IJRRAS 6 (3) Sepember 03 www.rppress.com/volumes/vol6issue3/ijrras_6_3_0.pdf THE COMINED FORECASTING MODEL OF GRAY MODEL ASED ON LINEAR TIME-VARIANT AND ARIMA MODEL Xi Long,, Yong Wei, Jie Li 3 & Zho Long School of Compuer Science in Sichun Universi of Science & Engineering, Chin Mhemics nd informion insiue in Chin Wes Norml Universi, Chin 3 School of Economics nd Finnce in Guizhou Universi of Finnce nd Economics, Chin ASTRACT As o he esblished gr model bsed on he liner ime-vrin nd individul predicion model of ARIMA, his ricle consrucs he combined forecsing model bsed on he gr model nd he ime series model b mens of relive error weighing. This predicion indices h boh he gr model nd ARIMA model exer efficien funcion on he Torpedo developmen cos predicion, nd he combinion forecs model improve he precision of predicion. Kewords: discree Verhuls model, ARIMA model, combined forecsing model.. INTRODUCTION Tcling d cn conribue o finding ou he lw underling d. Predicion pls role in inferring he pproching vriion of developmen of hings ccording o he exisen d. Differen d process modes emerge differen modeling pproches. The gr model presens in he w of cumuling he developmen rend of gr energ ccumulion, which full visulizes he poenil lw impling in he disordered originl d. However, mos of d re in reli h pril messge re nown nd pril messge re unnown. Gr model is jus bou he process objec of len smple nd smll informion [-3]. The ime series, such s ARIMA model [4], requires deque d nd hen i migh revel he inheren regulr perns underling he d. Thus, he gr model [6-9] is cpble of ming heir respecive dvnges complemenr o ech oher in he respec of informion dequc nd so on. This ricle is bou he consruc of discree Verhuls model (LTDVM model)bsed on he liner ime-vrin nd he building of ARIMA model, boh of which re used o predic he Torpedo developmen cos. A ls, he consruc of he combined forecsing model formed from he gr model nd he ime series is lso used for predicion nd he predicion ccurc is expeced o be efficien nd relible.. THE COMINED FORECASTING MODEL FORMED FROM GRAY MODEL AND ARIMA MODEL..The Discree Verhuls Model(LTDVM Model)sed On The Liner Time vrin 0 0 0 Assuming he non-negive primiive sequence s 0 X x, x,, x n, in which,,,, n ; 0 The -AGO sequence of X is: X x, x,, x n, in which, 0 x x i i,,,, n ; Z is he djoining nd generive men vlue sequence of Z,,3,, n. 0.5 x x... Gr Verhuls model [] There re four definiions. Definiion Clls 0 z b z X : Z z, z 3,, z n x () s Gr Verhuls model. When he primiive d hemselves embod he shpe of S, he originl d re en s 0 s X, he consruc of Verhuls model is helpful for direc simulion of X. Definiion Clls dx x bx d (), in which, X nd he -AGO 340
IJRRAS 6 (3) Sepember 03 Long & l. The Combined Forecsing Model of Gr Model s winerizion equion of gr Verhuls model. From he process of solving his equion,i is no hrd o see h he essence of equion soluion is he shif of primiive funcion ino bcwrd funcion. The inheren error of Gr Verhuls model derives from he pproximel non-sric mching of he gr model nd he winerizion equion. Thus, his ricle firsl considers he reciprocl remen of he primiive d nd hen he consruc of he discree gr predicion model.... Liner Time-vring Prmeers Discree Gr model [] Definiion 3 Clls x x 3 4 (3) s he discree gr predicion model of liner ime vring prmeers (TDGM(,)model). In prcicl pplicions, he ssem behviors sequence iself nd he inercion of differen behvior sequences represen compliced nonliner relions. esides, s ime goes, he prmeers nd srucures of he ssem behvior sequence m lso chnge consisenl. Therefore, i is difficul o pu simulion nd predicion ino effec b he ligh of consn prmeer model. For h reson, he echer Zhng Ke builds he DGM(,) model of he liner ime-vrin prmeers b mens of he subsiuion of liner ime for he consn prmeer of primiive discree gr model in he documen wo. Wih regrd o he simulion predicion model of oscilling sequence, we r o urn he primiive sequence ino reciprocl nd hen e i ino TDGM(,) model, equll he recombinion of discree Verhuls model nd liner ime-vrin model, h is, puing forwrd pe of discree Verhuls model bsed on he liner ime-vrin...3. sed on he Liner Time-vring Discree Verhuls Gre Model (LTDVM model). Definiion 4 ssumes he observion vlue of one priculr behvior chrcerisic sequence of he ssem s 0 0 0 0 X x x x 0 0 0,,, n. Y is he reciprocl sequence of X, h is,,, n 0 nd x Y is n ddiive sequence of 0 Y,nming,,,, n 3 4 s discree Verhuls model bsed on he liner ime-vrin(ltdvm model). The soluion process of such model is s follows: I. ming use of les squre mehod o ge he model prmeers, 3 4,, : 3 4 =, =, 3 =, 4 ; A A A A II. Ting dvnge of he recursion formul sequence ; 0 0 (4) 3 4 o ge he III. Uilizing he formul x o ge he model predicion vlue of primiive sequence... ARIMA Model [4] ARIMA model (Summion uo-regressive moving verge model) is brodl used in he field of predicive conrol. I is clssicl nlsis model of ime series s well s common predicion model in prcice. In fc, vi he uocorrelion of sisics, he ARIMA model is emploed in he predicion of d, he sbili of ime sequence, or he nonsionr ime series h is rending o smooh b mens of difference. Assuming s one nonsionr ime series, s is residul sequence, d s difference sequenc, nd bringing he del operor, here is in which is lg phse. Ming d order difference operion of is sble, ARIMA model cn be se up. The formul is s follows: (5) d -, suppose h he differenied sequence 34
IJRRAS 6 (3) Sepember 03 Long & l. The Combined Forecsing Model of Gr Model d u, in which p ( ) P q ( ) q (6) This is he uo-regression coefficien polnomils nd smoohing coefficien polnomils of ARMA (p,q) respecivel. ARMA(p,d,q)model is cull he combinion budge of ARMA(p,q) model nd d order difference operion. Is model-building process is s follows:() esing he sionri of observion sequence () If he observion sequence is no sble, ming d order difference operion for he primiive sequence.(3) ccording o sionr series, fiing ARMA(p,d,q)model (4) recognizing esimed prmeers (5) residul es (6) ccording o he d, predicing..3. The opiml combinion of he consrucion of weigh predicion model [5] For he sme predicion problem, here re wo or more hn wo single forecsing mehods,, he Ah predicion model is e,,, n;,,,,, smbolizes he number of predicion formul. Assuming h he relive error of in No. nd he combined model weigh is nd he combined predicion model is s..min X X (7) in which, X is he objecive funcion of opiml combined forecsing model. The objecive funcion ops error forms. This ricle chooses he bsolue vlue of relive error nd minimizes i. The weigh is bsed on he following formul: e e in which, e funcions s he men bsolue difference of he relive errors. In order o chieve he opiml predicion resul, ccording o he principle h smller weigh is for lrger relive error nd lrger weigh is for smller one, one opiml combined model cn be consruced: f (9) The individul predicion model in his ricle is Verhuls model bsed on he liner ime-vrin nd ARIMA model, eiher of which is represened b nd correspondingl. Thus, he combined forecsing model bsed on individul predicion model shows s he following formul: f (0) 3. APPLIED CASES: TORPEDO DEVELOPMENT COST PREDICTION Torpedo developmen is compliced ssem projec which is clssified ino seven phses: fesibili phse, preliminr design phse, deiled design phse, ril-mnufcure phse, experimenl phse, vlidion phse nd produc definiion phse. This ind of Torpedo developmen is highl volile on ccoun of differen degrees of complexi nd developmen cos ech sge. Generll speing, wih he developmen s going deeper, developmen cos requires more unil pe of developmen cos ppers. Thereupon, he developmen cos srs o diminish down. One pe of Torpedo developmen cos shows s he following wo bles: Tble. one Torpedo developmen cos [en-housnd un] ers 995 996 997 998 999 000 00 00 003 004 cos 496 779 87 05 488 55 57 0 87 79 [, 3] (8), 34
IJRRAS 6 (3) Sepember 03 Long & l. The Combined Forecsing Model of Gr Model Tble. -AGO of one Torpedo developmen cos [en-housnd un] ers 995 996 997 998 999 000 00 00 003 004 cos 496 75 46 3487 3975 430 4387 4497 4584 4663 () As he sisics bove displs, he cumulive curve h he Torpedo developmen cos presens is pproximive o he shpe of "S". I consrucs new discree Verhuls model (LTDVM model). Hence, he d in he Tble re used o me simulion nd predicion ccording o discree Verhuls model (LTDVM model) bsed on he liner 0 ime-vrin.,,, n 0,in which 0 i,,,, n x ; on he bsis of definiion4, les squre mehod cn be en o ge model prmeers i 0.30 0.0076 0.000 0. 00, 3 4,, nd hen he predicion formul comes ou:, whose compued resul is represened in Tble 3. ()The observion vlue sequence in originl smples is esed o be non-sble sequence b ADF b he ligh of eviews6.0, so he resul of difference operion displs h ADF sisicl mgniude is lrger hn mrginl vlue nd he second order difference sequence is sble sequence. Afer ming second order difference, he P vlue in uocorrelogrm conspicuousl embodies he nure of order runcion, h is, pril correlion coefficien is provided wih remrble chrcerisics of he second order difference(fig ). Overll, considering h his ricle dels wih originl sequence b mens of he second order difference, we len o he fiing originl sequence ARIMA(,,). Afer ming second order difference for he sequence fiing model, he consequence is s follows: 75.7.480 0.440 0.58 0.9949,where i( i 0,,) is he cumulive developmen cos of one pe of Torpedo er -i, nd ii ( 0,,) is he Roo Men Squred Error of he cumulive developmen cos of one pe of Torpedo er -i. (3) Figure. Self-correlogrm nd pril correlogrm According o he fiing resul, in he process of predicing, Theil s 0.000, n unequl coefficien, indices h his model hs good predicion bili. Thereino, he covrince proporion is 0.7503, which demonsres n idel predicion resul of his model(figure ). The clculion resul is showed in Tble 3. Figure. one Torpedo developmen cos in 999-004 (4)From he discree Verhuls model bsed on he liner ime-vrin nd he ARIMA model, we cn in new predicive model, simpl nmed s combinion model in which represens snds for he gr model nd 343
IJRRAS 6 (3) Sepember 03 Long & l. The Combined Forecsing Model of Gr Model ARIMA model. sed on he principle h smller weigh is for lrger relive error nd lrger weigh is for smller one, here re wo weigh models in Eq.8: 0.048 0.765 0.95; 0.8048. On he bsis of Eq.0, he new 0.765 0.048 0.765 0.048 combined forecsing model is f 0.95 0.8048. 3 Tble 3. comprison of he clculed resuls of four pes of models Gr Verhuls Combined forecsing LTDVM model ARIMA model model[,3] model ers d Simulion Relive Simulion Relive Simulion Relive Simulion Relive vlue error (%) vlue error (%) vlue error (%) vlue error (%) 995 496 996 75 9.5.6 74.8908 0.0086 75 0 75 0 997 46 6.07 4.053 465.006 0.3 46 0 46 0 998 3487 377.486 8.876 3473.565 0.3970 3487 0 3487 0 999 3975 393.739.54 3983.05 0.068 3973.793 0.0304 3975.6334 0.059 000 430 486.8.38 44.65 0.748 46.400 0.085 49.37 0.048 00 4387 4444.806.38 4387.46 0.005 438.463 0.035 4383.4375 0.08 00 4497 4507.36 0.30 4487.390 0.37 4495.4905 0.0336 4493.9094 0.0687 003 4584 453.77.50 4575.949 0.756 4579.0858 0.07 4578.4736 0.06 004 4663 4540.3.63 467.344 0.789 466.79 0.059 4663.6558 0.04 Averge relive error(%) 4.87 0.765 0.048 0.0350 Aenion: he relive errors in he chr e bsolue vlue. From he clculed resul in Tble 3, i is es o see h he LTDVM model nd he ARIMA model displs he bes simulion effec nd he verge relive error is 0.035%, while he Gr Verhuls model shows he wors simulion nd he verge relive error is 4.87% [3]. 4. CONCLUSION The predicion resul indices h boh he relive error of gr model bsed on he liner ime vriion nd he relive error of ARIMA model re smller hn he one of rdiionl Verhuls model. These wo models cn in good predicion effecs. Through opimizing he combined model, he error of predicion resul grdull diminishes. esides, he predicive error of combined model is smller hn he individul predicion error. Hence, he predicion ccurc is bound o be improved. 5. REFERENCES []. Sifeng Liu, Yoguo Dng,Zhigeng Fng nd Niming Xie, The Gr Ssem Theor nd Is Applicion, fifh ed., science press, ei Jing,00. []. Ke Zhng, Sifeng Liu, Ssems Engineering-Theor & Prcice,30(00)650-657. In Chinese. [3]. Lizhi Cui, Sifeng Liu, Zhiping Li, Ssems Engineering nd Elecronics,33(0)590-593.In Chinese. [4]. Rui Dn, Shuhu Wng, Lingling Li nd Donglin Go, Journl of Lioning Technicl Universi (JCR Science Ediion), (0)8-. In Chinese. [5]. Yu Di, The opiml combinion forecs model consrucion nd pplicion. Economic mhemics, (00) 9-98. In Chinese. [6]. Qingwei Ling, owei Song,Yue Ji, Journl of Ssem Simulion, 7(005) 57-58. In Chinese. [7]. Zhengxin Wng,Yoguo Dng,Sifeng Liu, Ssem Engineering-Theor & Prcice, 9(009)38-44. In Chinese. [8]. Mingxi Lu, Gungle Yn, Sisics nd decision, 7(009)5-53. In Chinese. [9]. Weiguo Li, Aiqing Zhng, Sisics nd Decision, (007)-. In Chinese. 344