A New Type of Combination Forecasting Method Based on PLS

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1 American Journal of Operaions Research, 2012, 2, hp://dx.doi.org/ /ajor Published Online Sepember 2012 (hp:// A New Type of Combinaion Forecasing Mehod Based on PLS The Applicaion of I in Cigaree Sales Forecasing * Biao Luo, Liang Wan, Weiwei Yan, Jiejie Yu School of Managemen, Universiy of Science and Technology of China, Hefei, China wanl001@mail.usc.edu.cn Received June 5, 2012; revised July 10, 2012; acceped July 21, 2012 ABSTRACT Cigaree marke is a kind of monopoly marke which is closed loop running, i depends on he plan mechanism o schedule producing, supplying and selling, bu he bullwhip effec sill exiss. So i has a fundamenal significance o do sales forecasing work. I needs o considerae he double rend characerisics, hisory sales daa and oher main facors ha affec cigaree sales. This paper depends on he panel daa of A province s cigaree sales; firs we esablished hree single forecasing models, afer geing he prediced value of hese single models, hen using he combinaion forecasing mehod which based on PLS o predic he province s cigaree sales of he nex year. The resuls show ha he predicion accuracy is good, which could provide a cerain reference o cigaree sales forecasing in A province. Keywords: PLS; ARMA; Time Series Mehod; Combinaion Forecasing Mehod; Sales Forecas 1. Inroducion China has he world s larges cigaree producion and consumer, he conribuion of obacco excise ax and oher axes have been living in he forefron of he Indusry over he years, i plays an imporan role on he naional s economic developmen and he srucure sabiliy of ax revenue. A he same ime, his indusry is proeced by he Tobacco Monopoly Law, during a longerm i is in a monopoly business sae, closed and planning is he main operaional feaures of i. As a monopoly marke, scienific predicion of cigaree sales can eliminae he bullwhip effec in he obacco supply chain effecively, paricularly, i s imporan in he background of enerprises separaion of indusrial and commercial in China s obacco indusry. Sales forecasing is he premise of he enerprise s budge managemen, i is also a benchmark of he performance plan, and he predicion accuracy grealy affec he scienific naure of he decision-making behavior. Therefore, in recen years, many scholars have researched he predicing mehod of cigaree sales, such as muliple linear regression mehod, ARMA predicion mehod, ime series forecasing mehod, exension clusering mehod, arificial neural neworks, gray model [1], hese mehods mine he essence * ACKNOWLEDGEMENT: This paper is founded by he youh science fund projecs of he naional naural science foundaion, No And he naional innovaion research group projecs of naional naural science foundaion, No of sales forecasing in differen aspecs, however, due o he limiaions of a single mehod, and he daa processing accuracy is no high enough, i is necessary o propose a new predicion mehod, in order o furher improve he accuracy of sales predicion. Based on he above discussion, his paper considers he characerisics of double rend (long-erm rend facors and seasonal facors for cigaree sales), hisory sales daa and oher main facors ha affec cigaree sales. Then depends on A province s cigaree sales panel daa, firs we esablished hree single forecasing models (ARMA model, Holer-Winer season produc model, ime series decomposiion model), afer geing he prediced value of hese single models, hen using he combinaion forecasing mehod which based on PLS o predic he province s cigaree sales of he nex year, in order o furher increase he sales forecasing accuracy. 2. Lieraure Review Sales forecasing is rying o use a specific mehod o idenify specific sales law, bu in a seemingly chaoic hisorical sales daa. There are many radiional forecasing mehods such as linear predicion mehod, moving average mehod, muliple regression mehod, Box-Jenkins mehod, Markov chain predicion mehod and so on [2]. The applicaion of hese mehods have specific requiremens for he sample iself (linear sequence law or he sample size is large enough), however, in pracical ap

2 B. LUO ET AL. 409 plicaions, he hisorical sequence of produc sales isn ofen fully saisfy he linear law or has a insufficien sample size, so he forecasing effec will be affeced by a cerain degree of influence. In China cigaree sales mee he obvious characerisics of ime series, monhly sales daa influenced by consumer groups which have obvious seasonal characerisics, herefore i ofen consisen wih he ypical characerisics of double rend, and he prediced value ends o have a high correlaion wih he hisory of cigaree sales. Predicion mehods such as auoregressive moving average model, Holer- Winer season produc model [3] and ime series decomposiion mehod [4] can solve he problems of cigaree sales forecasing beer in pracice. Holer-Winer season produc model and ime series decomposiion mehod are radiional ime series forecasing analysis mehods, building a ime series model appropriaely can effecively eliminae he uncerainies relaed wih he fuure. Time series mehod requires he predicion objec has a correlaion wih he curren economic rends, a he same ime predicion analysis has an exremely shor-erm meaning, herefore he effec of hese wo ime series analysis mehods used for mulisep predicion will be affeced [5]. ARMA model is a kind of commonly used sochasic ime series model, foreign scholars Rober Yaffee and Monnie McGee said, o shor-erm forecasing which have sable and seasonal characerisics, ARMA model can ge a more reliable predicion resul, and can achieve he desired confidence inerval [6]. ARMA model has srong flexibiliy, i can express he various properies of ime series ha appear in he acual work, bu he predicion ofen requires relaively large amoun of daa. In addiion, o some smoohing mehods, when new daa can be aken advanage of, updaing he ARMA model parameers is no an easy ask [7]. These hree single forecasing models all have heir own characerisics, advanages and disadvanages, when used alone, we can only use a cerain poin of he effecive informaion. Furhermore, single model will also be affeced by he model s se condiions and oher facors, herefore, when in predicions, i ofen demonsraes ha he range of informaion sources is no enough, and he forecasing accuracy is difficul o mee he requiremen [8]. The choice of a forecasing mehod should consider he predicion objec, forecasing range, as well as he daa and oher facors, so we can comprehensive use he advanages of he single predicion mehod, in his case, he imporance of he combinaion forecasing mehod has been pu on he agenda. Combinaion forecasing mehod is rying o inegrae he predicion resuls of a variey of forecasing mehods hrough he esablishmen of a combined forecasing model, so as o ge a more accurae forecasing range, and for sysem analysis or decision-making using. In 1969, J.M. Baes and C.W.J. Granger published he aricle named combinaion forecasing on Operaional Research Quarerly [9], hey sysemaic sudied he combinaion forecasing mehod for he firs ime, and i had caused he aenion of many forecasing scholars. Afer he 1970s, combinaion forecasing research gradually caused he aenion of scholars, and published a series of papers abou combinaion forecasing, comparing wih individual predicion mehods, i has a incomparable advanage in improving he predicion accuracy, and in recen years i received an exensive aenion from scholars [8]. The mos concerned of combinaion forecasing is how o calculae he weighed average coefficien, so ha he combinaion forecasing model can be more effecive o improve he forecasing accuracy, domesic and foreign scholars have proposed a series of combinaion forecasing mehod, commonly has minimum difference mehod, no consrain leas square mehod, consrained leas squares mehod, Bayes mehod, recursive combinaion of predicion mehod and so on [10]. However, he above combinaion predicion mehods ofen give differen weighed average coefficien depending on he ypes of single forecasing mehod. To a single predicion mehod he weighed average coefficien a each poin is he same. However, single predicion mehod may have high predicion accuracy a one poin, lower in forecas accuracy a anoher poin, so i is necessary o discuss a new combinaion forecasing model, so ha he empowermen of he new model can base on he level of fiing accuracy a each poin of he single forecasing mehod. In addiion, because of ARMA model, Holer-Winer season produc model and ime series decomposiion mehod all belong o he conex of he ime series mehod, he predicion resuls of hese hree individual forecasing mehods generally have a higher correlaion, Parial Leas Squares mehod (PLS) can modeling in he case of a limied amoun of daa and effecively solve his problem [11,12], while absorbing he advanages of each individual predicion model, i also ries as much as possible o improve he predicion accuracy. This paper presens a new combinaion forecasing mehod based on PLS, is basic idea is o consider he single predicion mehod predicion value as he independen variables, o consider he predic objec as he dependen variable, hen building he parial leas squares regression, is bigges advanage is o consider he independen variable, while also aking ino accoun he informaion of he dependen variable, so we could ge a more reliable combinaion forecasing resuls. In his paper, he general idea is shown in Figure 1, in order o ensure he forecasing accuracy of he resuls in a greaer exen, we esablished hree single forecasing models, afer geing he prediced value of hese single

3 410 B. LUO ET AL. Figure 1. The forecasing process of cigaree sales in his paper. models, hen using he combinaion forecasing mehod which based on PLS o predic he province s cigaree sales of he nex year, so as o furher improve he predicion accuracy. 3. Mehods and Principles 3.1. The Principle of ARMA Model ARMA model is a kind of commonly used sochasic ime series model, proposed by G.E.P. Box and G.M. Jenkin, i is also known as he B-J mehod [13,14]. The basic idea of his mehod is alked a below, some ime series is a se of random variables depends on he ime, alhough he single sequence values ha consiue he ime series is uncerainly, however, he changes of he enire sequence has a paern o follow, analyzing i hrough he esablishmen of corresponding mahemaical model, we can ge a beer undersanding of he srucure and he characerisics of ime series, and o achieve he opimized predicion resuls wihin he minimum variance [15,16]. There are hree basic ypes of ARMA model, auo-regressive (AR) model, moving average (MA) model, auo-regressive moving average (ARMA) model [17-20]. If he ime series y is he random error of is curren and is early value, as well as he linear funcion of is

4 B. LUO ET AL. 411 early value, hen he (p, q) order auoregressive moving average model is called ARMA (p, q) model, he general form of i is, y = y + y u u + + y + u p p u q q In (1), he real parameer 1, 2,, p, is he auoregressive coefficien, θ 1, θ 1,, θ q is he moving average coefficien, i s he model s parameers o be esimaed. Inroducing he lag operaor B, he above equaion can be abbreviaed as, By Bu (2) 3.2. The Principle of Holer-Winer Season Produc Model Holer-Winer season produc model belongs o he mehod areas of muli-parameer exponenial smoohing, including non-seasonal model and seasonal produc model. Cigaree sales daa boh have he characerisics of rend and seasonal flucuaions, herefore, his sudy seleced he Holer-Winer season produc model [21]. The model has hree smoohness indexes α, β and γ (0 α, β, γ 1). The predicion model is, In (3.3), y a bk c, for all k 1 (3) ˆ k a sk y 1aa b 1 1 b 1 (1) a 1 1 (4) c s b a a (5) y c 1c a s y a b k c, For all k 1 (7) ˆT k T T Tsk (6) s is he lengh of he seasonal cycle, in here i sands for monhly daa, he cycle is 12 monhs, ha is s = 12. If = T (The las cycle), hen he predicion formula is, In (7), a is he inercep, b is he slope, c is seasonal facor (seasonal index). I can be seen ha hey are obained by smoohing. Because of he rend erm and he season erm of his model are obained by calculaing he smoohed value, so his mehod is similar o oher smoohing mehods, mainly reflecs he changes of recen daa, so i is suiable for shor-erm forecass The Principle of Time Series Decomposiion Mehod Tradiional ime series analysis [22] due he flucuaion of ime series o four facors, rend variaion (rend, T), seasonal variaion (seasonal, S), cyclical variaion (circle, C) and irregular variaion (irregular, I). Among hem, he cyclical variaion refers o he cycle changes of years, i usually refers o he economic cycle. Irregular variaion namely random variaion, he relaions beween he four variaions and he original sequence are summarized as wo models, muliplicaive model, Y = TSCI addiive model, Y = T + S + C + I Among hem, muliplicaive model is suiable for he case of ha T, S and C is inerrelaed, addiive model is suiable for he case of ha T, S and C is independen of each oher. In his paper we use he muliplicaive model o predic he cigaree sales. The main purpose o decomposiion ime series is o break down he flucuaions facors of ime series, so ha he specific circumsances of each par of he flucuaions can be clearly observed, in order o provide he condiions for he in-deph sudy of he variaion of he various pars of flucuaions The Principle of PLS Parial Leas Square (PLS) analysis, is a new ype of mulivariae saisical daa analysis mehods which is exraced from he applicaion field, i was proposed by S. Wold and C. Albano (1983) [23], his analysis mehod is mainly applied o linear regression modeling beween muli-dependen variables and muli-independen variables, and i can effecively solve many complex problems ha an ordinary muliple linear regression can no solve. S. Wold [24] and Agnar [25] poined ou ha i no only can solve he exisence of muliple correlaion problems in he muliple regression independen variable sysem, bu also can creae he regression model when he sample size is less han he number of variables. The PLS regression mehod is a improvemen of he Principal Componen Regression (PCR) analysis mehod, in he process of exracing componen, PLS no only akes he informaion of he independen variables ino accoun, bu also aking ino accoun he informaion of he dependen variables, i overcomes he adverse effecs of mulicollineariy in sysem modeling, i can ge a more reliable analysis resuls. Because his research mainly alks abou he predicion of cigaree sales, involving only one dependen variable (cigaree sales), so i is necessary o inroduce he regression mehod of single dependen variable of PLS. Given ha he dependen variable is Y and P numbers of independen variables form he independen variables se X = [x 1, x 2,, x p ], firs, PLS regression mehod exracs 1 from he marix, i is required ha i should carry he variaion informaion in X as far as possible, i should also have a grea relevance wih Y. If he regression equaion has reached a saisfacory accuracy, hen he

5 412 B. LUO ET AL. algorihm erminaes. Oherwise, using he residual informaion ha X has been explained by 2 and he residual informaion ha Y has been explained by 1 o do he second round of componen exracion, so back and forh, unil you can reach a saisfacory saisical accuracy. Assuming ha we finally exrac m componens from X, hey are 1, 2,, m, hen PLS will esablish he regression beween Y and 1, 2,, m, finally expressed as a regression equaion of Y on he original variable X. 4. Daa Collecion We obained A province s acual cigaree annual sales daa and monhly sales daa (January 2004 o December 2011) from A province s Tobacco Monopoly Bureau (company). Before 2004, he obacco sysem s informaion managemen is no well organized, sales daa was manual recording and incomplee, and some daa is los, so he previous daa is no used in his sudy. To do monhly cigaree sales forecasing, we use a sample of daa from January 2004 o December 2011, namely 8 years, a oal of 96 samples, and we also use he daa from January 2004 o December 2011 o do comparaive analysis. 5. Daa Processing To ARMA model, Holer-Winer season produc model, and ime series decomposiion mehod, we use EViews 6.0 [15] for daa processing. To PLS combined model, his sudy use SPSS16.0 and SIM CA-P12.0 for daa processing The Forecasing Resuls of Single Predicion Models Considering ha he forecasing seps of a single predicion model is a lo, in view of he reasons of space, in his paper, we give he finally predicion effec fiing figure or oher core seps of he hree single predicion models The Forecasing Resuls of ARMA Model By calculaing, he sofware oupus he fiing effec figure of he acual values and he prediced values (Figure 2), as can be seen from he Figure 2, using ARMA model o predic, he projeced rends and he acual developmen rends is very consisen, we can make an inuiive judgmen ha he forecasing resuls is saisfacory. Afer furher calculaions, he annual forecasing absolue error range of ARMA model is beween 60,419 o , he mean relaive error is 2.37%, range beween 3.16% o 1.45% The Forecasing Resuls of Holer-Winer Season Produc Model By calculaing, he annual forecasing absolue error range of Holer-Winer season produc model is beween o , he mean relaive error is 0.27%, range beween 0.75% o 0.05%, from he Figure 3 we can also make an inuiive judgmen ha he forecasing resuls is saisfacory, namely he predicion 440, , , , , , , , ,000 80, M M M M07 YF Y Figure 2. The fiing effec figure of he acual values and he prediced values sands for monhly sales sands for monh forecas values 2004M M M M M M M M M M M M M M M M M M07 Figure 3. The fiing effec figure of Holer-Winer season produc model.

6 B. LUO ET AL. 413 accuracy of Holer-Winer seasonal produc model is good The Forecasing Resuls of Time Series Decomposiion Model By calculaing, he annual forecasing absolue error range of ime series decomposiion model is beween o , he mean relaive error is 0.393%, range beween 0.853% o 0.202%, from he Figure 4 we can see ha he forecasing resuls is saisfacory, namely he predicion accuracy of Holer-Winer seasonal produc model is good Comparing beween he Predicion Resuls of he Three Single Predicion Models We use A Province Tobacco Company s sales daa during o as he researching objec of his sudy, firs we esablished hree single forecasing models o predic he province s cigaree sales of he nex year, o compare he predicion accuracy of he hree single predicion models, we use forecas absolue error sum of squares as he index o measure he predicion accuracy of he model, predicion error sum of squares is an good indicaor o reflec he predicion accuracy, he comparison resuls of predicion accuracy of he hree single forecasing models are shown in Table 1. From he comparison resuls we can see ha among he single forecasing models, Holer-Winer model s predicion accuracy is close o he model of ime series decomposiion, bu ARMA model s accuracy is relaively poor Combinaion Forecasing Model Based on PLS As we use all he hree single predicion models o predic A province s cigaree sales, so here mus have a high degree of correlaion beween he hree models predicion resuls and he acual sales, in his paper Y 1 sands for he predicion resuls of Holer-Winer season sands for monh forecas values sands for monhly sales Jan-04 Jun-04 Nov-04 Apr-05 Sep-04 Feb-06 Jul-06 Dec-06 May-07 Oc-07 Mar-08 Aug-08 Jan-09 Jun-09 Nov-09 Apr-10 Sep-10 Feb-11 Jul-11 Dec-11 May-12 Oc-12 Figure 4. The fiing effec figure of ime series decomposiion model. Table 1. The comparison resuls of predicion accuracy of he hree single forecasing models 1. H-W Time Series ARMA E Y 1,763,770 1,803,680 1,866,767 1,911,036 1,929,118 Ŷ 1,750,546 1,803,511 1,856,475 1,909,440 1,962,404 e 13, , ,286 e2 174,858,014 28, ,922,176 2,546,436 1,107,957,796 1,391,997,774 Ŷ 1,761,824 1,812,414 1,863,004 1,913,594 1,964,184 e ,066 e2 3,786,674 76,277,981 14,162,321 6,545,233 1,229,624,356 1,540,048,353 Ŷ 1,730,314 1,777,383 1,807,824 1,850,616 1,886,675 e , e2 1,119,262, ,515,010 3,474,284,204 3,650,460,757 1,801,383,293 10,736,905,448 * Y sands for he acual sales for he year; 5 2 Ŷ sands for he predicion resuls; e = Ŷ Y, sands for absolue error; E= e 1 sands for he sum of absolue error square. 1 In view of he reasons of space, his paper jus chose he nearly five years daa o compare.

7 414 B. LUO ET AL. produc model, Y 2 sands for he predicion resuls of ime series decomposiion mehod, Y 3 sands for he predicion resuls of ARMA model, Y sands for he acual sales, we use SPSS16.0 o analysis he correlaion beween acual values and prediced values, resuls are shown in he Table 2, we can see ha here is a high de gree of correlaion beween he predicion resuls of hree single predicion models and he acual sales, and here is also a high degree of linear correlaion beween he prediced values, furher proof ha use PLS for combinaion forecasing is necessary. Afer a preliminary analysis, his aricle akes he predicion resuls of he hree single forecasing models as he independen variable, akes he acual sales as he dependen variable, hen building he parial leas squares regression model, when we exrac one componen, we have exraced 99.9% informaion of he independen variable and 97.5% informaion of he dependen variable, a his ime he error sum of squares is minimum, hen we esablish he combinaion forecas model, predicion resuls are shown in he Figure 5. We can see from he above figure ha he daa poins which is consiued by prediced values and he original observaion values are disribued near he diagonal of he regression graph, furher calculaion shows ha he regression coefficien of original observaions and prediced values is , his indicaes ha he difference beween prediced values and original values is small, he fiing resuls is good, furher demonsraing ha he effec of using his model o analyze A province s cigaree consumpion demand is very good, i can grealy improve he predicion accuracy of he cigaree sales in he com- ing year. Afer furher calculaion, we can ge he absolue error and absolue error sum of squares of his combined model, in Table 3 we can see ha he absolue error sum of squares of he combinaion forecasing model based on PLS is 498,982,516, his value is jus he 35.8%, 32.4% and 4.65% of Holer-Winer season produc model, ime series decomposiion model and ARMA model. I is hus clear ha, comparing wih he single predicion model, he predicion accuracy of he combinaion forecasing model based on PLS is improved obviously, especially o he ARMA model, PLS combinaion forecasing model grealy improve is predicion accuracy. A las, we use PLS combinaion forecasing model o predic he cigaree sales of A province in 2012, resuls are shown in he Table 4. Table 2. Resuls of correlaion analysis. Y 1 Y Y 1 Y 2 Y 3 Y ** 1 Y ** ** 1 Y ** ** ** 1 1** In confidence (double measure) of 0.01, he correlaion is significan; 1* In confidence (double measure) of 0.05, he correlaion is significan. Table 4. The forecasing resuls of cigaree sales of A province in Year Y 1 Y 2 Y 3 Y ,928,537 1,984, y=4.36e+0.04x+1.724e+0.06 R 2 = YPredPS[1](Y) YVarPS(Y) Figure 5. The predicion resuls of PLS. Table 3. PLS predicion resuls analysis E Y 1,763,770 1,803,680 1,866,767 1,911,036 1,929,118 PLS Ŷ 1,765,770 1,811,570 1,857,510 1,897,550 1,941,970 e e2 4,002,020 62,252,384 85,694, ,862, ,170, ,982,516 2 Y 4 sands for he predicion resuls of PLS.

8 B. LUO ET AL Conclusions In his sudy, we overview he combinaion forecasing mehods and models, on he basis of summing up he achievemens of previous sudies, his paper presens a new combinaion forecasing mehod based on PLS, and we use i o predic he cigaree sales of A province in The resuls here show ha, he fiing index beween predicion resuls and acual sales of each single predicion model is good. Comparing wih he single predicion model, he predicion accuracy of he combinaion forecasing model based on PLS is improved obviously, shows ha his mehod can effecively inegraed he informaion of single forecasing mehods, i is effecive and feasible. In addiion, a major innovaion of his aricle is he way o build he combinaion forecasing model, i is differen from he radiional mehod of calculaing he weighed average coefficien, in his paper, we use he PLS mehod o esablish he regression model, saring o build a new combined model base on he predicion resuls of he single predicion model, no jus a simple calculaion of he weighing coefficien, bu o give a full consideraion o he individual predicion model and ry o opimize i, i can be said i is a new way of building he combinaion forecasing model. Through his sudy, we can see ha he predicion accuracy of eiher single or combinaion model has reached a high level, despie his, i should be noed ha he expeced cigaree sales given in his sudy is only a reference value, when in he formulae of he acual sales ask, we can no be compleely according o i, we should combine wih he hisory sales and he expeced developmens of A province o developmen an appropriae and reasonable sales ask or scheduler. REFERENCES [1] P. Yin, Z.-J. Wang and D.-Y. Xiao, Predicion of Companies Failure Risk Based on Various Precision Rough Neural Neworks, Journal of Managemen Science, Vol. 23, No. 4, 2010, pp [2] D. Waers, Invenory Conrol and Managemen, 2nd Ediion, John Wiley & Sons, New York, [3] W. H. Greene, Ecomomeric Analysis, Prenice-Hall Inernaional Inc., New York, [4] W. Wimam and S. Wei, Time Series Analysis Univariae and Mulivariae Mehods, Addison-Wesley Pub., California, [5] W. R. Bell and S. C. Hillmer, Issues Involved Wih he Seasonal Adjusmen of Economic Time Series, Journal of Business and Economic Saisics, Vol. 21, No. 1, 1984, pp [6] M. L. Peer, S. A. John, H. D. William, e al., Is Cigaree Smoking in Poorer Naions Highly Sensiive o Price, Journal of Healh Economics, Vol. 23, No. 1, 2004, pp doi: /j.jhealeco [7] J. E. Hanke and D. W. Wichern, Business Forecasing, Tsinghua Universiy Press, Beijing, [8] Z. Xiao and W. Wu, The Applicaion of Combining Forecasing Based on PSO-PLS o GDP, Journal of Managemen Science, Vol. 21, No. 3, 2008, pp [9] J. M. Baes and C. W. J. Granger, Combinaion of Forecass, Operaions Research Quarerly, Vol. 20, No. 4, 1969, pp. 45l-468. [10] H.-Y. Chen, Combinaion Forecas Mehod Validiy Theory and Is Applicaion, Beijing Science Press, Beijing, [11] H.-W. Wang, Z.-B. Wu and J. Meng, The Linear and Nonlinear Mehod of Parial Leas-Square Regression, Naional Defence Indusry Press, Beijing, [12] C. Fornell and J. Cha, Parial Leas Squares, Advanced Mehods of Markeing Research, Blackwell, Cambridge, [13] S.-Q. Wei, Y. He and W. Jiang, The Combinaion Predicion of GDP Based on ARMA-ARCH, Journal of Saisics and Decision, Vol. 14, No. 7, 2007, pp [14] K. Gilber, An ARIMA Supply Chain Model, Managemen Science, Vol. 51, No. 2, 2005, pp doi: /mnsc [15] D.-H. Yi, Daa Analysis and EViews Applicaion, China Renmin Universiy Press, Beijing, [16] G. Galman and S. M. Disney, Sae Space Invesigaion of Bullwhip wih ARMA (1,1) Demand Processes, Inernaional Journal of Producion Economics, Vol. 35, 2006, pp [17] Y.-H. Ou, The Predicion of Real Esae Price Index Based on ARMA Model, Journal of Saisics and Decision, Vol. 14, No. 7, 2007, pp [18] Y. Feng and J.-H. Ma, Demand Forecasing in Supply Chains Based on ARMA (1,1) Demand Process, Indusrial Engineering Journal, Vol. 11, No. 5, 2008, pp [19] Y.-H. Xun, The Applicaion of ARMA Model in Esimaes he Impac of Major Emergencies, Journal of Saisics and Decision, Vol. 14, No. 7, 2006, pp [20] S.-J. Li, Applying ARMA Models o Forecas he Price Index of Ships, Ship Engineering, Vol. 29, No. 6, 2007, pp [21] R. S. Pindyck and L. Daniel, Rubinfeld Economeric Model and Economic Forecass, 4h Ediion, McGraw- Hill Companies, New York, [22] Y. Wan, Applied Time Series Analysis, China Renmin Universiy Press, Beijing, [23] S. Wold, C. Albano and M. Dun, Paern Regression Finding and Using Regulariies in Mulivariae Daa, Analysis Applied Science Publicaion, London, [24] S. Wold, C. Albano and M. Dun, Modelling Daa Tables by Principal Componen and PLS: Class Paerns and Quaniaive Predicive Relaions, Analysis, Vol. 12, 1984, pp

9 416 B. LUO ET AL. [25] A. Höskuldsson, A PLS Regression Mehods, Journal of Chemoerics, Vol. 2, 1988, pp doi: /cem

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