# A Comparative Study of Linear and Nonlinear Models for Aggregate Retail Sales Forecasting

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3 Y = T + S + ε (2) where T i he rend componen and S i he eaonal componen. ε i he error erm ofen aumed o be uncorrelaed. Thi addiive model i appropriae if he eaonal variaion i relaively conan. If he eaonal variaion increae over ime hen he muliplicaive model hould be more appropriae. In hi cae however he logarihmic ranformaion can be ued o equalize he eaonal variaion. Tradiionally he rend componen can be modeled by polynomial of ime of ome low order. Here we conider he linear rend model: T = β + 1 (3) 0 β On he oher hand he eaonal componen can be modeled by eiher eaonal dummy variable or rigonomeric funcion. Wih he eaonal dummy variable I i defined a I i =1 if ime period correpond o eaon i and I i =0 oherwie we have S = ω I + ω I + L+ ω I (4) When combining (3) and (4) ino model (2) i i neceary o eiher omi he inercep β 0 or e one of he eaonal parameer ω o zero in order for he parameer o be eimaed. The eaonal componen S can alo be modeled a a linear combinaion of rigonomeric funcion S m 2πi = Ai in + φ i= 1 i where A i and φ i are he ampliude and he phae of he ine funcion. m i he number of ine funcion ued o repreen he eaonal variaion. In many cae m = 1 or 2 i ufficien o repreen complex eaonal paern (Abraham and Ledoler 1983). An equivalen form o (5) which i ofen ued in pracice i S = m i= 1 1i 2πi in + ω 2i 2πi co ω (6) I i imporan o noe ha unlike ARIMA model regreion model are deerminiic in ha model componen or coefficien are conan over ime. Thu he behavior of he regreion mehod can be quie differen from ha of he ochaic model. If he model componen are changing a in many economic and buine ime erie he ochaic ARIMA model may be more appropriae. (5) The Nonlinear Modeling Approach A number of nonlinear ime erie model have been developed in he lieraure bu few are pecifically for eaonal modeling. Moreover mo of hee model are parameric and he effecivene of he modeling effor depend o a large exen on wheher aumpion of he model are aified. To ue hem he model form mu be pre-pecified and uer mu have knowledge on boh daa propery and model capabiliy. Thi i he major roadblock for general applicaion of hee nonlinear model. Neural nework are he mo veraile nonlinear model which can repreen boh noneaonal and eaonal ime erie. The mo imporan capabiliy of neural nework compared o oher nonlinear model i heir flexibiliy of modeling any ype of nonlinear paern. The mo popular feedforward hree-layer nework for forecaing problem ha he following pecificaion: y q p 0 + α j f βijxi + β0 j ) j= 1 i= 1 = α ( + ε (7) where p i he number of inpu node q i he number of hidden node f i a igmoid ranfer funcion uch a he logiic: 1 f ( x) =. { j j = n 1+ exp( x) β ij i = m; j = α } i a vecor of weigh from he hidden o oupu node and { n } are weigh from he inpu o hidden node. α 0 and β 0j are weigh of arc leading from he bia erm which have value alway equal o 1. The inpu variable x i i = 1 2 p are he lagged pa obervaion if he direc ime erie daa are ued in model building. In hi cae model (7) ac a a nonlinear AR model. To model eaonaliy eaonal lagged obervaion (obervaion eparaed by muliple of eaonal period ) hould be ued. However elecing an appropriae NN archiecure or more imporanly lagged variable may require ome experimenal effor and radiional modeling kill (Faraway and Chafield 1998). Noe we can alo ue he eaonal dummy variable or rigonomeric erm a predicor variable in which cae he neural nework (7) i equivalen o a nonlinear regreion model.

4 METHODOLOGY The daa ued in hi udy are monhly reail ale compiled by he U.S. Bureau of he Cenu. The ampling period examined i from January 1985 o December Alhough longer daa erie are available a pilo udy how ha larger ample are no necearily helpful in overall forecaing performance. The daa e i ued o illurae he applicaion of neural nework and oher radiional mehod in an effor o beer foreca aggregae reail ale. In addiion everal reearch queion are addreed: I i helpful uing auxiliary variable uch a eaonal dummy variable or rigonomeric variable in forecaing performance? Wha i he be way o model eaonal ime erie wih neural nework? I eaonal adjumen i ueful o improve forecaing accuracy? Doe he increaed modeling power of he nonlinear neural nework model improve he ou-of-ample forecaing performance for reail ale? Thee iue are inveigaed hrough a comparaive udy of ou-of-ample forecaing beween linear and nonlinear eaonal model dicued in he previou ecion. Three linear model ARIMA regreion wih dummy variable and regreion wih rigonomeric variable are buil uing he in-ample daa. The forecaing performance of each model i hen evaluaed by reul from he ou-of-ample which i hidden in he in-ample model fiing proce. Since he reail ale erie exhibi boh rend and eaonaliy he following regreion model i eablihed: Y = β + β + ω I + ω I + L + ω I (8) where he eaonal dummy variable I i defined a I i =1 if ime period correpond o monh i and I i =0 oherwie; β L ω are model parameer. Noe ha he dummy variable for December i no explicily defined. 0 β1 ω1 ω2 11 We alo conider he following regreion model wih 2-erm and 4-erm rigonomeric funcion repecively: Y Y 2π 2π = β 0 + β1 + ω1 in + ω 2 co (9) 2π 2π 4π 4π = β 0 + β1 + ω1 in + ω2 co ω1 + in + ω2 co (10) Uing model (8)-(10) require ha he ime erie have conan eaonal variaion. Since he reail ale preen a clear increaing eaonaliy he naural log-ranformaion i performed o abilize he eaonal variaion. The model are hen fied o he ranformed daa. Finally he foreca are caled back o he original uni. On he nonlinear model ide we ue he andard fully conneced hree-layer feedforward nework. The logiic funcion i ued for all hidden node a he acivaion funcion. The linear acivaion funcion i employed for he oupu node. Bia erm are employed for boh oupu and hidden node. For a ime erie forecaing problem neural nework model building i equivalen o deermining boh he number of inpu node and he number of hidden node. The inpu node are he pa lagged obervaion hrough which he underlying auocorrelaion rucure of he daa can be capured. However here i no heoreical guideline ha can help u pre-pecify how many inpu node o ue and wha hey are. Idenifying he proper auocorrelaion rucure of a ime erie i no only a difficul ak for nonlinear modeling bu alo a challenge in he relaively imple world of linear model. On he oher hand i i no eay o pre-elec an appropriae number of hidden node for a given applicaion. Alhough he neural nework univeral approximaion heory indicae ha a good approximaion may require a large number of hidden node only a mall number of hidden node are needed in many real applicaion (Zhang e al. 1998). Following he common pracice we ue he radiional cro-validaion approach o elecing he be neural nework archiecure. Tha i he in-ample daa are pli ino a raining e and a e e. The raining e i ued o eimae he model parameer and he e e i ued o chooe he be neural nework model. In hi udy we conider 10 differen level of inpu node: and 36 and 7 hidden node level from 2 o 14 wih an incremen ize of 2. Thu a oal of 70 differen nework are experimened in order o find he be neural nework model. To ee he effec of eaonal adjumen on he forecaing performance of neural nework we ue he mo recen Cenu Beaure X12-ARIMA eaonal adjumen program o deeaonalize he original erie. Then neural nework are fied o deeaonalized daa and. Finally he foreca reul baed on he noneaonal daa are ranformed back o original cale uing he foreca eaonal indice provided by he program. We alo inveigae he iue of wheher uing dummy or rigonomeric variable can enhance neural nework capabiliy of modeling eaonal variaion. Correponding o dummy regreion model a 12-inpu neural nework wih he ame predicor variable ued in (8) i conruced. The ame idea i ued o build neural nework model wih inpu variable correponding o hoe in rigonomeric mo del (9) and (10). Of coure in hee eing ince he inpu node are idenified he only hing lef decided in neural nework model i he number of hidden node. The ame experimenal deign a in he ime erie modeling decribed earlier will be carried ou o deermine hi parameer. Neural nework raining i carried wih a GRG2 baed yem (Subramanian and Hung 1993). On he oher hand we ue Foreca Pro o conduc he ARIMA model fiing and forecaing. In paricular we ue he auomaic model idenificaion feaure of Foreca Pro o chooe he be model. The mehodology i baed on he augmened Dickey-Fuller e and he Bayeian Informaion Crierion (BIC).

5 To enure ha he oberved difference of performance beween variou model are no due o chance we ued a five-fold moving validaion cheme wih five ou-of-ample period from 1995 o 1999 in he udy. Each ou-of-ample (or validaion ample) conain 12 obervaion repreening 12 monhly reail ale in a year. The lengh of he correponding in-ample period i fixed a 10 year. Tha i for each validaion ample he previou 10 year of daa are ued a in-ample for model developmen. Thi "moving" validaion approach wih muliple overlapped in-ample daa and differen ou-of-ample can provide ueful informaion of he reliabiliy of a forecaing model wih repec o changing underlying rucure or parameer over ime. In hi cro-validaion analyi all of he model are rebuil and re-eimaed each ime a new validaion ample i examined. To evaluae and compare he forecaing performance of differen model we ue hree overall error meaure in hi udy. They are he roo mean quared error (RMSE) he mean abolue error (MAE) and he mean abolue percenage error (MAPE). Since here i no univerally agreed-upon performance meaure ha can be applied o every forecaing iuaion muliple crieria are herefore ofen needed o give a comprehenive aemen of forecaing model. RESULTS We fir give a deailed analyi of variou model on heir performance for he 1999 reail ale forecaing. The in -ample period for model fiing and elecion i from 1989 o 1998 while he ou-of-ample coni 12 period in A menioned before for NN ime erie modeling he la wo year in-ample daa are ued a he eing ample and he re of obervaion are ued for model eimaion. The model wih he be performance in he eing ample will be eleced a he final model for furher validaion in he ou-of-ample. All model comparion are baed on he reul for he ou-of-ample. We find ha overall he ARIMA model follow he eaonal paern exhibied in he ale daa. However i doe no provide good foreca for he reail ale in 1999 becaue mo of he forecaed value are below he acual a clear under-forecaing iuaion. On he oher hand we find ha he direc NN model perform even wore han he ARIMA model a almo all foreca are relaively far below he acual value. Furhermore he neural nework model buil wih he deeaonalized daa can improve he foreca accuracy dramaicall alhough foreca are ill generally lower han he acual. Comparion i made beween linear and nonlinear regreion model wih eaonal dummy variable and rigonomeric funcion repecively. However none of hee model provide aifacory foreca alhough he nonlinear regreion model perform generally beer han heir linear counerpar. The dummy regreion model conienly underforeca while foreca from rigonomeric model are no able. In addiion while dummy regreion model follow he general eaonal paern fairly well rigonomeric model fail o capure he eaonal paern in he ou-of-ample. For hi paricular ampling period we find ha linear model a well a mo of he nonlinear model do no perform well judged by all hree crieria hough nonlinear model in general ouperform linear model. The be model i he neural nework buil on deeaonalized daa. Thi model ha he mo accurae foreca becaue of he lowe error meaure. Moreover compared o ime erie model boh linear and nonlinear regreion model wih dummy or rigonomeric variable yield much wore foreca for he 12 monh in Becaue of he poible change in he daa rucure or parameer of he model over ime we ue a five-fold moving validaion cheme decribed earlier. In hi validaion analyi all of he model are rebuil and re-eimaed each ime a new validaion ample i examined. I i imporan o noe ha he model doe change each ime a differen in-ample i ued o build ARIMA and NN model wih regard o model rucure and/or he parameer. For example he ARIMA model ued for validaion ample of 1995 and 1996 are ARIMA(011)(011) 12 he ame a ha for 1999 bu all wih differen parameer and he model for boh 1997 and 1998 are ARIMA(011)(013) 12 bu again wih differen parameer. When comparing ARIMA and NN ime erie model acro he five-year validaion period we find ha judged by all hree overall accuracy meaure and acro five validaion ample neural nework wih deeaonalized daa (NN-deeaon) perform he be overall while ARIMA and neural nework modeled wih original daa (NN-direc) perform abou he ame. The overall uperioriy of he deeaonalized NN model i furher confirmed by he ummary aiic uch a he mean and he andard deviaion and he reul from Wilcoxon rank-um e. The mean difference meaure beween NN-deeaon and ARIMA and beween NN-deeaon and NN-direc are all ignifican a he 0.05 level while here i no ignifican difference beween ARIMA and NN-direc. Noe ha he ue of eaonal adjumen ignificanly reduce he variabiliy of he neural nework model in predicion a indicaed by he andard deviaion of he performance meaure. We noice ha he forecaing performance of he NN-deeaon model i no a good a ha of he ARIMA model in 1996 acro all hree accuracy meaure alhough i ouperform he ARIMA in oher four validaion ample. Thi obervaion ugge ha hough an overall be model can provide mo accurae predicion over everal forecaing horizon judging by a pecific porion of he forecaing horizon i i poible ha anoher model perform beer. I furher ugge ha no forecaing model i alway he be for all iuaion. Therefore he imporance of uing muliple cro-validaion ample o compare differen forecaer become clearer. Comparing linear and nonlinear regreion model we find ha excep for he 1999 cae in which boh linear and nonlinear regreion fail o predic well he dummy variable are very helpful in improving forecaing performance. In fac dummy regreion model foreca much beer han all hree linear and nonlinear ime erie model in he validaion ample from 1995 o However dummy regreion model performed very poorly in forecaing 1999 ale wih all hree error meaure more han

6 doubling he ize of hoe for ime erie model. Therefore eaonal dummy variable may be ueful for reail ale forecaing bu he model may no be robu. In addiion wih a flexible nonlinear model he forecaing error can be coniderably reduced. No only he nonlinear neural nework ouperform heir linear regreion counerpar in almo all validaion ample heir reul are alo more able a refleced by he andard deviaion meaure. The mean difference in hree error meaure beween linear and nonlinear model are all poiive and ignifican a he 0.05 level wih Wilcoxon rank-um e uggeing he advanage of nonlinear model over i linear counerpar. Reul from uing rigonomeric variable are ignificanly wore han hoe obained wih he ime erie and dummy regreion model. Thu uing rigonomeric funcion o predic reail ale i no helpful a all. Alhough i i almo alway he cae ha a nonlinear NN model perform beer han i linear counerpar and a four-erm rigonomeric regreion model i beer han a wo-erm one none of he rigonomeric regreion model examined i able o provide adequae foreca for reail ale in all of he validaion ample. CONCLUSIONS Thi paper preen a comparaive udy beween linear model and nonlinear neural nework in aggregae reail ale forecaing. Accurae foreca of fuure reail ale can help improve effecive operaion in reail buine and reail upply chain. Since reail ale daa preen rong eaonal variaion we inveigae he effec of differen eaonal modeling raegie and echnique on heir forecaing accuracy. Boh ime erie approach and regreion approach wih eaonal dummy and rigonomeric variable are examined in he udy. Our reul ugge ha he nonlinear mehod i he preferred approach o modeling reail ale movemen. The overall be model for reail ale forecaing i he neural nework model wih deeaonalized ime erie daa. In he forecaing lieraure i i an eablihed fac ha no ingle forecaing model i he be for all iuaion under all circumance. Therefore he be model in mo real world forecaing iuaion hould be he one ha i robu and accurae for a long ime horizon and hu uer can have confidence o ue he model repeaedly. To e he robune of a model i i criical o employ muliple ou-of-ample o enure ha he reul obained are no due o chance or ampling variaion. The uefulne of hi raegy in comparing and evaluaing forecaing performance of variou model i clearly demonraed in our experimen. REFERENCES Abraham B. and Ledoler J Saiical Mehod for Forecaing John Wiley & Son New York. Agrawal D. and Schorling C Marke hare forecaing: An empirical comparion of arificial neural nework and mulinomial logi model Journal of Reailing 72 (4) Barkdale H. C. and Hilliard J. E A cro-pecral analyi of reail invenorie and ale Journal of Buine 48 (3) Box G. E. P. and Jenkin G. M Time Serie Analyi: Forecaing and Conrol San Francico: Holden-Day. Callen J. L. Kwan C. Y. Yip C. Y. and Yuan Y Neural nework forecaing of quarerly accouning earning Inernaional Journal of Forecaing De Gooijer J. G. and Frane P. H Forecaing and eaonaliy Inernaional Journal of Forecaing Elkaeb M. M. Solaiman K. and Al-Turki Y A comparaive udy of medium-weaher-dependen load forecaing uing enhanced arificial/fuzzy neural nework and aiical echnique Neurocompuing Faraway J. and Chafield C Time erie forecaing wih neural nework: A comparaive udy uing he airline daa Applied Saiic McKenzie E General exponenial moohing and he equivalen ARIMA proce Journal of Forecaing Menzer J. T. and Bienock C. C Sale Forecaing Managemen SAGE Thouand Oak CA. Nelon M. Hill T. Remu T. and O'Connor M Time erie forecaing uing neural nework: Should he daa be deeaonalized fir? Journal of Forecaing Nungo C. and Boyd M Commodiy fuure rading performance uing neural nework model veru ARIMA model The Journal of Fuure Marke 18 (8) Prybuok V. R. Yi J. and Michell D Comparion of neural nework model wih ARIMA and regreion model for predicion of Houon' daily maximum ozone concenraion European Journal of Operaional Reearch Sharda R. and Pail R. B Connecioni approach o ime erie predicion: An empirical e Journal of Inelligen Manufacuring Subramanian V. and Hung M. S A GRG2-baed yem for raining neural nework: Deign and compuaional experience ORSA Journal on Compuing Thall N Neural foreca: A reail ale booer Dicoun Merchandier 23 (10) Zhang G. Pauwo E. P. and Hu M. Y Forecaing wih arificial neural nework: The ae of he ar Inernaional Journal of Forecaing

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