In gene al, indi idual models a e combined unde wo main s a egies: se ies and pa allel. While i has beenp o en ha heses a egiescanimp o e o e all o ecas ing accu acy, he li e a u e on ime se ies o ecas ing emains ague on he choice o an app op ia e s a egy o gene a e a mo e accu a e hyb id model. Me hods: The e o e, his s udy’s key aim is o e alua e he pe o mance o se ies and pa allel s a egies o de e mine a mo e accu a e one. Resul s: Acco dingly, he p edic i e capabili ies o i e hyb id models a e cons uc ed on he basis o se ies and pa allel s a egies compa edwi heacho he andwi h hei base models o o ecas s ock p ice. To do so, au o eg essi e in eg a ed mo ing a e age (ARIMA) and mul ilaye pe cep ons (MLPs) a e used o cons uc wo se ies hyb id models, ARIMA-MLP and MLP-ARIMA, and h ee pa allel hyb id models, simple a e age, linea eg ession, and gene ic algo i hm models. Conclusion: The empi ical o ecas ing esul s o wo benchma k da ase s, ha is, he closing o he Shenzhen In eg a ed Index (SZII) and ha o S anda d and Poo ’s500 (S&P 500), indica e ha al hough all hyb id models pe o m be e han a leas one o hei indi idual componen s, he se ies combina ion s a egy p oduces mo e accu a e hyb id models o inancial ime se ies o ecas ing. Keywo ds: Se ies and pa allel combina ion s a egies, Mul ilaye pe cep ons, Au o eg essi e in eg a ed mo ing a e age, Financial ime se ies o ecas ing, S ock ma ke s Backg ound Real ime se ies o ecas ing wi h a high deg ee o accu acy is gaining inc easing impo ance in many domains, pa icula ly he inancial ma ke s, and hus, a ious a emp s ha e been made o de elop mo e accu a e echniques. The objec i e o inancial ime se ies o ecas ing is o p o ide inancial analys s and in es o s wi h eliable guidance on asse managemen . Thus, imp o ing o ecas ing accu acy and in oducing eliable o ecas ing me hods can acili a e mo e p o i able inancial ma ke in es men s by lead in es o s and inancie s. To his e ec , choosing a me hod ha Financia l Inno a ion © The Au ho (s). 2017 Open Access This a icle is dis ibu ed unde he e ms o he C ea i e Commons A ibu ion 4.0 In e na ional License (h p://c ea i ecommons.o g/licenses/by/4.0/), which pe mi s un es ic ed use, dis ibu ion, and ep oduc ion in any medium, p o ided you gi e app op ia e c edi o he o iginal au ho (s) and he sou ce, p o ide a link o he C ea i e Commons license, and indica e i changes we e made. Khashei and Haji ahimi Financial Inno a ion (2017) 3:24 DOI 10.1186/s40854-017-0074-9 pe o ms well in inancial ime se ies o ecas ing is impe a i e. To p o ide mo e accu - a e esul s, s udies on ime se ies o ecas ing and modeling widely use a combina ion o di e en models and me aheu is ic op imiza ion app oaches. Conside able esea ch has adop ed op imiza ion me hods such as gene ic algo i hm (GA; Aghay Kaboli e al. 2016a, 2016b, Kaboli e al. 2016), pa icle swa m op imiza ion (PSO; Aghay Kaboli e al. 2016a, 2016b), and gene exp ession p og amming (GEP; Aghay Kaboli e al. 2017a, 2017b; Aghay Kaboli e al. 2016a, 2016b). Kaboli e al. (2016) p oposed he a i icial co- ope a i e sea ch (ACS) algo i hm o o ecas long- e m elec ici y ene gy consump ion and nume ically con i med he e ec i eness o he algo i hm using o he me aheu is ic algo i hms including he GA, PSO, and cuckoo sea ch. Modi i-Delshad e al. (2016) p esen ed a back acking sea ch algo i hm (BSA) and e i ied he eliabili y o he me hod in sol ing and modeling he economic dispa ch (ED) p oblem. Aghay Kaboli e al. (2016a, 2016b) es ima ed elec ici y demand using GEP, a gene ic-based me hod, as an exp ession-d i en app oach and showed ha GEP ou pe o ms he mul ilaye pe - cep on neu al (MLP) ne wo k and mul iple linea eg ession models. Recen s udies on ime se ies o ecas ing la gely ocus on combina ion me hods gi en he dis inguishing ea u es o hyb id models (e.g., unique modeling capabili y o each model), d awbacks in using single models, and he esul an imp o emen s in o ecas ing accu acy. The key concep o combina ion heo y is employing he unique me i s o indi idual models o ex ac di e en da a pa e ns. Impo an ly, he li e a u e con i ms ha no indi idual model can uni e sally de e mine da a-gene a ion p ocesses. In o he wo ds, all cha ac e is ics o unde lying da a canno be ully modeled by one model and he e o e, combining di e en models o using hyb id ones helps analyze complex pa e ns in da a mo e accu a ely and comple ely. Fu he , combining a ious models simpli ies he selec ion o a model ha is app op ia e o p ocess di e en o ms o ela ionships in he da a and educes he isk o choosing an ine icien one. Se e al app oaches ha e been p oposed o combine linea and nonlinea models. These combina ion me hods a e gene ally di ided in o wo p ima y classes: se ies and pa allel. In a se ies combina ion me hod, a ime se ies is decomposed in o linea and nonlinea pa s. Acco dingly, in he i s s age, he model is used o p ocess one ime se ies componen and hen, he ob ained alues a e used as inpu s o he second model o analyze ano he componen . On he o he hand, in he pa allel combina ion me hod, he o iginal da a a e simul aneously conside ed o be inpu s o di e en models and hen, he linea combina ion o he o ecas ed esul s acili a es inal hyb id o ecas ing. The li e a u e on se ies linea o nonlinea combina ion models has d ama ically expanded since he ea ly wo k o Zhang (2003). Fo ins ance, Pai and Lin (2005) p oposed a se ies hyb id me hodology o exploi he unique s eng h o au o eg essi e in eg a ed mo ing a e age (ARIMA) and suppo ec o machines (SVMs) o o ecas s ock p ice and indica ed ha a hyb id model ou pe o ms i s componen s. Chen and Wang (2007) cons uc ed a se ies combina ion model ha inco po a es seasonal au o- eg essi e in eg a ed mo ing a e age (SARIMA) and SVMs o seasonal ime se ies o ecas ing and achie ed mo e accu a e esul s han bo h componen s. Zeng e al. (2008) p esen ed a se ies combina ion o he ARIMA and MLP models o p edic sho - e m a ic low. Thei expe imen al esul s o he eal da ase s indica ed ha he p oposed hyb id model can be an e ec i e in imp o ing o ecas ing accu acy Khashei and Haji ahimi Financial Inno a ion (2017) 3:24 Page 2 o 24 achie ed by ei he componen . Zhou and Hu (2008) conduc ed expe imen s using a hy- b id modeling and o ecas ing app oach, which was based on he G ey and Box–Jenkins ARMA models, and showed ha hei p oposed model had highe o ecas ing p ecision han i s single componen s. Pao (2009) p oposed a hyb id se ies model inco po a ing a i icial neu al ne wo k (ANN) and di e en ypes o gene alized au o eg essi e condi ional he e oscedas ici y (GARCH) models o o ecas ene gy consump ion. Table 1 lis s o he ecen s udies on se ies linea and nonlinea hyb id models. Ba es and G ange (1969) in oduced he concep o a pa allel combina ion, which was subsequen ly used by many esea che s such as Mak idakis e al. (1982), G ange and Ramana han (1984), Bunn (1989), and De Menezes and Bunn (1993). Wedding and Cios (1996) p oposed a pa allel combina ion model using adial basis unc ion ne wo ks and he Box–Jenkins ARIMA model. Mo e ecen ly, se e al pa allel hyb id o ecas ing models ha e been p oposed o combine linea and nonlinea models. Fo ins ance, Wang e al. (2012) p esen ed a pa allel hyb id model using GA and by employing ARIMA, exponen ial smoo hing (ES), and back p opaga ion neu al ne wo k (BPNN) models. Thei nume ical esul s showed ha he p oposed model ou pe o ms all adi ional models, including he ESM, ARIMA, BPNN, equal weigh hyb id (EWH) model, and andom walk (RWM) model. Fo ecas ing s ock e u ns, Ra he e al. (2015) p oposed a no el hyb id model ha me ges p edic ions by h ee indi idual models: ES, ecu en neu al ne wo k (RNN), and ARIMA; he op imum weigh s o each model a e iden i ied using GA. Yang e al. (2016) p esen ed a combined o ecas ing model using BPNN, adap i e ne wo k-based uzzy in e ence sys em (ANFIS), and SARIMA models, and hus, used a di e en ial e olu ion me aheu is ic algo i hm o op imize he weigh s o a hyb id model. Thei expe imen al case s udy showed ha hei p oposed me hod pe o med be e han he h ee indi idual me hods and had highe accu acy. In sum, se e al gene al conclusions can be d awn om he li e a u e using hyb id models o explo e ime se ies o ecas ing. Fi s , in ecen yea s, he e has a g owing numbe o s udies in es iga ing he impac o using combina ion heo y on o ecas ing accu acy; hei objec i e is o enhance o ecas ing accu acy by combining di e en Table 1 Li e a u e on se ies linea o nonlinea models o ime se ies o ecas ing Au ho (s) Linea model Nonlinea model Yea Field Ghasemi e al. (2016) ARIMA SVM 2016 Elec ici y p ice and load o ecas ing Ba ow (2016) SMA MLP 2016 In aday call a i als o ecas ing Ka is and Daskalaki (2015) FARIMA MLP 2015 In e ne a ic o ecas ing Chaâbane (2014) FARIMA MLP 2014 Elec ici y p ice o ecas ing Adhika i and Ag awal (2013) RWM MLP 2013 Financial ime se ies o ecas ing ene gy Wang and Meng (2012) ARIMA MLP 2012 Consump ion o ecas ing Khashei e al. (2012) ARIMA PNNs 2012 Time se ies o ecas ing Nou ani e al. (2011) SARIMAX MLP 2011 Rain all– uno p ocess modeling Wu and Chan (2011) ARIMA Time delay neu al ne wo k (TDNN) 2011 Hou ly sola adia ion o ecas ing Aladag e al. (2009) ARIMA Elman’s ecu en neu al ne wo ks 2009 Time se ies o ecas ing Khashei and Haji ahimi Financial Inno a ion (2017) 3:24 Page 3 o 24 models. The nume ical esul s o he e iewed pape s e idence ha he p edic i e cap- abili y and accu acy o hyb id models a e be e han hose o single models. Mo eo e , hyb id models ha e ecen ly become a domina ed ool o ime se ies o ecas ing. Sec- ond, schola s ha e in oduced se ies and pa allel combina ion me hodologies o con- nec he componen s o hyb id models. Howe e , he ques ion o how o combine single models, ha is, which combina ion yields mo e accu a e esul s, emains un- answe ed. In o he wo ds, he li e a u e has neglec ed o compa e he wo ypes o hyb id me hods o in oduce a mo e accu a e one and ocused on imp o ing o ecas- ing accu acy by employing hyb id models a he han hei cons i uen s. Thi d, he li - e a u e e iew e ealed ha among he linea and nonlinea models, ARIMA and MLPs ha e a ac ed o e whelming a en ion and pe o m well when pa o hyb id models gi en hei unique ea u es. ARIMA models a e one o he mos impo an o ecas ing models ha ha e been success ully applied in modeling and o ecas ing. The popula i y o he ARIMA model can be a ibu ed o i s s a is ical p ope ies and he well-known Box–Jenkins (Box and Jenkins 1976) me hodology in he model- building p ocess. The model assumes a linea co ela ion be ween he alues o a ime se ies and hus, pe o ms well in linea modeling. MLP is he mos well-known a i icial neu al ne wo k ha p ocesses nonlinea pa e ns in da a wi hou any assump ion and does no equi e he de e mina ion o a model’s o m. MLPs a e lexible compu ing amewo ks and uni e sal app oxima o s wi h a high deg ee o accu acy and can be applied o a wide ange o o ecas ing p oblems. The key ad an age o he neu al ne wo ks is hei lexible nonlinea modeling. Gi en ha he li e a u e on ime se ies o ecas ing emains ambiguous on he choice o combina ion s a egy, he co e objec i e o his s udy is o in oduce an e ec i e com- bina ion me hodology and elucida e how indi idual models can be combined o imp o e inancial ime se ies o ecas ing. Acco dingly, his s udy p esen s a comp ehensi e discus- sion on se ies and pa allel combina ion me hods and hen, cons uc s a model using bo h echniques o combine MLP as a nonlinea model and ARIMA as a linea model. Then, using wo combina ion s a egies, ARIMA-MLP and MLP-ARIMA, he se ies and pa allel hyb id models, comp ising simple a e age (SA), linea eg ession (LR) and gene ic algo i hm (GA), a e compa ed wi h hei indi idual componen s. To e alua e he e ec- i eness o he hyb id models and in oduce a mo e accu a e and eliable hyb id me hod, wo benchma k da ase s, he closing o Shenzhen In eg a ed Index (SZII) and ha o S anda d and Poo ’s 500 (S&P 500), a e selec ed o he o ecas ing and modeling. The emainde o his pape is o ganized as ollows. Sec ion Me hods p esen s he basic concep s and modeling p ocedu es o he ARIMA and MLP modes o ime se ies o ecas ing. Sec ion Se ies combina ion me hod o ARIMA and MLP models and Pa allel combina ion o ARIMA and MLP desc ibe he se ies and pa allel combina ion echniques and he hyb id models cons uc ed using hese me hods. Sec ion Resul s and discussion epo s he empi ical esul s o he hyb id se ies and pa allel models o a o ecas ing benchma k da ase . Sec ion Compa ison o o ecas ing esul s compa es he pe o mance o he models o he o ecas ing benchma k da ase . Sec ion Conclusions concludes. Me hods This sec ion in oduces he basic concep s and modeling p ocedu es o he ARIMA and MLP models and se ies and he pa allel hyb id me hods o ime se ies o ecas ing. Khashei and Haji ahimi Financial Inno a ion (2017) 3:24 Page 4 o 24 ARIMA model ARIMA is one o he mos widely used app oaches o p edic he u u e alue o ime se ies by ex ac ing and modeling linea pa e ns in da a. The e o e, he classic model is sui able o linea pa e ns. In ARIMA models, he u u e alue o a a iable is assumed o be a linea unc ion o he pas alues and e o e ms. y ¼u þφ1y −1þ…… þφpy −p−ε −θ1ε −1−……−θqε −qð1Þ whe e (y ) is ac ual alue in ime and ε is whi e noise, which is assumed o be independen ly and iden ically dis ibu ed wi h a mean o ze o and cons an a iance o σ 2 .pand qa e he in ege numbe s o au o eg essi e and mo ing a e age e ms in he ARIMA model and φ i (i =1,2,...., p) and θ i (j =1,2,....,q) a e he model pa ame e s o be es ima ed. The modeling p ocedu e o he ARIMA models, which is based on he Box–Jenkins me hodology, comp ises h ee i e a i e s eps: model iden i ica ion, pa am- e e es ima ion, and diagnos ic checking. In he iden i ica ion s ep, da a ans o ma ion is o en equi ed o ende he ime se ies s a iona y, which is a necessa y condi ion when building an ARIMA model o o ecas ing. A s a iona y ime se ies is cha ac e - ized by a cons an mean and au oco ela ion s uc u e o e ime. When he obse ed ime se ies p esen s a end and he e oscedas ici y, di e encing and powe ans o m- a ion a e applied o he da a o emo e he end and s abilize he a iance be o e he ARIMA model can be i ed. Once a en a i e model is iden i ied, he es ima ion o he model pa ame e s is s aigh o wa d. The pa ame e s a e es ima ed such ha an o e all measu e o e o s is minimized, which can be accomplished using a nonlinea op imiza ion p ocedu e. The inal s ep is he diagnos ic checking o model adequacy, which de e mines i he model assump ions abou e o s a a e sa is ied. Se e al diagnos ic s a is ics and esidual plo s can be used o examine he goodness o i o a en a i ely adop ed model o he his o ical da a. I he model is deemed inadequa e, a new en a i e model is iden i ied, which is also subjec ed o pa ame e es ima ion and model e i ica ion. Diagnos ic in o ma ion can help de e mine al e na i e model(s). This h ee-s ep model-building p ocess is ypically epea ed se e al imes un il a sa is ac o y model is iden i ied. The inal model is hen used o he p edic ion. MLP model Compu a ional in elligence sys ems, mo e speci ically, ANNs, which in ac , a e a ee dynamics model, a e being widely used o he app oxima ion o unc ions and o ecas ing. In he case o eal-wo ld p oblems, neu al ne wo ks a e an e ec i e ool o ecognize nonlinea pa e ns. ANNs a e uni e sal app oxima o s ha app oxima e a la ge class o unc ions wi h a high deg ee o accu acy, which is a c ucial ad an age o e o he classes o nonlinea models (Zhang e al. 1998). Thei powe is de i ed om he pa allel p ocessing o in o ma ion om he da a and no p io assump ion is e- qui ed in he model-building p ocess. Ins ead, he ne wo k model is la gely de e mined by he da a cha ac e is ics. MLPs o single hidden laye eed- o wa d neu al ne wo ks a e key and commonly used model o ms o ANNs o ime se ies modeling and o ecas ing. The model is cha ac e ized by a ne wo k o h ee laye s o simple p ocess- ing uni s connec ed by acyclic links (Fig. 1). The ela ionship be ween he ou pu (y ) Khashei and Haji ahimi Financial Inno a ion (2017) 3:24 Page 5 o 24 and (y −1 ,…,y −p ) inpu s has he ollowing ma hema ical ep esen a ion (Khashei and Bija i 2010): y ¼w0þX q j¼1 wj⋅gw 0;jþX p i¼1 wi;j⋅y −i ! þε ;ð2Þ whe e w i, j (i=0,1,2,..., p, j =1,2,...,q) and w j (j=0,1,2,...,q) a e model pa ame e s o en e med connec ion weigh s, pis he numbe o inpu nodes, and qis he numbe o hidden nodes. The ac i a ion unc ions ake se e al o ms. The ype o ac i a ion unc ion is indica ed by he condi ion o he neu on wi hin he ne wo k. In a majo i y o cases, inpu laye neu ons do no ha e an ac i a ion unc ion because hei ole is o ans e inpu s o he hidden laye . The mos widely used ac i a ion unc ion o he ou pu laye is he linea unc ion because a non-linea one may dis o he p edica ed ou pu . The logis ic unc ion is o en used as a hidden laye ans e unc ion, as shown in Eq. (3). O he ac i a ion unc ions can also be used such as linea and quad a ic unc ions, each wi h a a ie y o modeling applica ions. Sig xðÞ¼ 1 1þexp −xðÞ :ð3Þ Fig. 1 MLP neu al ne wo k a chi ec u e Khashei and Haji ahimi Financial Inno a ion (2017) 3:24 Page 6 o 24 Thus, he ANN model in Eq. (2) pe o ms a nonlinea unc ional mapping om he pas obse a ions o he u u e alue y, ha is, y ¼ ðy −1;…;y −p;wÞþε ;ð4Þ whe e wis a ec o o all pa ame e s and (.) is a unc ion de e mined by he ne - wo k s uc u e and connec ion weigh s. Thus, he MLP is equi alen o a nonlinea au o eg essi e model. The simple ne wo k in Eq. (2) is unexpec edly powe ul, ha is, i can app oxima e he a bi a y unc ion as a numbe o hidden nodes when q is su icien ly la ge. In p ac ice, a simple ne wo k s uc u e wi h a small numbe o hidden nodes o en wo ks well in ou -o -sample o ecas ing, possibly because o he o e - i ing e ec ypically ound in he MLP modeling p ocess. An o e - i ed model has a good i o he sample used o model building bu poo gene alizabili y o ou -o -sample da a. The choice o qis da a dependen and he e is no sys ema ic ule in deciding his pa ame e . In addi ion o choosing an app op ia e numbe o hidden nodes, selec ing he numbe o lagged obse a ions, p, and dimensions o he inpu ec o is an impo an ask in he ANN modeling o a ime se ies. This is, pe haps, he mos impo an pa ame e o be es ima ed in an ANN model because i plays a majo ole in de e mining he (nonlinea ) au oco ela ion s uc u e o he ime se ies. Se ies combina ion me hod o ARIMA and MLP models In he se ies linea o nonlinea combina ion models, a ime se ies is di ided in o a linea and nonlinea pa , as ollows: y ¼X n ¼1 L þX n ¼1 N ð5Þ whe e L and N deno e he linea and nonlinea pa s es ima ed om he da a. Then, hese wo componen s a e sequen ially p ocessed by ARIMA and MLP models. Thus, in he i s s age o his me hod, he ARIMA o MLP model is selec ed o iden i y linea o nonlinea pa e ns in he o iginal da a. Then, o disco e he emaining pa e ns ha a e no cap u ed by he i s model, he ou pu ob ained in he i s s age is used as an inpu o he second model. The basic concep is ha one model is insu icien o cap u e all ela ionships in he da a. Mo eo e , ully iden i ying and modeling he da a cha ac e is ics in he eal ime se ies is di icul and some imes, e en impossible. Thus, using an indi idual model such as he ARIMA (MLP) model, undoub edly, e eals nonlinea (linea ) pa e ns ha a e no comple ely ecognized. Consequen ly, he MLP (ARIMA) model is employed in he second s age o cap u e he emaining nonlinea (linea ) pa e ns. A summa ion o he ou pu s ob ained om he wo s ages is conside ed he inal combined o ecas . On he basis o he sequence o model selec ion, wo hyb id models (ARIMA-MLP and MLP-ARIMA) a e p esen ed in he nex sec ion. Khashei and Haji ahimi Financial Inno a ion (2017) 3:24 Page 7 o 24 ARIMA-MLP model In line wi h he se ies modeling p ocedu e, in he i s s age o he ARIMA-MLP model, he ARIMA is applied o model he linea componen . Le e deno e he esidual o he ARIMA model a ime : e ¼y − ^ L ð6Þ whe e ^ L is he o ecas ing alue o ime om he ARIMA model based on o iginal da a. In he second s age, he esiduals o he i s s age a e used as inpu da a o he MLP model, allowing o he iden i ica ion o nonlinea ela ionships. Wi h ninpu nodes, he MLP model o he esiduals will be. e ¼ ðe −1;e −2;…;e −nÞþε ⇒c N′ ¼^ e ¼ ðe −1;e −2;…;e −nÞð7Þ whe e is a nonlinea unc ion de e mined by he MLP, N′ is he o ecas ing alue o ime in he MLP model based on esidual da a, and e is he andom e o . The ame- wo k o he ARIMA-MLP model is displayed in Fig. 2a. No e ha i model is inapp op ia e, he e o e m is no necessa ily andom; he e o e, he co ec iden i ica ion is c i ical. In his way, he combined o ecas will be as ollows: ^ y ¼ ^ L þc N ′ ð8Þ MLP-ARIMA model Simila o he ARIMA-MLP model, he MLP-ARIMA model has wo main s ages. In he i s s age, he MLP model is used o model he nonlinea pa o he ime se ies. Le e′ deno e he esidual o he MLP model a ime . Then, (a) (b) Fig. 2 F amewo k o (a) ARIMA-MLP and (b) MLP-ARIMA model Khashei and Haji ahimi Financial Inno a ion (2017) 3:24 Page 8 o 24 Fig. 8 Es ima ed alues o GA-based hyb id model o SZII Fig. 7 Es ima ed alues o SA-based hyb id model o SZII Khashei and Haji ahimi Financial Inno a ion (2017) 3:24 Page 15 o 24 S anda d and Poo ’s 500 da ase S anda d and Poo ’s 500 (S&P 500) da ase includes 2349 daily closing s ock p ices om Oc obe 1998 o Feb ua y 2008 (Zhang and Wu 2009). The S&P 500 da ase is plo ed in Fig. 10. Acco ding o p e ious s udies, he S&P da ase is di ided in o aining and es da ase s. The i s obse a ions (abou 80% o he sample) a e used as he aining sample o o mula e he models and he las 470 obse a ions a e applied as a es sample o e alua e he pe o mance o he cons uc ed models. ARIMA-MLP se ies hyb id model S age I - (Linea modeling): Simila o he linea modeling phase, ARIMA(1,0,0) is designed and he esiduals o his s ep a e used in he nex s ep. Table 2 Pe o mance o models o SZII using ain and es da ase s Model T ain Tes MAE MAE MAPE RMSE MAE MSE MAPE RMSE ARIMA-MLP 212.39 91,955 7.22% 303.24 1082.88 1,915,716 9.52% 1384.09 MLP-ARIMA 210.91 86,221 7.23% 293.63 1064.91 1,915,422 10.16% 1383.98 SAHM 217.78 95,399 7.12% 308.86 1123.87 1,997,323 9.91% 1388.69 GAHM 215.38 94,577 7.08% 307.53 1102.75 1,969,593 9.79% 1403.42 LRHM 215.54 94,573 7.09% 307.52 1074.87 1,928,479 9.64% 1388.69 ARIMA 224.45 99,255 7.30% 315.04 1166.17 2,221,776 10.26% 1490.56 MLP 215.38 94,577 7.30% 307.53 1102.33 1,974,479 9.80% 1405.16 Fig. 9 Es ima ed alues o LR-based hyb id model o SZII Khashei and Haji ahimi Financial Inno a ion (2017) 3:24 Page 16 o 24 S age II - (Nonlinea modeling): In his s age, he esiduals o he p e ious s ep a e used as inpu o he MLP model and a ne wo k wi h h ee inpu , i e hidden, and one ou pu neu on is i ed o ex ac he emaining nonlinea s uc u es. S age III - (Combina ion): He e, he o ecas ed alues o p e ious wo s ages a e combined o gene a e he inal combined o ecas . The es ima ed alues o he ARIMA-MLP model agains he ac ual alues o all da a a e plo ed in Fig. 11. MLP-ARIMA se ies hyb id model S age I - (Nonlinea modeling): In he nonlinea modeling phase, a ne wo k wi h h ee inpu , h ee hidden, and one ou pu neu ons is designed o cap u e he nonlinea ela ionships in he ime se ies gene a ed and he gene a ed esiduals a e used in he nex s ep. S age II: (Linea modeling): In his s ep, an ARIMA (3, 0, 3) model is i o p ocess he linea s uc u es ha a e no modeled by he MLP model. S age III: (Combina ion): In he inal s ep, he o ecas ed alues om s ages I and II a e combined. The es ima ed alues o he MLP-ARIMA model agains he ac ual alues o all da a a e plo ed in Fig. 12. Pa allel hyb id models S age I - (Linea and nonlinea modeling): Simila o he p e ious sec ion, o cap u e he linea and nonlinea pa e ns in he da a o he S&P ime se ies, he ARIMA (1, 0, 0) and MLP models wi h h ee inpu , h ee hidden, and one ou pu neu on a e designed. Fig. 10 Daily S&P 500 s ock closing p ices, Oc obe 1998–Feb ua y 2008 Khashei and Haji ahimi Financial Inno a ion (2017) 3:24 Page 17 o 24 Fig. 12 Es ima ed alues o MLP-ARIMA model o S&P500 Fig. 11 Es ima ed alues o ARIMA-MLP model o S&P 500 Khashei and Haji ahimi Financial Inno a ion (2017) 3:24 Page 18 o 24 Fig. 13 Es ima ed alues o SA-based hyb id model o S&P 500 Fig. 14 Es ima ed alues o GA-based hyb id model o S&P 500 Khashei and Haji ahimi Financial Inno a ion (2017) 3:24 Page 19 o 24 S age II - (Ini ializing weigh s): In his s a e, he op imum weigh s a e de i ed by applying he LR, GA, and SA weigh ing me hods. No e ha he OLS app oach and GA a e designed using E iwes and MATLAB so wa e. S age III - (Combina ion): Acco ding o he modeling p ocedu e o he pa allel hyb id models, he combined o ecas is made using he alues ob ained om he p e ious wo s ages. The es ima ed alues o he hyb id models based on pa allel SA, GA, and LR agains he ac ual alues o all da a a e plo ed in Figs. 13,14,15, espec i ely. Table 3 summa izes he pe o mance o he se ies and pa allel hyb id models in p edic ing he S&P 500 s ock p ice using ain and es da ase s. The able shows ha he se ies models, ARIMA-MLP and ARIMA-MLP, a e no compa able wi h each o he and epo s signi ican ly high p edic ion pe o mance when he se ies hyb id me hodology is used ins ead o he pa allel me hods and hei base models. Fig. 15 Es ima ed alues o LR-based hyb id model o S&P 500 Table 3 Pe o mance o models o S&P 500 using ain and es da ase s Model T ain Tes MAE MSE MAPE RMSE MAE MSE MAPE RMSE ARIMA-MLP 9.94 176.76 0.01% 13.30 9.02 159.33 0.64% 12.62 MLP-ARIMA 9.96 180.17 0.01% 13.42 9.20 157.69 0.65% 12.56 SAHM 9.98 180.69 0.01% 13.44 9.22 164.48 0.65% 12.83 GAHM 9.97 180.31 0.65% 13.42 9.23 159.33 0.85 12.62 LSHM 9.97 180.28 0.01% 13.43 8.98 152.55 0.63% 12.35 ARIMA 10.02 182.21 086% 13.50 9.36 173.85 0.66% 13.19 MLP 9.97 180.31 0.01% 13.43 9.24 159.34 0.65% 12.62 Khashei and Haji ahimi Financial Inno a ion (2017) 3:24 Page 20 o 24 Compa ison o o ecas ing esul s This sec ion compa es he p edic i e capabili ies o he hyb id models cons uc ed by applying he se ies and pa allel combina ion me hods wi h ei he o hei componen s, MLP, and ARIMA, using he wo abo emen ioned da ase s. The compa a i e analysis is conduc ed om wo iewpoin s: compa ison o se ies and pa allel hyb id models and analysis o a e age pe cen age imp o emen in he se ies and pa allel hyb id models in compa ison wi h hei componen s. Two pe o mance indica o s, MAE and MSE, a e employed o compa e he o ecas ing pe o mance o he hyb id models and hei componen s. In he i s s ep, he o e all pe o mance o he se ies and pa allel hyb id models is compa ed. Tables 4 and 5 p esen he o e all pe o mance o he hyb id models and hei componen s o he SZII and S&P 500 da ase s. The compa ison e eals ha he a e age o ecas ing e o o MAE and MSE in he ain and es da ase s is lowe in he ARIMA-MLP and MLP-ARIMA hyb id models cons uc ed using he se ies combina ion echnique han hose in he pa allel hyb id models. Fo example, in he SZII da ase , in MAE and MSE e ms, he o ecas ing esul s o he se ies models using es da ase imp o ed by 2.41% and 2.51% compa ed o hose o he pa allel hyb id models. Tables 4 and 5 compa e he pe o mance and accu acy o he se ies and pa allel hyb id me hods. In he second s ep, he pe o mance o he hyb id models is compa ed wi h hose o hei base models. In o he wo ds, he a e age pe cen age imp o emen o he se ies and pa allel hyb id models is compa ed wi h ha o hei base models. The esul s show ha applying all i e hyb id models, on a e age, im- p o es he o ecas ing accu acy o e a leas one model o he ARIMA and MLP neu al ne wo k models. This con i ms he hypo hesis ha indi idual models do no cap u e all ela ionships in he da a and combining he wo models can be e ec i e in o e coming hei limi a ions and imp o ing o ecas ing accu acy. Tables 6 and 7 p esen he a e age imp o emen pe cen age o he se ies and pa allel hyb id models o he SZII and S&P 500 da ase s compa ed o he ARIMA and MLP Table 4 O e all pe o mance o se ies and pa allel models o SZII Model T ain Tes MAE MSE MAE MSE Se ies models 211.65 89,088.05 1073.89 1,915,569.00 Pa allel models 216.23 94,849.02 1100.49 1,965,131.66 ARIMA 224.45 99,255.51 1166.17 2,221,776.70 MLP 215.38 94,577.00 1102.33 1,974,479.94 Lowe o ecas ing e o a e in bold Table 5 O e all pe o mance o se ies and pa allel models o S&P 500 Model T ain Tes MAE MSE MAE MSE Se ies models 9.95 178.46 9.11 158.51 Pa allel models 9.97 180.42 9.14 158.78 ARIMA 10.02 182.21 9.36 173.85 MLP 9.97 180.31 9.24 159.34 Lowe o ecas ing e o a e in bold Khashei and Haji ahimi Financial Inno a ion (2017) 3:24 Page 21 o 24 models. The esul s sugges ha , on a e age, he se ies hyb id models ha e a highe imp o ing impac on he ARIMA and MLP models han he pa allel hyb id models. Fo example, when using he S&P 500 da ase , he se ies hyb id models imp o e he MLP models in e ms o MSE by 1.02% in he case o ain da a, while his imp o e- men is −0.21 o he pa allel models. Acco ding o he analy ical esul s, he accu acies o he se ies hyb id models a e be e han hose o he pa allel hyb id models in bo h o e all pe o mance and a e age imp o emen pe cen age o e he base models. F om he abo e compa a i e analyses, he models can be anked as ollows: (i) se ies hyb id models (ii) pa allel hyb id models, and (ii) indi idual models. Conclusions Fo ecas ing eal-wo ld ime se ies, pa icula ly inancial ime se ies, is a c i ical ask ha has ecen ly ecei ed o e whelming a en ion. Gi en he impo ance o accu a e o ecas ing, se e al ela ed me hods ha e been p oposed in he li e a u e. In addi ion o single me hods, s udies ha e combined di e en me hods o gene a e mo e accu a e e- sul s and con i med ha combining di e en models enhances o ecas ing accu acy and accoun s o he unique ea u es o indi idual models. Al hough nume ous s udies ha e used se ies o pa allel me hods o cons uc hyb id models and con i m ha com- bining di e en models educes o ecas ing e o and o e mo e accu a e esul s, hey emain ague on he p ecise combina ion ha p oduces a mo e accu a e hyb id model. Thus, his s udy p oposed a mo e e icien echnique o o ecas inancial ime se ies and hen, conduc ed a comp ehensi e compa ison o he p edic i e capabili ies o he se ies and pa allel combina ion echniques ha we e combined wi h linea and nonlin- ea models, such as ARIMA and MLP, along wi h hei indi idual componen s. Fi s , he se ies and pa allel hyb id models we e compa ed, ollowed by a compa ison o he hyb id models’a e age pe cen age imp o emen wi h hose o hei base models. The empi ical esul s o he wo benchma k da ase s, SZII and S&P 500, indica ed ha all hyb id models cons uc ed using he wo combina ion me hods gene a ed supe io esul s han a leas one o hei indi idual componen s. The esul s also show ha he se ies me hod gene a e mo e accu a e hyb id models and has a highe imp o emen Table 6 A e age imp o emen in se ies and pa allel models o e ARIMA and MLP models o SZII Hyb id model ARIMA MLP T ain (%) Tes (%) T ain (%) Tes (%) MAE MSE MAE MSE MAE MSE MAE MSE Se ies models 5.70 10.24 7.91 13.78 1.72 5.80 2.57 2.98 Pa allel models 3.65 4.43 5.64 11.55 −1.11 −0.40 0.27 0.47 Lowe o ecas ing e o a e in bold Table 7 A e age imp o emen in se ies and pa allel models o e ARIMA and MLP models o S&P 500 Hyb id model ARIMA MLP T ain (%) Tes (%) T ain (%) Tes (%) MAE MSE MAE MSE MAE MSE MAE MSE Se ies models 0.69 2.05 2.66 8.82 0.3 1.02 1.40 1.25 Pa allel models 0.49 0.63 1.45 6.37 0.00 −0.21 1.04 0.52 Lowe o ecas ing e o a e in bold Khashei and Haji ahimi Financial Inno a ion (2017) 3:24 Page 22 o 24 pe cen age han he pa allel me hod. The e o e, he se ies combina ion me hod can be conside ed an e icien al e na i e o cons uc mo e accu a e hyb id models in bo h analy ical app oaches o o ecas ing inancial ime se ies. Fu u e wo ks should conside implemen ing he se ies and pa allel hyb id me hodologies o de elop an app oach wi h h ee o mo e indi idual models and acco dingly, compa e and analyze he ob ained esul s. Resea che s can also examine o he s a is ical and in elligen models, such as GARCH and SVM models, o cons uc se ies and pa allel hyb id models o o ecas inancial ime se ies. Acknowledgemen s The au ho s exp ess hei g a i ude o D . Fa imah Mokha ab Ra iei, associa e p o esso o indus ial enginee ing a he Ta bia Moda es Uni e si y o Teh an, and D . Mehdi Bija i, p o esso o indus ial enginee ing a Is ahan Uni e si y o Technology, o hei insigh ul and cons uc i e commen s, which ha e helped conside ably imp o e his pape . Funding The au ho s ha e no unding o epo . A ailabili y o da a and ma e ials No applicable. 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