Applying the Theta Model to Short-Term Forecasts in Monthly Time Series

Transcription

1 Applyng he Thea Model o Shor-Term Forecass n Monhly Tme Seres Glson Adamczuk Olvera *, Marcelo Gonçalves Trenn +, Anselmo Chaves Neo ** * Deparmen of Mechancal Engneerng, Federal Technologcal Unversy of Paraná, Va do Conhecmeno, km 1, Pao Branco, Brazl + Deparmen of Elecrcal Engneerng, Federal Technologcal Unversy of Paraná, Va do Conhecmeno, km 1, Pao Branco, Brazl ** Deparmen of Sasc, Federal Unversy of Paraná, Cenro Polécnco - Jardm das Amércas, Curba, Brazl Emal: glson@ufpr.edu.br; marcelo@ufpr.edu.br; anselmo@ufpr.br Absrac Ths paper addresses demand forecasng for monhly daa usng he unvarae Thea mehod. The model s relavely recen and was developed by Nkopoulos and Assmakopoulos (000). I s based on he concep of modfyng he local curvaures of he me seres, obaned by a hea () coeffcen. The cenral dea s o decompose he me seres no a leas wo hea lnes L() represenng a long erm perod and he oher, a shor-erm one. The Forecas s a combnaon of he forecass obaned by adjusng he hea lnes acheved n he decomposon. The model was appled o he hsorcal record of four Producs from he mechancal meal secor usng L( = 0) and L( = ). The choce of hese wo values for he (0 and ) coeffcens s he smples case for he model, whch was used n he M3-compeon of Makrdaks and Hbon (000). The forecass were compared wh radonal mehods and he model performed well, as presened average MAPE s of 5.99%, 14.44%, 16.53% and 7.78% respecvely, for he four Producs suded over he las hree monhs whch were separaed for valdaon. Keywords: demand forecasng; me seres; decomposon; combnng forecass; hea model. 1 Inroducon One of he major dffcules of he secor ha manages he area of maerals n a company les n he need o ake decsons abou fuure acves. Normally here s a hsorcal record of daa, and aemps are made o forecas daa for he near fuure, and he shor and medum erm. Generally, n ndusral envronmens, demand forecasng and sales of producs s necessary and fundamenal for maerals plannng and sraegc and operaonal decsons. The Demand Forecas s he bass for he sraegc plannng of producon, sales and fnances of any busness. From hese daa, companes can draw up plans for capacy, cash flow, sales, producon and socks, manpower, purchases and so forh. In he modern world companes face ferce compeon. Wh ha hey need o seek more nformaon abou he scenaros hey may face. The use of forecass assss admnsraors n decson makng, especally n unceran marke. Sascal ools are of grea mporance for hs ype of acvy, no beng allowed smply nformal predcons. The forecasng models based on me seres are hen recommended. Beer forecass allow he creaon of plannng more conssen and relable. Medum- and small-szed busnesses do no always have he human resources and sofware avalable ha would enable hem o use me seres models nvolvng grea deph sascs. For hs reason, companes fnd dffcul o mplemen he more complex forecasng models of me seres. In hs conex, smpler models whch are known for her effcency are desred and supply hs need. In addon o smoohng models, a varaon known as he Thea Model (Assmakopoulos; Nkolopoulos, 000) s presened as a good alernave. The smples case of he hea model nvolves he decomposon of he me seres no wo perods. A long-erm perod, whch seeks o nvesgaes he rend from he enre hsorcal record. A second shor ID5.1

2 ICIEOM 01 - Gumarães, Porugal perod nvesgaes he laes changes. Based on hs cenral dea, he forecas s mached by he endency of he long-erm wh he local changes whch have been occurrng a he presen me. Ths sudy ses ou o dscuss he hea model, by seekng o prove s performance, and by makng forecass n sales seres, whch were obaned from a company n he mechancal meal secor. To prove hs, we used hsorcal daa on monhly sales of producs from a company company whch handles sanless seel producs. Employng an elecronc spreadshee, here was a predcons performance comparson beween model hea and radonal models of me seres forecasng, sad auomac. The forecas horzon s hree monhs. As a performance creron, we used he absolue average percenage error (MAPE). Secon 1 shows why he ssue s mporan; Secon gves a bref heorecal framework on he subjec. The hea model s descrbed n Secon 3. Secon 4 presens he mehodology and he applcaon n he company s descrbed n Secon 5. The fndngs are repored n Secon 6. Theorecal Framework Tme seres are observaons ordered n me, a sequence of values ha do no follow a non-random order (Moren; Toló, 004). The analyss of me seres s based on he hypohess ha he fuure s a connuaon of he pas, a leas of he recen pas, n whch he rends of growh or declne observed should reman n he fuure, as well as he seasonaly or cyclcaly observed n he pas. Ths can be consdered a emporal model (Corrêa; Ganes; Caon, 001). The represenaon of me seres s normally done by consderng he me seres (a realzaon of a sochasc process, namely, processes conrolled by probablsc laws), such as X, n whch observaons are recorded a dscree nsans and equally dsrbued, wh a noaon of X, X... X. The classcal form 1 n of wrng a seres s: X f ( ) (1) a where f () s a compleely deermned funcon (a sysemac, deermnsc par) and a s a random sequence, ndependen of f (), beng he case ha he random varables a are no correlaed and have a zero mean and consan varance. The random varable a s also called whe nose when he dsrbuon s normal,.e., a ~ N(0, a ). Many of he properes observed n a me seres X can be capured and ake on he followng form of classcal decomposon follows: X T S C a () n whch T s he componen of rend; S he seasonal componen; C s a cyclcal componen and a s a random or nose componen, whch s he non-explaned par whch s expeced o be purely random. The seasonal and cyclcal componens are repeaed a each fxed nerval, n whch perodc varaons can be capured by hese componens. Gven a se of me seres observaons colleced up o he nsan and a model ha represens hese phenomena, he forecas of he value of he seres a he me h seres can be obaned. Several procedures have been drawn up for me seres decomposon. Each mehod seeks o remove he non-observable componens as precsely as possble. These procedures, afer each of he componens has been denfed, enable he behavor of he seres o be beer undersood and hus enable more accurae values o be forecas (Wheelwrgh; Makrdraks, 1985). There are wo man objecves n me seres analyss. The frs s o denfy he naure of he phenomenon generang he sequence of observaons. The second s o forecas fuure values of he me seres. To acheve hese goals, here s a need o denfy he paern ha generaes he daa of he sequence ID5.

3 Applyng he Thea Model o Shor-Term Forecass n Monhly Tme Seres observed. Havng denfed he paern of behavor, s possble o exrapolae he paern denfed and predc fuure evens (Sasof, 007). Generally wha s done s o denfy he srucure of he process generang he seres, whn a class of predefned models, and under some condons he parameers are esmaed and forecass made. Or moreover, he sequence s modeled accordng o he premses of sofenng he varaon and adjusng he updaed rend n me and hen fnally he forecass are made. The mehods of me seres forecasng can be dvded no wo groups: auomac ones, whch can be drecly appled wh he ad of a compuer; non-auomac ones, whch requre he nervenon of a specals so ha hey can be appled (Moren; Toló, 004). The man ones are: movng average, smple exponenal smoohng, lnear exponenal smoohng and seasonal exponenal smoohng and Wner s lnear. Among he auomaed mehods, he exponenal smoohng models sand ou because hey are smple o mplemen and he good resuls acheved. Among he non-auomac ones, whch requre he user o have a greaer sascal base and knowledge, menon should be made of he auoregressve movng average models (AR, MA and ARMA), auoregressve negraed movng averages (ARIMA), Kalman and AEP flers, ARARMA models, and mulvarae ARMA models (MARMA). Among he nonauomac ones, ARIMA models are he man represenaves (Moren; Toló, 004). An opporune revew of he leraure on exponenal smoohng, from he orgnal sudes by Brown and Hol, can be found n Gardner (1985) and laer updaed n Gardner (006). Box and Jenkns (1976) presen he ARIMA mehodology, whch s wdely used n me seres analyss, and has grea promnence gven ha has been wdely announced and s flexble. Goojer and Hyndman (006) revewed he pas wenyfve years of me seres forecass, and hs sudy s a prmary source of references for sudes on he subjec, besdes whch hey found ha here had been remendous progress n several areas, bu many opcs sll requre furher sudes. The accuracy of forecas s no based only on he forecas horzon desred, bu s also srongly nfluenced by he characerscs of he observaons of he seres under sudy. The combnaon of forecass of more han one model ncreases relably and reduces large devaons n he projeced values, no necessarly beng he mos sophscaed models ha oban he bes resuls (Wheelwrgh; Makrdraks, 1985). In varous sudes here s conroversy beween he resuls obaned wh smple models and hose obaned usng more complex ones whch requre users o have greaer knowledge. Dependng on he applcaon and who wll use he model, he smples mehod s he one ha has mos o recommend (FILDES e a.l, 1998) (Makrdaks; Hbon, 000). On concludng he M3-Compeon, Makrdaks and Hbon (000) confrmed ha mehods hey consdered were smpler obaned good resuls n he compeon and n some cases were even beer n relaon o ohers ha are regarded as beng sascally sophscaed such as ARIMA and ARARMA. In hs conex he hea model s presened as a quck and smple alernave. Ths can be consdered an auomac mehod, snce he choce of he coeffcens s fxed. The hea model nvolves he decomposon of he me seres no a se of new seres. The resulng seres are called hea lnes L(), and manan he average and he slope of he orgnal daa, bu no her curvaures. Ths s based on modfyng he local curvaure of a me seres seasonally adjused by he hea coeffcen ().The coeffcen s appled drecly o he second dfference of he me seres. Each of he hea lnes s exrapolaed separaely and he forecass are combned wh equal weghs (Assmakopoulos, Nkolopoulos, 000). Dfferen combnaons of hea lnes can be used for each forecas horzon. Hyndman and Koehler (006) clam ha he hea model n parcular presened very good forecass n he M3-Compeon (Makrdaks; Hbon, 000), as obaned one of he bes performances among all oher models (Goojer; Hyndman, 006). Hyndman and Bllah (003) consder ha he hea model s equvalen o smple exponenal smoohng wh drf (SES-d). The drf consdered corresponds o half he value of he slope of a lnear regresson adjused o he daa. Thus, provdes a form of conrol ha lms he possbly of he model producng naccurae forecass (Koehler, 006). Nkolopoulos and Assmakopoulos (005) dsagree wh hs ID5.3

4 ICIEOM 01 - Gumarães, Porugal consderaon, and sae ha he hea model s more generc han exponenal smoohng. The model s descrbed n he followng secon. 3 The hea model The challenge of he mehod proposed was o ncrease he degree of explong he useful nformaon embedded n he daa before applyng a forecasng mehod. Ths nformaon from an nuvely pon of vew has shor and long erm componens. These componens are denfed n he hea model and are hen exrapolaed separaely (Assmakopoulos; Nkopoulos, 000). The model s based on he concep of modfyng he local curvaures of he me seres. Ths aleraon s obaned by usng he hea co-effcen, or smply, whch s appled drecly o he second dfference of he me seres. The X daa of he seres can be wren n as: 1 " X X1 ( 1).( X X1) ( ). X 1 (3) where n he me : X X (4) " X 1 X The Y pons of a hea lne by defnon are: 1 Y1 ( 1).( Y Y1 ). ( ). " X 1 Y (5) The problem falls back on mnmzng quadrac errors, namely: mn( e ) mn( ( Y X ) ). (6) As shown n Assmakopoulos and Nkopoulos (000) he resul of hs mnmzaon mples ha boh he mean and he slope of he hea lnes are equal o hose of he orgnal seres. Dfferen combnaons of hea lnes can be used for each forecas horzon. One of he smpler cases s he combnaon of wo hea lnes wh =0 and =,.e., he daa can be decomposed as follows: 1 ( L( 0) L( )) X (7) The L( = 0) s he lnear regresson of he daa. Ths s shown n Assmakopoulos and Nkopoulos (000). The L( = ) can be obaned n a smplfed form usng he prevous equaon. Thus: L( ). X L( 0) (8) The L( = 0) descrbes he seres as a lnear rend. Snce he L( = ) doubles he local curvaures by exendng he shor-erm acon. In he forecass L( = 0) s exrapolaed by a lnear rend and L( = ) s exrapolaed by smple exponenal smoohng (SES). The smple combnaon of wo forecass gves he fnal Xˆ ( h) forecas for he hea model, namely: 1 Xˆ ( h) ( Lˆ( 0) h Lˆ( ) h ) (9) where L ˆ( ) lnes. 0 h and L ( ) h ˆ are he forecass by lnear regresson and SES of he respecve hea ID5.4

5 Applyng he Thea Model o Shor-Term Forecass n Monhly Tme Seres Fgure 1 shows he lnes hea and he respecve forecass and real daa of seres 30 of he M3- compeon. Use of he coeffcens (0 and ) was made o produce he forecass esmaes n he M3- Compeon of Makrdaks and Hbon (000). The seps followed were: Uns Sep 0: es of seasonaly. The creron used was he -es for auo-correlang daa wh a oneyear lag (1 observaons for monhly seres). The T value was calculaed compared wh = o a 10% sgnfcance level Sep 1: seasonal decomposon of he daa by he classcal mulplcave mehod Sep : decomposon of he seres no L( = 0) and L( = ) Sep 3: exrapolaon. The L( = 0) s exrapolaed by lnear regresson and L( = ) by SES Sep 4: combnaon. The forecass produced n he exrapolaon of he hea lnes are combned wh equal weghs Sep 5: The seasonal ndces obaned n he decomposon of sep 1 are ncorporaed no he forecass Forecass Tme Ν0030 Fgure 1: Seres 30 of he M3-compeon and hea forecass. Source: Assmakopoulos and Nkopoulos (005) Assmakopoulos and Nkopoulos (005) sugges how he model wh L( = 0) and L( = ) can be appled usng an elecronc spreadshee: Sep 0: seasonal decomposon of he daa by he classcal mulplcave mehod, f necessary Sep 1: Apply he lnear regresson of he daa L( = 0), prepare he regresson lne and forecass Sep : Prepare he values of L( = ) usng L( ). X L( 0) Sep 3: exrapolae he L( = ) wh SES, opmzed by he Mcrosof Excel Solver) or by anoher smpler mehod such as movng averages Sep 4: combne forecass wh equal weghs obaned by lnear regresson and SES. θ=0 θ= 4 Mehodology In order o apply he model, he choce of coeffcens, due o he good performance obaned n he M3-Compeon, was resrced o wo hea lnes L( = 0) and L( = ). The oher mehodologcal seps and decsons adoped n hs sudy were: Seasonaly: no seasonaly ess were carred ou. Daa were analyzed graphcally where a check was made ha here was no sgnfcan seasonaly n he hsorcal daa used Sequence of applcaon of he mehod: seps 1-4 were followed as descrbed n em 3.1 for he elecronc spreadshee Measuremen of accuracy: o evaluae he forecass obaned he creron of MAPE (mean absolue percenage error) was adoped. Alhough some auhors crcze hs creron n he M3- Compeon, hs was he man creron, wh he advanage of beng able o spulae an average MAPE beween he seres analyzed, whch show order of magnude dfferences ID5.5

6 ICIEOM 01 - Gumarães, Porugal n x xˆ x MAPE (10) n Comparson wh oher mehods: he forecass obaned were compared o auomac mehods whch were adjused by usng sascal sofware 5 Applcaon The meal mechanc secor company provded he hsorcal record monhly sales of four producs A, B, C and D. The company s regarded as beng of medum o large, locaed n Parana sae, souhern Brazl. The company operaes n he marke for over 0 years. I has approxmaely 300 employees. Manufacures and sells fnshed producs o he lnes of resdenal kchen furnure, servng he Brazlan marke and expor. Is producs are geared for boh he hgher socal classes and for he mos popular. Was no obaned permsson from he company o dsclose more nformaon abou s srucure and also of s producs. Producs A, B and C are radonal and have long been on he marke. They are shown n Fgure. Produc D has a more recen hsorcal record and s shown n Fgure Uns Produc A Produc B Produc C Tme Fgure : Hsorcal record of Producs A, B and C from January/005 o March/007. Source: company suded Uns Produc D Tme Fgure 3: Hsorcal record of Produc D from January/005 o March/007. Source: company suded ID5.6

7 Applyng he Thea Model o Shor-Term Forecass n Monhly Tme Seres The hsorcal record suppled by he company covers 30 monhs sarng n Jan/005. The monhs from Aprl o June 007 were separaed for valdaon and comparson wh auomac mehods adjused by he smalles roo mean square error (RMSE). Tables 1, and 3 show, respecvely, he real values, he forecass usng he hea model and he forecass usng oher auomac mehods adjused by he hsorcal daa. Table 1: Real demand values for he Producs from Aprl o June 007 Perod Produc A Produc B Produc C Produc D 04/ / / Source: Auhors Tables and 3 enable o be affrmed ha he hea model, n s smples applcaon, wh L( = 0) and L( = ), obaned resuls ha are a leas equvalen o hose of oher auomaed mehods. In a comparson wh he MAPE s of he four Producs, an average of 11.19% was obaned usng he hea model agans 1.07% usng auomac mehods auomac adjused by he smalles RMSE. Table : Forecass and MAPE s obaned usng he hea model Perod Produc A Produc B Produc C Produc D Forecas MAPE Forecas MAPE Forecas MAPE Forecas MAPE 04/ % % % % 05/ % % % % 06/ % % % % Average MAPE 5.99 % % % 7.78 % Source: Auhors Table 3: Forecass and MAPE's obaned wh auomac mehods adjused by he smalles RMSE Perod Produc A Produc B Produc C Produc D (lnear endency) (lnear endency) (lnear endency) (quadrac endency) RMSE = RMSE = RMSE = RMSE = Forecas MAPE Forecas MAPE Forecas MAPE Forecas MAPE 04/ % % % % 05/ % % % % 06/ % % % % Average MAPE % % % 9.00 % Source: Auhors ID5.7

8 ICIEOM 01 - Gumarães, Porugal The hsorcal daa of he produc A and he forecass made for mes 8 o 30, respecvely X (1), 7 X 7 (), X 7 (3), are shown n he las hree columns of able 4. L( = ) was exrapolaed by smple exponenal smoohng (SES) wh α = 3,44804E-03. Table 4: Daa used n he analyss of demand for produc A (X ) L( = 0) (y=85, ,) L( = ) L( ). X L( 0) , , , , , , , , , , ,58 566, , , , , , , , , , , , , ,0 7037, , , , , , , , , , , , , , , , , , , , , , , , , ,1 5784, , ,44 X (1) , ,59 X 7 () , ,59 X 7 (3) , ,59 Source: Auhors ID5.8

9 Applyng he Thea Model o Shor-Term Forecass n Monhly Tme Seres For nsance, he forecasng for he momen 8, beng n me 7, can be obaned by equaon (9) and compued by (11): 1 X 7 (1) (9180, ,59) = 8411 (11) To he effec of he mehod see Fgure 4 whch shows he real seres of produc A and he hea lnes L( = 0) and L( = ) and forecasng (daa n Table 4) Uns Produc A L(Θ = 0) L(Θ = ) Tme Fgure 4: Produc A seres and hea forecass. Source: company suded 6 Conclusons The am of hs sudy whch was o apply he hea model was acheved. The hea mehod wh wo lnes (= 0 and = ) s smple and does no requre exensve ranng or knowledge before can be appled. I s flexble because can be appled, wh good resuls, n saonary monhly seres or wh a rend wheher or no combned wh seasonaly. I also brngs he advanage of one beng able o apply by usng a spreadshee. The forecass obaned were compared wh radonal mehods and he model performed well, as presened average MAPE s of 5.99%, 14.44%, 16.53% and 7.78% respecvely for he four producs suded n he las hree monhs separaed for valdaon. The MAPE's obaned n he forecass confrm he resuls of he M3-Compeon of Makrdaks and Hbon (000), whch suggess he hea mehod s a fas and effcen mehod for obanng good forecass. A dsadvanage ha may be ced s he lack of confdence nervals for he forecass n he sudy by Nkopoulos and Assmakopoulos (000). Hyndman and Bllah (003) sugges how confdence nervals can be bul n. One suggeson for furher sudes s o use more han wo hea lnes and (or) wh dfferen values of = 0 and =. Ths should be carefully analyzed n praccal applcaons o ensure ha he advanage of he smplcy of he mehod s no dmnshed. Acknowledgemens We are graeful o CAPES and he Araucara Foundaon, for her fnancal suppor whch enabled hs sudy o be conduced. ID5.9

10 ICIEOM 01 - Gumarães, Porugal References ASSIMAKOPOULOS, V.; NIKOLOPOULOS, K. The hea model: a decomposon approach o forecasng. Inernaonal Journal of Forecasng v.16 p , 000. ASSIMAKOPOULOS, V.; NIKOLOPOULOS, K. Fahomng he Thea Model. 5h Inernaonal Symposum on Forecasng, ISF, San Anono, Texas, USA, 005. BOX, G. E. P.; JENKINS, G. M. Tme Seres Analyss: forecasng and conrol, Ed. Holden Day, CORRÊA, H. L.; GIANESI, I. G. N.; CAON, M. Planejameno, Programação e Conrole da Produção. 4h ed., São Paulo: Alas, 001. FILDES, R.; HIBON, M.; MAKRIDAKIS, S.; MEADE N. Generalsng abou unvarae forecasng mehods: furher emprcal evdence. Inernaonal Journal of Forecasng, v.14, p b, GARDNER, E. S., Jr. Exponenal smoohng: The sae of he ar. Journal of Forecasng, vol. 4, p. 1 8,1985. GARDNER, E. S., Jr. Exponenal smoohng: The sae of he ar Par II. Inernaonal Journal of Forecasng, vol., p , 006. GOOIJER, J.G.; HYNDMAN R.J. 5 years of me seres forecasng. Inernaonal Journal of Forecasng, Vol, p , 006. HYNDMAN, R.J.; BILLAH, B. Unmaskng he Thea mehod. Inernaonal Journal of Forecasng, v.19 (), p , 003. HYNDMAN, R.J.; KOEHLER, A.B. Anoher look a measures of forecas accuracy. Inernaonal Journal of Forecasng, v., p , 006. KOEHLER, A. Dscusson. Inernaonal Journal of Forecasng, v., p , 006. MAKRIDAKIS, S.; HIBON, M. The M3-Compeon: resuls, conclusons and mplcaons. Inernaonal Journal of Forecasng, v. 16, p , 000. MORETTIN, P.A.; TOLOI, C.M.C. Análse de Séres Temporas. São Paulo: Edgard Blücher, 004 NIKOLOPOULOS, K.; ASSIMAKOPOULOS, V. Fahomng he Thea model. 5h Inernaonal Symposum on Forecasng, ISF, San Anono, Texas, USA, 005. STATSOFT, INC. Elecronc Sasc Texbook. Tulsa, OK: Sasof,. Avalable a: hp:// Consuled on 10/08/007. WHEELWRIGHT, S. C.; MAKRIDAKIS, S. Forecasng Mehods for Managemen. 4h ed., New York: John Wley & Sons Inc., ID5.10