The Optimal Combination Forecasting Based on ARIMA,VAR and SSM

Transcription

1 Advaces Compuer, Sgals ad Sysems (206) : 3-7 Clausus Scefc Press, Caada The Opmal Combao Forecasg Based o ARIMA,VAR ad SSM Bebe Che,a, Mgya Jag,b* School of Iformao Scece ad Egeerg, Shadog Uversy, Ja, Cha, 2500 a sducbb@63.com, *bcorrespodg auhor: jagmgya@sdu.edu.c Keywords: he opmal combao forecasg model; Arfcal Bee Coloy algorhm; he Maufacurers Shpmes; ARIMA; VAR; SSM Absrac: I order o overcome he defecs of a sgle me seres forecasg model ad o mprove he predco accuracy, hs paper proposes a mproved opmal combao model wh Arfcal Bee Coloy algorhm used o solve he opmal wegh coeffce auomacally. Tag he Maufacurers' Shpmes as a example o aalyze, we use ARIMA VAR ad SSM o forecas he shpmes respecvely. Based o hese hree models, we cosruc he opmal combao forecasg model. By speco, s superor o he oher hree models accuracy.. Iroduco Oe of he purpose of me seres aalyss s o forecas, whch meas predcg he fuure values of a seres based o he hsory of ha seres ad, possbly, oher relaed seres of facors [].I shows ha o sgle mehod ca be appled o he predco of all me seres. I order o overcome he lmao of he sgle forecasg model, people combe varous forecasg models ad use he comprehesve formao gve by hem proper maer o ge he fal forecasg resul. The maufacurg dusry reflecs a coury's producvy level, playg a sgfca role he ecoomc developme of a coury. More accurae forecasg of he Maufacurers' Shpmes s very mpora. I hs paper, hree me seres models of ARIMA VAR ad SSM were used o forecas respecvely frs. The based o hese hree models, we auomacally ge he opmal weghs va Arfcal Bee Coloy algorhm so as o cosruc he opmal combao model o forecas. 2. Revew of relaed mehodologes 2. ARIMA model The sgle egraed auoregressve movg average (ARIMA) model whch s also ow as Box-Jes model because of s smplcy, feasbly ad flexbly. If he rasferred seres s saoary, we ca use ARIMA model o forecas. For modelg of seasoal me seres besde o-seasoal seres, ARIMA( p, d, q)( P, D, Q) s ow as mulplcave ARIMA model s defed as follows: ( B 2 B 2 p B p )( Φ B s Φ2 B 2 s ΦP B Ps )( B) d ( B s ) D X ( B 2 B 2 q B q )( Θ B s Θ2 B 2 s ΘQ B Qs ) 3 ()

2 where s he radom varable, s s he perodc erm, B s he dfferece operaor as s D d B ( X ) X, ( B ) s he Dh seasoal dfferece measure s, ( B) s he dh o-seasoal dfferece, p ad q s he order of o-seasoal auoregressve model ad movg average model respecvely, P ad Q s he order of seasoal auoregressve model ad movg average model respecvely, ad s he parameer of o-seasoal auoregressve model ad movg average model respecvely, Φ ad Θ s he parameer of seasoal auoregressve model ad movg average model respecvely [2]. 2.2 VAR model The vecor auoregressve (VAR) model s o based o ecoomc heory, whose basc dea s ha each equaos of he model, he edogeous varables regress her lagged values so as o esmae he log-erm dyamc relaoshp bewee varables [3] as follows: X A X Ap X p, (2) where X ad varable, p s he lagged ra, 2.3 SSM s he edogeous ad exogeous varable vecor respecvely, A,, Ap ad are he coeffce marx o be esmaed. s he radom Sae space model (SSM) s ofe used o esmae he me varable ha ca o be observed ecoomercs. I esablshes he relaoshp bewee he observable varables ad he eral sysem, ad ca acheve he purpose of aalyss ad predco by esmag he dffere sae vecors. The geeral lear ormal sae space model s defed as follows: F K, (3) Y H, (4) where s he varable ha ca o be observed, Y s he observed value, ad are he radom varables, ad he hree sysem marx of F, K ad H deerme he srucure of he model. Eq. (3) s he sae equao, descrbg he sae space evoluo of a sochasc dyamcal sysem. Eq. (4) s he observao equao, showg ha he m-dmesoal measureme Y s subjec o a lear rasformao of he hdde sae ad s furher corruped by a measureme ose process. 3. Combao Forecasg Model The core ssue of combg forecasg s how o dsrbue he wegh of each sgle predco model, order o mprove he predco accuracy effecvely. Commo mehods are arhmecal average mehod, varace recprocal mehod [4], mea square recprocal mehod [5], sadard devao mehod, ec. 3. The Opmal Combao Forecasg model Ths paper uses he opmal weghed mehod o deerme he wegh, whose basc dea s o cosruc he objecve fuco of predco error accordg o cera rules, ad o deerme he opmal wegh he combao forecasg model by solvg he opmal value of he objecve 4

3 fuco uder cera cosras. I a cera predco problem, we se x s he observao objec ad here are ds of mehods for forecasg : x, x2,, x. The combao forecasg model ca be represeed as follows: ) ( ) x ( ) (,2,, ) (5) where x () ad x () s he predced value of he sgle model ad he combao forecasg model mome of respecvely, () s he wegh of he sgle mome of ad mees ( ), ( ) 0 (,2,, ) (6) O he prcple of mmzg he absolue value of he combed predco error of he sample pos, he mahemacal expresso of opmal combao forecasg model s as follows: m J e s.. ) xˆ( ) ( ), ( ) 0 ( ) e ( ) where e () ad e () s he predced error of he sgle forecasg model ad he combao forecasg model mome of respecvely, x ˆ( ) s he acual value mome of. 3.2 Solvg opmal weghs by ABC algorhm The soluo o he model above s a cosraed opmzao problem, he Arfcal Bee Coloy (ABC) algorhm s used o solve he opmal problem o accous of s hgh parallelsm, radomess, self-adapably ad ease of mplemeao. The ma seps of ABC algorhm are gve below [6]: Sed he scous oo he al food sources REPEAT Sed he employed bees oo he food sources ad deerme her ecar amous Calculae he probably value of he sources wh whch hey are preferred by he olooer bees Sed he olooer bees oo he food sources ad deerme her ecar amous Sop he exploao process of he sources exhaused by he bees Sed he scous o he search area for dscoverg ew food sources,radomly Memorze he bes food source foud so far UNTIL(requremes are me) 4. Smulao I order o verfy predco performace of he combao forecasg model, we comm smulao wh he model. The raw daa, whch comes from he U.S. Deparme of Commerce Web se, are used he smulao. The daa clude he Maufacurers' Shpmes, New Orders ad Toal Iveory from February 992 o Jue 205. The daa se s dvded o rag se ad es se, whch he rag se daa s from February 992 o March 205,whle he es se daa s from Aprl 205 o Jue 205. By predcg he es se ad calculag he predco error of each sgle (7) 5

4 model, we use he ABC algorhm o solve he opmal wegh accordg o Eq. (7) so as o esablsh he combao forecasg model o he prcple of mmzg he absolue value of he combed predco error of he sample pos. The opmal combao wegh obaed by usg ABC algorhm s ( ).To es predco effec, we roduce he followg error dcaors. The mea absolue perce error (MAPE), MAPE e ) xˆ( ) xˆ( ) xˆ( ) ad he roo mea square error (RMSE), RMSE [ ) xˆ( )] 2 where x () ad x ˆ( ) s predced value ad acual value mome of respecvely, s sample umber. The comparso of forecasg resuls by dffere models s showed he Tab.. From he Tab., we ca see ha SSM acheved he hghes precso amog he hree sgle model, bu he opmal combao forecasg model ca furher rase he accuracy of smulao,.e. he RMSE has decled by 449.3, ad he MAPE has decled by 0.53 perce, whch meas he predco accuracy s mproved by 90 perce. So he proposed model has successfully realzed he muual suppleme wh each oher erms of advaages of ARIMA VAR ad SSM. 5. Coclusos Table Comparso of forecasg resuls by he four models error aalyss ARIMA VAR SSM he opmal combao forecasg model MAPE 0.37% 0.2% 0.7% 0.07% RMSE Ths paper preses a mproved opmal combao forecasg model based o ARIMA, VAR ad SSM. Whe deermg he weghs of he model, he Arfcal Bee Coloy algorhm s used o solve he opmal wegh coeffce auomacally. Tag he Maufacurers' Shpmes as a example o carry o he emprcal aalyss, we use ARIMA VAR ad SSM o forecas he shpmes respecvely. Based o hese hree models, we cosruc he opmal combao model o forecas. By speco, he combao model esablshed hs paper s superor o he oher hree sgle models. I ca ge beer resuls MAPE ad RMSE, mprovg he predco accuracy ad sably of he model. So he proposed model s effecve ad has cera praccal value. Acowledgemes Ths wor was suppored by he Shadog Naural Scece Foudao (ZR204FM039). Refereces [] Clare, Hogyu Pa, e al. Aalyss ad Applcao of Tme Seres: R laguage [M]. Bejg: Mechacal Idusry Press, 20. (8) (9) 6

5 [2] Karamouz M., Araghejad sh. Advace Hydrology [M]. Amrabr Uversy of Techology Press, 202. [3] Shyu L, Fe Zhag, Zhegl Wag. Daa Aalyss: real R laguage [M]. Bejg: Elecroc Idusry Press, 204. [4] Chegfag Fa, Jam Sh. Aalyss of Gra Produco Cos Forecas based o Hol-Wers ad Tred ARMA Combed Model Tag Cor ad Whea of Shadog Provce for Example [J]. Chese Joural of Agrculural Resources ad Regoal Plag, vol.3, o.3, pp.45-5, 204. [5] L Zheg, Fegb Lu, Dogme Dua, ec. Iegrao forecas of Chese por cosumpo demad Emprcal based o ARIMA VAR ad VEC models [J]. Sysems Egeerg Theory & Pracce, vol.4, o.4, pp , 203. [6] Mgya Jag, Dogfeg Yua. Arfcal Bee Coloy algorhm ad s applcaos [M]. Bejg: Scece press,