Data Mining Algorithms and Statistical Analysis for Sales Data Forecast

Transcription

1 22 ifh Inernaional Join Conference on Compuaional Sciences and Opimizaion Daa Mining Algorihms and Saisical Analysis for Sales Daa orecas Lin Wu; JinYao Yan;YuanJing an Deparmen of Compuer and Nework Communicaion Universiy of China, Beijing, 24, China Absrac This paper develops and compares differen models o forecas new produc sales daa wih increasing sales rend and muliple predicor inpus. In order o analyze new produc wih increasing sales rend, we developed and evaluaed muliple ime series forecasing mehods, including Exponenial Smoohing model, Hol s Linear model, ARMA model, and ARMA wi linear rend models. urhermore, we creaed muliple Causal acor orecasing models o incorporae various dependen inpu facors such as sale person s quoes, produc pricing, produc seasonaliy facors, o furher reduce forecasing error. We analyzed original daa regression model, rend and residual regression model, and ARMAV wi linear rend model o consider inpu facors. We discovered ha ARMAV wi linear rend model gives bes forecasing accuracy and lowes RSS (Residual Sum of Square). In conclusion, ARMAV wih linear rend mehod is he bes benchmark model o forecas sales daa for new produc wih rend and wih sales person s inpus. Keywords-orecas; Time-Series orecasing; Causal acor orecasing; ARMA; ARMAV I. INTRODUCTION In consumer elecronics indusry, he normal produc selling cycle is wo o hree years. During his selling period, produc goes hrough iniial produc inroducion and maured selling period. orecasing sales of consumer elecronics producs faces challenges. irs, in he new produc inroducion sage, he demand may have upward rend. Second, consumer elecronics sales may experience seasonal selling paern. Third, shor selling period resrics he daa se size, which is a big challenge in ime series forecasing. Even a he end of 2 nd year, monhly sales only have 24 daa poins, which is much smaller han he radiional ime series problem []. Prior research on sales daa forecas is inensively focusing on forecasing wih large hisorical daa. However, new produc sales forecas wih limied daa poins remains as new research area. In addiion, prior paper on sales daa forecas mosly uses ime-series models wih no inpu facors. This paper explores new mehod in forecasing new produc sales daa using 24 monhs hisorical daa, and incorporaed muliple inpu facors o improve he forecas accuracy [2]. We analyze boh pure ime-series forecasing models and ime-series forecasing model wih causal facor forecasing mehod. We found ha wih causal facor inpus, he forecasing resul can be grealy improved [3]. We concluded ha causal facor inpus can compensae for he less accurae forecas due o limied daa in new produc sales forecasing. II. TIME SERIES ORECASTING MODELS A. Exponenial Smoohing (ES) Mehod The Exponenial Smoohing mehod fis a rend model such ha he mos recen daa are weighed more heavily han daa in he early par of he series. I has a weigh parameer α, which is beween and. The larger he alpha, he new forecas is influenced more by he recen daa. The reason ha i is called exponenial smoohing is ha he weigh of an observaion is a geomeric (exponenial) funcion of he number of periods ha he observaion exends ino he pas relaive o he curren period [4]. The model for ES is: + = α( α) j Y j= Or = αy + ( + α) (2) Here, is he prediced daa a ime, and is he acual daa a ime. ES mehod can be easily implemened. Wih iniial prediced daa = Y, he predicion of all monhs can be generaed by equaion (2). α is a decision variable o choose wih he objecive o minimize he RSS. The whole model ca be implemened in Excel, and solved by Excel solver. Min RSS By changingα, Subjec o α The opimal α ha minimize RSS is α * =.58. The minimum RSS in ES mehod is RSS = 3.2 RSS=3.2 () /2 $2. 22 IEEE DOI.9/CSO

2 L weigh of he prediced daa + b. The (4) gives he igure. Acual Sales and orecas by ES Mehod igure shows he acual and forecasing daa using ES mehod. We observe ha ES mehod ends o underesimae in increasing rend. I consisenly under esimaes he daa from ebruary 28 o Ocober 28 during he produc inroducion period. In fac, under-esimaion in increasing rend and over-esimaion in decreasing rend are generally observed in ES algorihm. This is because ha ES algorihm does no considers he rend facors, and only smoohing previous daa poins o generae nex daa poin. This smoohing inerpolaion mehod canno adjus o increasing or decreasing rend, so he predicion always shows a lag behind he increasing or decreasing rend [5]. The pros and cons of ES mehod are summarized as follows: Pros: Easy o undersand and implemen Predicion is relaively accurae if he daa is no oo complicaed Cons: Purely based on hisorical ime-series daa, canno inpu explanaory facors, e.g. quoes, seasonaliy, ec. No good a handling daa wih rend: ends o underesimae in increasing rend; over-esimae in decreasing rend. Bad daa in recen monhs can cause large errors in forecas B. Hol s Mehod As menioned in previous session, ES mehod ends o under-esimae in increasing rend and over-esimae in decreasing rend. Hol s linear mehod is inroduced o beer esimae daa wih rend. Hol s linear mehod b inroduces a parameer o esimae he slope of he rend. The algorihm of Hol s mehod is given by he following hree equaions: Level: L = α + ( α)( L + b ) (3) Trend: b = β( L L ) + ( β) b (4) orecas: + m= L+ mb (5) Highcas: + m= L+ mb () The equaion (5) is used o forecas m sep ahead predicion + m using level L and he slope b imes he number of period o predic m. The equaion (3) gives he base level ha forecas sars from. Similar as ES mehod, L is given by a cerain weigh of acual daa and cerain b slope of he rend in each ime period. The slope is also a weighed sum of acual slope and prediced slope. Hol s mehod can also be easily implemened. The b = Y Y L iniial values can be given by 2 and = Y. The predicion of all monhs can be generaed by equaion L (5), and in he equaion (5), and b are given by equaion (3) and (4). α and β are he decision variables o choose in order o minimize he RSS. The whole model ca be implemened in Excel, and solved by Excel solver. Min RSS By changingα, β Subjec o α, β The opimal α and β ha minimizes RSS are α* =.4, β* =. The minimum RSS in he Hol s mehod is 2.38, which is lower han he RSS in he ES mehod. igure2. Acual Sales and orecas using Hol s Linear Mehod igure 2 shows he acual and forecasing daa using Hol s mehod. We can see ha he forecas can race he increasing rend much beer han ES mehod. Since he new inroduced produc has increasing rend in he daa. Hol s mehod can fi he daa beer han ES mehod, hus decreases he RSS []. The pros and cons of Hol s mehod are summarized as follows: Pros: Incorporaes rends in he forecas Can be efficien compared o oher mehods even wih less daa Cons: Purely based on hisorical ime-series daa, canno inpu explanaory facors, e.g. quoes, seasonaliy, ec. Ouliers can significanly impac he forecas C.ARMA wih Linear Trend Since he daa displays an increasing rend, ARMA model wih linear rend is esed o see wheher i can furher improve RSS. irs, he original daa is fied ino a linear rend line. The bes fied linear rend line is 578

3 igure shows he original sales daa fiing he linear rend line. daa) s impac in he sales daa forecasing. Causal facors forecasing considers inpu facors, and uses hem o improve he forecasing accuracy. Afer fiing he linear line, he daa model becomes A. Original daa regression = In his mehod, we direcly regress he acual sales daa. Nex, he residual is wih respec o quoes daa (wih, 2, 3, 4, 5 monhs prior o fied ino ARMA model. Using Malab code, we found ha he sales monh) and seasonaliy index. The seasonaliy ARMA(4,2) is he adequae model. index is implemened by giving dummy variables o = represen + a.799 each a monh. +.8 The aresul 2 proves ha he only significan facor is he quoes daa 3 monhs prior o he sales monh), wih -sa=3.3. All oher quoes daa and seasonaliy daa are insignifican. The four characerisics roos are: The regression forecasing model is λ,2 =.89±.25 i, = Q 3 (7) λ3,4 =.247 ±.737i Q Here, is he quoe daa a ime, and Q 3 is he quoes daa wih 3 monhs prior o he sales monh. No seasonaliy is observed in his model. igure 5 gives he acual daa and forecas daa using forecasing model (7). The RSS is 3.4, which is higher han all he previous mehod. rom igure 8 we observe ha he forecas displays an over-forecas in he earlier monhs and under-forecas in he laer monhs. This is because ha he regression mehod does no consider he increasing rend. Original Daa Regression orecas vs. Acual Sales 3 igure 3. i Acual Sales ino a Trend Line 25 2 igure 4 gives he acual daa and forecas daa comparison using ARMA(4,2) plus linear rend model, as described in igure 3. The RSS is 2.9, which is lower han ES mehod, bu higher han Hol s mehod and ARMA wih no rend model. This resul ells ha he linear rend does no help in he sales predicion for ARMA model [7]. Unis 5 5 eb-8 Mar-8 Oc-8 Nov-8 Dec-8 Jan-9 eb-9 Mar-9 Apr-9 May-9 Jun-9 Jul-9 Aug-9 Sep-9 Oc-9 Nov-9 Dec-9 Jan- eb- Monhs Acual Sales orecas igure 4. Acual Sales and Linear Trend + ARMA Model orecas III. CAUSAL ACTORS ORECASTING In above sessions, we forecas purely based on hisorical daa, bu do no consider inpu facor (quoes igure 5 Acual Sales and Original Daa Regression orecas B. Trend and residual regression As we observed in he previous session, he regression mehod gives he wors RSS, and one reason is ha he regression does no consider he increasing rend. In his session, I firs fi he daa o a linear rend, as wha has been done in he ARMA + Linear Trend session. Then I ake he residual, and perform a regression of he residual wih respec o he quoes daa and seasonaliy [8]. As menioned, afer fiing he linear rend line, he Y model becomes = + +. Nex, regress wih respec o quoes daa (wih, 2, 3, 4, 5 monhs prior o he sales monh) and seasonaliy index. The resul shows ha he only significan facor is he quoes 579

4 daa 3 monhs prior o he sales monh), wih -sa=7.. All oher quoes daa and seasonaliy daa are insignifican. The residual regression forecasing model is = Q 3 So he complee residual regression plus linear rend model is = Q 3 igure shows he acual and forecas daa using residual regression plus linear rend mehod. The RSS is.85, which is much lower han all he previous mehods. This is because his mehod no only considers he increasing rend, bu also considers he inpu facors impac. Unis 29 9 Residual and Trend orecas vs. Acual Sales 9 - eb-8 Mar-8 Oc-8 Nov-8 Dec-8 Jan-9 eb-9 Mar-9 Apr-9 May-9 Jun-9 Jul-9 Aug-9 Sep-9 Oc-9 Nov-9 Dec-9 Jan- eb- Monhs Acual Sales Residual+Trend orecas igure. Linear Trend + Residual orecas C. ARMAV wih Linear Trend Since Trend + Residual Regression mehod can grealy improve he forecasing accuracy, my hypohesis is ha vecor ARMA (ARMAV model) wih Linear Trend may work even beer han Trend + Residual Regression model. The reasons are as follows: ARMAV wih Linear Trend mehod considers he increasing rend; ARMAV has a more complicaed algorihm han regression, hus can capure more inpu facor impacs [9]. As menioned, he model wih linear rend is = Nex, I model he residual Q using ARMAV model, and make quoes daa as inpu facor. Due o he limiaion of daa sample size, I canno run full ARMAV model. Therefore I limied he parameer of ARMAV o ARMAV(2,2,) model. The ARMAV model ha I found is = Q 3.23Q 4+ a a igure demonsraes he acual daa and forecas daa using ARMAV plus linear rend model. The RSS is.7, which is he lowes among all he mehods. This resul aligns wih our inuiion ha ARMAV + Linear Trend mehod is he bes mehod []. ARMAV + Linear Trend orecas vs Acual Sales Unis eb-8 Mar-8 Oc-8 Nov-8 Dec-8 Jan-9 eb-9 Mar-9 Apr-9 May-9 Jun-9 Jul-9 Aug-9 Sep-9 Oc-9 Nov-9 Dec-9 Jan- eb- Mar- Monhs ARMAV+Trend orecas Acual Sales igure 7. ARMAV + Linear Trend orecas IV. METHODOLOGY AND RESULTS In his paper, I colleced 2-year monhly sales daa for a consumer elecronics produc. Using -2 daa o forecas he fuure demand is very common o consumer elecronics indusry. Anoher se of daa ha I colleced is he 27 monhs quoe daa. Quoe daa is he number of quoes ha he sales people sen ou each monh. Some quoe daa end up wih an acual sales even, bu oher quoes may be los. Therefore, quoe daa gives an indicaion of final sales daa, bu is no compleely correlaed o he sales daa. In addiion, quoes daa may no have immediae impac o sales daa []. The marke may demonsraes sales daa be impaced by he quoes daa which sen ou one or several monhs ago. The purpose of his projec is o idenify he bes sales forecasing model considering hisorical sales daa and quoe daa [2]. igure 8 and 9 give he acual sales daa and quoe daa. Unis Acual Sales Oc-7 Nov-7 Dec-7 Jan-8 eb-8 Mar-8 Oc-8 Nov-8 Dec-8 Jan-9 eb-9 Mar-9 Apr-9 May-9 Jun-9 Jul-9 Aug-9 Sep-9 Oc-9 Nov-9 Dec-9 Jan- eb- Monhs igure 8. Acual Sales Daa igure 9. Quoes Daa In his paper, I would like o explore muliple forecasing models, and compare he RSS of hese models. 58

5 Tradiional forecasing mehodologies can be divided ino wo big caegories: Time-Series orecasing and Causal acors orecasing. In his paper, I will inroduce several models in each caegory, and compare he forecasing accuracies for hose models. The following liss he models ha I am discussing in his paper [3]:. Time series forecasing Exponenial Smoohing (ES) Mehod Hol s Linear Mehod ARMA ARMA wih linear rend 2. Causal facors forecasing Original daa regression Trend and residual regression ARMAV wih linear rend I exend his paper o he models beyond ARMA model. This is because alhough ARMA may provide bes forecasing resul because of is model complexiy, he model iself is no as inuiive as oher mehods, e.g. ES and Hol s mehod. In addiion, ARMA model is difficul o derive wihou programming and relevan sofware (e.g. MATLAB) [4]. Compared o ARMA, oher mehods (e.g. ES, Hol s, Regression) can be easily implemened in Excel. In his consideraion, ARMA may be aracive if i shows significanly advanage in improving forecasing accuracy compared o oher mehods [5]. If ARMA only shows slighly beer resul, companies may sill prefer simple mehods (e.g. ES, Hol s, Regression) because hey are easy o implemen. The RSS of differen models are summarized in Table Table. RSS of Muliple Models Models RSS Exponenial Smoohing (ES) Hol s Linear Mehod ARMA ARMA + Linear Trend Original Daa Regression Residual Regression + Linear Trend ARMAV + Linear Trend V. CONCLUSION.7 This paper sudies differen forecasing models, from simple exponenial smoohing model, Hol s linear model, o ARMA model, ARMAV + Trend model. Based on above resuls, we can give he following 5 conclusions:. ARMAV + Linear Trend model and Residual Regression + Linear Trend model consider boh he rend of he daa and he inpu facor (quoes daa). They give significanly beer forecasing accuracies. 2. ARMAV + Linear Trend model gives bes forecasing accuracy because ARMAV is a more comprehensive way o model inpu oupu relaionships han muliple regression. 3. In he ime-series model wihou inpu facors, boh ARMA model and Hol s mehod give good resuls. However, Hol s mehod is very simple o undersand and easier o implemen. 4. The forecas accuracy may improve if we have more daa for ARMA model. 5. ARMA wih linear rend does no improve he forecasing accuracy. In conclusion, o forecas new produc sales wih limied pas daa poins, ARMAV and linear rend model is he bes model. This is because his model can use inpu facors o compensae for he less accurae forecas due o limied daa in new produc sales forecasing. ACKNOWLEDGMENT This work is suppored by compuing Science by The Communicaion of China (No.NG44), he Naional Naural Science oundaion of P. R. China (No. 9727) and parly suppored by Program for New Cenury Excellen Talens in Universiy (NCET-9-79). REERENCES [] H. L. Willis and J. V. Aansoos, "Some unique signal processing applicaions in power sysem planning", IEEE Trans. Acous., Speech, Signal Processing, vol. ASSP-27, 979,pp [2] C. Komninakis, "A fas and accurae Rayleigh fading simulaor," in IEEE Globecom, vol., 23, pp [3] W.ie, L.Yu and S.Y.u, A new mehod for crude oil price forecasing based on suppor vecor machines, LECTURE NOTES IN COMPUTER SCIENCE, 3994, pp.44-45, June,2. [4] E.Williams, Energy inensiy of compuer manufacuring: hybrid assessmen combining process and economic inpu- oupu mehods. Environmenal science echnology. 24. [5] E.Williams and R.Kuehr; "Today's markes for used PCs - and ways o enhance hem." In: In: Kuehr, R. and Williams, E., Eds. 23. []. A. Sanhakumaran and V. Thangaraj, A Single Server Queue wih Impaien and eedback Cusomers, Vol., pp. 7-79, June,2. [7].Iravani and B.Balcoglu, On Prioriy Queues wih Impaien Cusomers, Vol. 58, pp ,July,28. [8] L.M. Liu, and Z. Kong, Daa Mining in Sales orecasing[j], Business Times, 27, pp.8-9. [9] C. He, Design and Technique research on ETL sysem[j], CompuerApp licaions and Sofware, 29, pp [] Allen, D.E., S. Cruickshank and N. Morkel-Kingsbury, "A Commen on 'The Informaion Conen of Earnings and Prices: A Simulaneous Equaions Approach' by W.H. Beaver, M.L. McAnally and C.H.Sinson ( 997)", Working Paper, School of inance and Business Economics, Edih Cowan Universiy, and School of Business and Economics, Monash Universiy. [] C.Lee, and Tsai,, "The ime-series relaion beween monhly sales and sock prices", Proceedings of he 9h Join Conference on Informaion Sciences. [2] G.Agrawal, Shared Memory Parallelizaion of Daa Mining Algorihms: Techniques, Programming Inerface Knowledge and Daa Engineering,25. [3] H.Du,B.Zhang and D.. Chen. Design and acualizaion of SOAbased daa mining sysem. Compuer-Aided Indusrial Design and Concepual Design, 28. [4] M.Quzzani and A.Bougueaya. Efficien Access o Web Services[J].IEEE Inerne Compuing, 24. [5] G.Aposolikas, On- line RBNN based idenificaion of rapidly imevarying nonlinear sysems wih opimal srucureadapaion.-j] Mahemaics and compuers in Simulaion,