A Hybrid Method to Improve Forecasting Accuracy Utilizing Genetic Algorithm and Its Application to Stock Market Price Data

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1 A Hyrd Mhod o Improv Forcsng Accurcy Ulzng Gnc Algorhm nd Is Applcon o Sock Mrk Prc D Ysuo Ish * Kzuhro Tkysu Dprmn of Mngmn Dsgn, Fculy of Busnss dmnsron Osk Inrnonl Unvrsy -5-, Sug, Hrk, Osk 57-9, Jpn Collg of Busnss Admnsron, Tokoh Unvrsy, 5,Oouch, Fu, Shzuok 47-8, Jpn Asrc. Focusng h h uon of h ponnl smoohng mhod (ESM) s uvln o (,) ordr ARMA modl uon, nw mhod of smon of h smoohng consn n h ponnl smoohng mhod ws proposd for y us whch ssfd h mnmum vrnc of forcsng rror. In hs ppr, w ulz h ov sd horcl soluon. Frsly, w smd h ARMA modl prmr nd hn sm h smoohng consns. Thus h horcl soluon s drvd n smpl wy nd my ulzd n vrous flds. Ths nw mhod shows h s usful for h m srs h hs vrous rnd chrcrscs. Th ffcvnss of hs mhod should mnd n vrous css. Kywords: mnmum vrnc, ponnl smoohng mhod, forcsng, rnd, gnc lgorhm. INTRODUCTION In hs ppr, ulzng ov sd mhod, rvsd forcsng mhod s proposd. In mkng forcs such s sock mrk prc d, rnd rmovng mhod s dvs d. Trnd rmovng y h comnon of lnr nd nd ordr non-lnr funcon nd rd ordr non-lnr funcon s cud o h sock mrk prc d of consumr lcroncs ndusry. Gnc Algorhm s ulzd o srch opml wghs for h wghng prmrs of lnr nd non-lnr funcon. For h comprson, monhly rnd s rmovd fr h. Thorcl soluon of smoohng consn of ESM s clculd for oh of h monhly rnd rmovng d nd h non monhly rnd rmovng d. Thn forcsng s cud on hs d. Ths s rvsd forcsng mhod. Vrnc of forcsng rror of hs nwly proposd mhod s ssumd o lss hn hos of prvously proposd mhod. Th rs of h ppr s orgnzd s follows. In scon, ESM s sd y ARMA modl nd smon mhod of smoohng consn s drvd usng ARMA modl dnfcon. Th comnon of lnr nd non-lnr funcon s nroducd for rnd rmovng n scon.th Monhly Ro s rfrrd n scon 4. Msurng mhod of forcsng ccurcy s hd n 5. GA modl o srch opml wghs for h wghng prmrs of lnr nd nonlnr funcon s nroducd nd forcsng s cud n scon 6, nd smon ccurcy s mnd.. DESCRIPTION OF ESM USING ARMA MODE [] In ESM, forcsng m + s sd n h followng uon. Hr, ˆ ˆ : forcsng : rlzd vlu : smoohng consn () s r-sd s: ˆ l l l () () () By h wy, w consdr h followng (,) ordr ARMA modl. Gnrlly, p p, ordr ARMA modl s sd s: (4) (5) Hr, : Smpl procss of Sonry Ergodc Gussn Procss,,, N, :Gussn Wh Nos wh mn vrnc MA procss n (5) s supposd o ssfy convrly condon. Ulzng h rlon h: E,, 5

2 w g h followng uon from (4). ˆ Oprng hs schm on +, w fnlly g: (6) (7) If w s, h ov uon s h sm wh (),.., uon of ESM s uvln o (,) ordr ARMA modl. Comprng (4) wh (5) nd usng () nd (7), w g: (8) From ov, w cn g smon of smoohng consn fr w dnfy h prmr of MA pr of ARMA modl. Bu, gnrlly MA pr of ARMA modl com non-lnr uons whch r dscrd low. (5) : p (9) () W prss h uocorrlon funcon of s r k nd from (9), (), w g h followng nonlnr uons whch r wll known [] r k r k k ( k ) ( k ) () For hs uons, rcursv lgorhm hs n dvlop d. In hs ppr, prmr o smd s only, so cn solvd n h followng wy. From (4) (5) (8) (), w g: If w s: r r () k r k h followng uon s drvd. r W cn g s follows. In ordr o hv rl roos, As s whn h rng of Fnlly w g 4 4 mus ssfy 4 () (4) (5) (6) (7) whch ssfy ov condon. Thus w cn on hor cl soluon y smpl wy.. TREND REMOVA METHOD As ESM s on of lnr modl, forcsng ccurcy for h m srs wh non-lnr rnd s no ncssrly good. How o rmov rnd for h m srs wh nonlnr rnd s g ssu n mprovng forcsng ccurcy. In hs ppr, w dvs o rmov hs non-lnr rnd y ulzng non-lnr funcon. As rnd rmovl mhod, w dscr lnr nd nonlnr funcon, nd h comnon of hs. [] nr funcon W s: y (8) s lnr funcon, whr s vrl, for mpl, m nd y s vrl, for mpl, sock mrk prc, nd r prmrs whch r smd y usng ls sur mhod. [] Non-lnr funcon W s: y c (9) y c d () s nd nd rd ordr non-lnr funcon. (,, c ) nd,, c, ) r lso prmrs for nd nd rd ( d 6

3 ordr non-lnr funcons whch r smd y usng ls sur mhod. [] Th comnon of lnr nd non-lnr funcon W s: y c c d (),, ( ) s h comnon lnr nd nd ordr non-lnr nd rd ordr non-lnr funcon. Trnd s rmovd y d vdng h orgnl d y (). Th opml wgh ng prmr α, α, α r drmnd y ulzng GA. GA mhod s prcsly dscrd n MONTHY RATIO For mpl, f hr s h monhly d of yrs s sd llow:,,,, whr R n whch mns monh nd mns yr nd s shppng d of -h yr, -h monh. Thn, monhly ro,, s clculd s follows. ( ) Monhly rnd s rmovd y dvdng h d y (). Numrcl mpls oh of monhly rnd rmovl cs nd non-rmovl cs r dscussd n FORECASTING ACCURACY Forcsng ccurcy s msurd y clculng h vrnc of h forcsng rror. Vrnc of forcsng rror s clculd y: N Whr, forcsng rror s prssd s: N N N (4) (5) (6) 6. NUMERICA EXAMPE 6. Applcon o sock mrk prc d Followng fv ypcl socks of consumr lcroncs ndusry r slcd. Shrp Corporon: "SHARP" Pnsonc Corporon: "Pnsonc" Sony Corporon: "SONY" Hch, d.:"hitachi" TOSHIBA CORPORATION:"TOSHIBA" Th ov mnond 5 compns for css from Aprl o Mrch 4 r nlyzd. Furhrmor, GA rsuls r comprd wh h clculon rsuls of ll consdrl css n ordr o confrm h ffcvnss of GA pproch. Frs of ll, grphcl chrs of hs m srs d r hd n Fgur , 8 6 4,5,,5, 5,5, 5 SHARP Fgur 6-: D of SHARP SONY Orgnl D Orgnl D Fgur 6-: D of SONY Pnsonc Orgnl D Fgur 6-: D of Pnsonc 7

4 , HITACHI Orgnl D Fgur 6-4: D of HITACHI TOSHIBA Orgnl D SONY 5,47 6. Pnsonc,468.8 HITACHI, TOSHIBA,79 8. Th cs monhly ro s no usd s smllr hn h cs monhly ro s usd concrnng h vrnc of forcsng rror n 8% compns. I my cus sock mrk prc dos no hv dfn ssonl rnd n gnrl. Th mnmum vrnc of forcsng rror of GA concds wh hos of h clculon of ll consdrl css nd shows h horcl soluon. Alhough s rhr smpl prolm for GA, w cn confrm h ffcvnss of GA pproch. Furhr sudy for compl prolms should mnd hrfr. N, opml wghs r hd n Tl 6-4, Fgur 6-5: D of TOSHIBA 6. Ecuon Rsuls GA cuon condon s hd n Tl 6-. Tl 6-: GA Ecuon Condon GA Ecuon Condon Populon Mmum Gnron 5 Crossovr r.7 Muon ro.5 Sclng wndow sz Th numr of ls o rn Tournmn sz W md ms rpon nd h mnmum of h vrnc of forcsng rror nd h vrg of convrgnc gnron r hd n Tl 6- nd 6-. Tl 6-: GA cuon rsuls (Monhly ro s usd) Cs Mnmum h vrnc Avrg of convrgnc of forcsng rror gnron SHARP 4, SONY 7, 8.5 Pnsonc, HITACHI 5,6 6.9 TOSHIBA,85. Tl6-:GA cuon rsuls(monhly ro s no usd) Mnmum h vrnc Avrg of Cs of forcsng rror convrgnc gnron SHARP 4, Tl 6-4: Opml wghs (Monhly ro s usd) Tl 6-5: Opml wghs (Monhly ro s no usd) In h cs monhly ro s usd, h comnon of lnr nd rd ordr non-lnr funcon modl s s n SHARP nd Pnsonc. On h ohr hnd, h comnon of nd + rd ordr non-lnr funcon modl s s n SONY, HITACHI nd TOSHIBA. In h cs monhly ro s no usd, h comn on of lnr nd rd ordr non-lnr funcon modl s s n SHARP nd TOSHIBA. On h ohr hnd, h comnon of lnr nd nd + rd ordr non-lnr fun con modl s s n SONY nd Pnsonc. And h comnon of lnr funcon modl s s n HITACHI. Cs SHARP SONY Pnsonc HITACHI TOSHIBA..67. Cs SHARP SONY.6.6. Pnsonc.96.. HITACHI... TOSHIBA.97.. Prmr smon rsuls for h rnd of uon () usng ls sur mhod r hd n Tl 6-6 for h cs of s o 4h d. 8

5 Tl 6-6: Prmr smon rsuls for h rnd of uon () c SHARP SONY Pnsonc HITACHI TOSHIBA c d SHARP SONY Pnsonc HITACHI TOSHIBA Trnd curvs r hd n Fgur , 8 6 4,5,,5, 5,5, SHARP Fgur 6-6: Trnd of SHARP SONY Fgur 6-7: Trnd of SONY Orgnl D Cs Cs Orgnl D Cs Cs Pnsonc Orgnl D Cs Cs, HITACHI Orgnl D Cs Cs Fgur 6-9: Trnd of HITACHI Orgnl D TOSHIBA Cs Cs Fgur 6-: Trnd of TOSHIBA Esmon rsul of h smoohng consn of mnmum vrnc for h s o 4h d r hd n Tl 6-7, 6-8. Tl 6-7: Smoohng consn of Mnmum Vrnc of uon (7) (Monhly ro s usd) ρ α SHARP SONY Pnsonc HITACHI TOSHIBA Tl 6-8: Smoohng consn of Mnmum Vrnc of uon (7) (Monhly ro s no usd) ρ α SHARP SONY Pnsonc HITACHI TOSHIBA Forcsng rsuls r hd n Fgur SHARP Orgnl D Cs Cs Fgur 6-8: Trnd of Pnsonc Fgur 6-: Forcsng Rsul of SHARP 9

6 ,,, Fgur 6-: Forcsng Rsul of SONY,5, 5 SONY Orgnl D Cs Cs Fgur 6-: Forcsng Rsul of Pnsonc, Pnsonc HITACHI Fgur 6-4: Forcsng Rsul of HITACHI Fgur 6-5: Forcsng Rsul of TOSHIBA Orgnl D Cs Cs Orgnl D Cs Cs TOSHIBA Orgnl D Cs Cs 6. Rmrks In 8% css, h monhly ro ws no usd hd r forcsng ccurcy (Tl7-,7-). SHARP nd TOSHIBA hd good rsul n s +rd ordr, SONY nd Pnsonc hd good rsul n s + nd + rd ordr. And HITACHI hd good rsul n s ordr. Th mnmum vrnc of forcsng rror of GA concds wh hos of h clculon of ll consdrl css nd shows h horcl soluon. Alhough s rhr smpl prolm for GA, w cn confrm h ffcvnss of GA pproch. Furhr sudy for compl prolms should mnd hrfr. 7. CONCUSION Focusng on h d h h uon of ponnl smoohng mhod (ESM) ws uvln o (,) ordr ARMA modl uon, nw mhod of smon of h smoohng consn n h ponnl smoohng mhod ws proposd for y us whch ssfd h mnmum vrnc of forcsng rror. Gnrlly, h smoohng consn ws slcd rrry. Bu n hs ppr, w ulzd h ov sd horcl soluon. Frsly, w md smon of ARMA modl prmr nd hn smd smoohng consns. Thus h horcl soluon ws drvd n smpl wy nd mgh ulzd n vrous flds. Furhrmor, comnng h rnd rmovl mhod wh hs mhod, w md o ncrs forcsng ccurcy. An pproch o hs mhod ws cud n h followng mhod. Trnd rmovl y lnr funcon ws ppld o h sock mrk prc d of consumr lcroncs ndusry. Th comnon of lnr nd non-lnr funcon ws lso nroducd n rnd rmovng. Gnc Algorhm s ulzd o srch h opml wgh for h wghng prmrs of lnr nd non-lnr funcon. For h comprson, monhly rnd ws rmovd fr h. Thorcl soluon of smoohng consn of ESM ws clculd for oh of h monhly rnd rmovng d nd h non monhly rnd rmovng d. Thn forcsng ws cud on hs d. Th nw mhod shows h s usful for h m srs h hs vrous rnd chrcrscs. Th ffcvnss of hs mhod should mnd n vrous css. REFERENCES []Kzuhro Tkysu nd Kzuko Ngo.(8) Esm on of Smoohng Consn of Mnmum Vrnc nd s Applcon o Indusrl D, Indusrl Engnr ng nd Mngmn Sysms, vol.7, no., pp []Hdksu Tokumru l. (98) Anlyss nd M surmn Thory nd Applcon of Rndom d H ndlng, Bfukn Pulshng. []Kngo Koysh. (99) Sls Forcsng for Bud gng, Chuokz-Sh Pulshng. [4]Pr R.Wnrs. (984) Forcsng Sls y Epon nlly Wghd Movng Avrgs, Mngmn Scn c,vol6,no., pp [5]Ksuro Md. (984) Smoohng Consn of Epo nnl Smoohng Mhod, Sk Unvrsy Rpor F culy of Engnrng, No.8, pp