In this paper the innovations state space models (ETS) are used in series with:

Transcription

1 Time series models for differen seasonal paerns Blaconá, M.T, Andreozzi*, L. and Magnano, L. Naional Universiy of Rosario, (*)CONICET - Argenina Absrac In his paper Innovaions Sae Space Models (ETS) are used o fi series wih: 1) a single seasonal period and 2) muliple seasonal periods. Sales daa of 1) axles and 2) suspensions of a meallurgical company from Alvear (Sana Fe, Argenina) are analyzed as series wih a single seasonal paern. To analyze series wih complex seasonal paerns, wo series were used: 3) vehicles passing daily hrough he oll booh on he Rosario-Buenos Aires highway (Argenina), and 4) daily average gas consumpion per cusomer measured in m 3 in Las Rosas (Sana Fe, Argenina). The main purpose of hese comparisons is o obain prediced values wih an accepable error and a conrollable level of uncerainy. Anoher reason for hese comparisons is ha Argeninean series show more variabiliy han he series in hose counries wih more sable developmen. In series wih a single seasonal paern, ETS models have a good possample forecasing performance. The ou-of-sample five-sep-ahead average forecas error is 9.4% for axles and 6.9% for suspensions, wih a conrollable level of uncerainy. BATS (Exponenial Smoohing Sae Space model wih Box-Cox ransformaion, ARMA errors, Trend and Seasonal Componens) and TBATS (Trigonomeric Exponenial Smoohing Sae Space model wih Box-Cox ransformaion, ARMA errors, Trend and Seasonal Componens) are inroduced o forecas complex seasonal ime series. The resuls show ha boh ypes of models are suiable o describe and predic he ime series of a daily number of vehicles. The TBATS model has some advanages over he BATS model such as: i) beer goodness of fi (lower AIC), ii) lower ou-of-sample forecas percenage for differen horizons (measured wih MAPE); iii) reducion in compuaion ime o esimae he model, given he smaller number of seed values used. However, for he gas demand daa, he performance of he proposed models is no as good as for axles and suspensions. The BATS model does no show a good fi and, alhough he TBATS model fis he daa well, i provides forecass wih more error han an SSM wih Spline. A possible explanaion for he lower qualiy forecass of he TBATS is ha while he SSM includes climaic variables as explanaory variables, his applicaion of TBATS models does no, and i is known ha climaic variables have much influence on uiliies demand series. However, given he simpliciy of hese models, hey canno be compleely discarded.

2 I. Inroducion There are several approaches o model series wih a single seasonal paern, such as exponenial smoohing (Hol-Winers, 1960), seasonal ARIMA models (Box and Jenkins, 1970), sae-space models (SSM, Harvey, 1989) and he innovaions ETS (Hyndman e al., 2008). However, models for muliple seasonal paerns are no as frequen, o cie some examples: SSM wih spline for daily series (Harvey and Koopman, 1993) exponenial smoohing models for double seasonaliy (Taylor and Snyder, 2009, Taylor 2010) and innovaions sae space models (ETS) for complex seasonal paerns (De Livera e al 2011), BATS (Exponenial Smoohing Sae Space model wih Box-Cox ransformaion, ARMA errors, Trend and Seasonal Componens) and TBATS (Trigonomeric Exponenial Smoohing Sae Space model wih Box-Cox ransformaion, ARMA errors, Trend and Seasonal Componens). In his paper he innovaions sae space models (ETS) are used in series wih: 1. A single seasonal paern and 2. Muliple seasonal paerns Sales daa of 1) axles and 2) suspensions of a mealworking firm in Alvear (Sana Fe, Argenina), which spans from January 2009 o Augus 2013, are analyzed as series wih a single paern. For he analysis of series wih complex seasonaliy, wo series are considered: 3) vehicles passing daily hrough he oll booh on he Rosario-Buenos Aires highway (Argenina) in he period April 22, 2010 o December 31, 2013, and 4) daily average gas consumpion per cusomer measured in m 3 in Las Rosas (Sana Fe, Argenina) in he period March 1, 2008 o Augus 31, The main purpose of hese comparisons is o obain prediced values wih an accepable error and a conrollable level of uncerainy. Anoher reason for hese comparisons is ha Argeninean series show more variabiliy han he series in hose counries wih more sable developmen. ETS, BATS and TBATS models are presened briefly in secion II. Applicaions over four ime series are shown in secion III and concluding remarks are saed in secion IV.

3 II. Mehodology Hol-Winers (HW) exponenial smoohing mehods are widely used o forecas ime series wih a single seasonal paern (addiive or muliplicaive) providing good resuls. However, his framework presens wo weaknesses: models canno be esimaed by maximum likelihood and calculaing predicion inervals is no possible. There are wo superior proposals: Sae Space Models wih muliple sources of error (SSM, Harvey, 1989) and innovaions Sae Space Models wih a single source of error (ETS, Hyndman e al, 2008). The exponenial smoohing mehods menioned above have been sudied in he framework of sae space models. The innovaions SS models include hose underlying he addiive and muliplicaive mehods of Hol Winers. Exending he mehod of HW over a seasonal paern, Taylor (2003) incorporaes a second seasonal componen. When he number of seasonal componens is long, i can be difficul o esimae he parameers and he seed values. Furhermore, if he seasonal period is long, he model obained is likely o be overparamerized. This can be simplified when a seasonal period is a muliple of he oher. The exponenial smoohing model assumes ha he process of whie noise is serially uncorrelaed. This assumpion is no always rue in pracice because i someimes behaves as an AR (1) process. De Livera e al. (2011) propose modificaions o he ETS models in order o include a wide variey of seasonal paerns and solve he problem of correlaed errors. To avoid falling ino nonlineariy problems, hese auhors resriced he models o hose homoskedasic and he Box-Cox ransformaion (Box and Cox, 1964) are used when here is some ype of specific nonlineariy. The model including he ransformaion of Box and Cox, ARMA errors and seasonal paerns can be expressed as follows y y 1, 0, (1.a) log y, 0 T w i 1 1 m i, (1.b) y b s d i1 b d, (1.c) b b b d, (1.d) i i m i s s d, (1.e) d p i d i i i i i1 i1 q. (1.f)

4 y is used o represen he Box and Cox ransformed observaion wih parameer, where y is he observaion a ime. m1, m2,..., m T are seasonal periods, is he local level in he period, b is he long-erm i rend, b is he shor-erm rend in he period, s represens he i-h seasonal componen a ime, d can be an ARMA model (p, q) and he process is a Gaussian whie noise process wih zero mean and 2 consan variance. Smoohing parameers are given by, and i for i 1,..., T. is he damping consan of he rend. This change ensures ha he value of he shor-erm b rend converges on he value b (Longerm rend), raher han on zero. These models are called BATS as an acronym for he key feaures of he model: Box and Cox ransformaion, ARMA errors, rend and seasonal componens. The argumens,, p, q, m1, m2,..., mt are he Box-Cox parameer, damping parameers, ARMA model parameers and seasonal periods respecively. The BATS model is he mos obvious generalizaion of he radiional seasonal innovaions models ha allows muliple seasonal periods. To provide a flexible and parsimonious approach De Livera e. al. (2011) inroduced a rigonomeric represenaion of he seasonal componens based on Wes and Harrison Fourier series form (1997) and Harvey (1989) in he following manner: s i k i j1 i s, (2.a) j, i i i * i i i cos, (2.b) s s s sen d j, j, 1 j j, 1 j 1 i i i i i i cos, (2.c) * * j, j, 1 j j, 1 j 2 s s sen s d where i and 1 i are he smoohing parameers and 2 i 2 describes he sochasic level of he i-h seasonal componen as Then, *i j, j j m i i s. s is sochasic growh of he i-h seasonal componen, needed o describe seasonal changes over ime. The number of harmonics required for he i-h seasonal componen is denoed by k. A new class of innovaions SS model is obained by replacing he seasonal componen in Equaion (1c) wih he rigonomeric formulaion, and hen he measuremen equaion is given by T w i (2.d) y b s d i1 i j,

5 This class of models is called TBATS, where he iniial T refers o rigonomerical ransformaion. If he argumens are lised, he TBATS,, p, q, m, k, m, k,...,, m, k, where each model is wrien: TBATS argumen has he same meaning as in he BATS models, and ki is he number of harmonics for he seasonal componen requires he esimaion of k k k T T s i T. The TBATS model iniial values. This number is generally smaller han he number of seed parameers in BATS models. Anoher advanage is ha he rigonomeric funcion can be used for models wih non-ineger seasonal frequency (eg daily for a year defined as days o conemplae leap years). In a general linear ETS approach, unknown smoohing and damping parameers are esimaed using he sum of squared errors of a Gaussian likelihood. In his conex we also have o esimae he Box and Cox ransformaion parameer, and ARMA coefficiens. The forecas disribuion for he fuure period in he ransformed space, given he final sae vecor and parameers, is Gaussian. The associaed random variable w yn h n has mean E y considering he Box and Cox ransformaion. w n h n The Akaike informaion crierion AIC L 2 j and variance w n h n V y, is used o perform he selecion, L is he likelihood and j is he number of parameers. This model is repeaedly adjused wih a gradual growh o deermine he number of harmonics for each seasonal componen mainaining all oher harmonics consan o achieve he minimum AIC. A wo-sage mehod is used o selec he orders p and q of he ARMA model. Firs of all, a suiable model is seleced regardless of he ARMA model for he residuals, hen he auomaic ARIMA algorihm of Hyndman and Khandakar (2008) is applied o deermine he orders of he residual ARMA model (assuming saionariy). The seleced model is fied again bu wih an ARMA pq, model for he error componen, where ARMA coefficiens are esimaed joinly wih he oher parameers. The ARMA componen is included in he model only if he AIC of he resuling model is lower han he model wihou i. Mean Average Percenage Error (MAPE) is used o esimae he efficiency of he models o forecas h seps ahead. MAPE h 1 h yˆ y h ' ni n ni 100. i1 yni

6 III. Applicaion 1 Firsly, wo ses of sales daa wih a single seasonaliy are analyzed using ETS models. Series 1: Number of axles sold by a meallurgical company Alvear (Sana Fe, Argenina) in he period January 2009 o Augus The series shows increasing rend and seasonaliy. The bes model from Akaike crierion is ETS (A, M d, A), which means addiive error, damped muliplicaive rend and addiive seasonaliy. The decomposiion properly represens he feaures of he ime series, Figure 1. Ou-ofsample five-sep-ahead forecass and he predicion inervals are presened in Figure 2. All acual values fall wihin he predicion inervals. Ou-of-sample error for five-monhs ahead is 9.4%. Figure 1. Decomposiion of he number of axles sold by a meallurgical company from Alvear (Sana Fe, Argenina). January 2009 o Augus Source: Own calculaions based on daa from a privae company. 1 R sofware is used in all he applicaion (Developmen Core Team, 2011)

7 Figure 2. Forecass of he number of axles sold by a meallurgical company from Alvear (Sana Fe, Argenina). Sepember 2013 o January Source: Own calculaions based on daa from a privae company. Series 2: Number of suspensions sold by a meallurgical company from Alvear (Sana Fe, Argenina). January 2009 o Augus Figure 3. Decomposiion of he number of suspensions sold by a meallurgical company from Alvear (Sana Fe, Argenina). Period January 2009 o Augus Source: Own calculaions based on daa from a privae company. This series is more irregular han series 1. I ends o flucuae over ime and i shows more irregular seasonaliy. The bes model according o he Akaike crierion is ETS (A, A, A), which means ha error, rend and seasonaliy are addiive. The decomposiion of he series is presened in Figure 3. As for series 1, all he values fall wihin he predicion inervals

8 excep for he four-sep-ahead one, which is in he upper limi of he range, Figure 4. The average error for five monhs ahead ou of sample, measured wih MAPE, is 6.9%. Figure 4. Forecass of number of suspensions sold by a meallurgical company from Alvear (Sana Fe, Argenina) (Sepember 2013 o January 2014). Source: Own calculaions based on daa from a privae company. Then wo series wih muliple seasonal (complex) componens are analyzed using BATS and TBATS models. Series 3: Toal number of vehicles passing per day hrough he oll booh General Lagos on he Rosario-Buenos Aires highway (Argenina) in he period from April 22, 2010 o December 31, The measuremens were made using a sensor ha couns he number of vehicles. This series shows annual and weekly seasonaliy. The annual seasonal period is no an ineger ( days), hus he shorer seasonal period - he week (7 days)- is no a divisor of he larger period. This seasonal behavior is complex and requires unconvenional models for he analysis. There is a missing value in he series due o a flaw in he measuring insrumen. Tha value is esimaed by inerpolaion. Oher flaws in he insrumen caused ouliers in November In order o ake ino accoun he behavior of he public holidays, a correcion o he daa is done prior o he model esimaion. Firsly, a regression model wih rend, a unique seasonal componen (weekly) and hree variables for he public holidays (F1: public holiday on a weekday; F2: he las day of a long weekend; and F3: he day before a long weekend) are fied o esimae he effec of hese days on he

9 daily number of vehicles passing hrough he oll. Only for his correcion he rend and seasonaliy are considered deerminisic. Then, he esimaed values are correced using he coefficiens obained for each ype of holiday. The models are fied using daa from April 22, 2010 o Ocober 31, 2012, leaving he monhs of November and December o assess he ou-ofsample forecasing performance of he models. Following he models proposed by De Livera e al (2011), TBATS and BATS models are fied: BATS 1, 0.998, p 3, q 1, m 7, m and TBATS 1, 0.999, p 1, q 2, m, k 7,3, m, k , 4, means ha he series does no need ransformaion, 1 indicaes ha he damping is very small and he harmonics in he TBATS model significanly reduce he seed values o be esimaed. The decomposiion of he series of he daily number of vehicles passing hrough he oll booh General Lagos using boh mehods shows a slighly increased level, a slighly declining growh rae and wo seasonal paerns, a weekly and an annual one. (Figures 5 and 6) The BATS model esimaes he componens wih more irregulariy han he TBATS model. Figure 5. Decomposiion of he number of vehicles passing daily hrough he oll booh General Lagos using BATS model. (April 22, 2010 o Ocober 31, 2012). Source: Auhor's calculaions based on daa from OCCOVI.

10 As menioned above, November and December 2012 are lef ou o evaluae he ou-of-sample forecass for boh models. The forecass for he days -November 11 and 12, were no aken ino accoun because hey can be considered ouliers due o a flaw in he sensor ha measures he passage of vehicles. The forecass were correced using he coefficiens obained for he public holidays. Figure 6. Decomposiion of he number of vehicles passing daily hrough he oll booh General Lagos using TBATS model (April 22, 2010 o Ocober 31, 2012). Source: Auhor's calculaions based on daa from OCCOVI Mos of he observed values fall wihin he 95% forecas inervals. Over he las days of December, he variabiliy of he observed values increased considerably and some of hem fall ouside he forecas inerval.

11 Figure 7. Forecass of he number of vehicles passing per day hrough he oll booh General Lagos wih heir respecive 95% predicion inervals for he model BATS (November and December, 2012) Source: Auhor's calculaions based on daa from OCCOVI. Figure 8. Predicions of he number of vehicles passing per day hrough he oll booh General Lagos wih heir 95% predicion inervals for he model TBATS (November and December 2012) Source: Auhor's calculaions based on daa from OCCOVI.

12 Boh models have a good performance, he TBATS model presening some advanages over he BATS model such as: i) lower AIC (AICTBATS = , AICBATS = ), ii) significanly less compuaion ime given he esimaion of less seed values and iii) lower error (MAPE TBATS(30 days) = 9.64% and MAPEBATS(30 days) = 10.66% and MAPE BATS(61 days) = 13.00% and MAPE TBATS(61 days) = 12.79%). Table 1. MAPE of BATS and TBATS models for differen horizons (h). Days (h) BATS TBATS Series 4: Daily average gas consumpion per cusomer measured in m 3 in Las Rosas (Sana Fe, Argenina) during he period March 1, 2008 o Augus 31, To fi he model he wo las monhs are lef ou (July and Augus) o assess he ou-of-sample goodness of forecass. This series also has wo ypes of seasonaliy, weekly (7 days) and annual ( days), so i can be considered as complex seasonaliy. Anoher feaure of his series is ha i is more volaile in winer han in summer due o he use of gas for heaing. This series was previously sudied by Acosa, P. (2013), using a saespace model (Harvey, Koopman, 1993), wih spline. This model considers he annual seasonaliy, and includes explanaory variables such as emperaure and public holidays. The series presens ouliers ha are aken ino accoun for he analysis. The forecass show a beer performance wih a MAPE of 5.25% for 31days and of 5.83% for 62 days. In he presen applicaion he BATS and TBATS models do no include explanaory variables. A possible loss of accuracy due o he use of simpler models is evaluaed. In order o ake ino accoun he paricular behavior of he public holidays, as for Series 3, a prior correcion of he daa was done using a regression model wih rend, a unique seasonal componen (weekly) and a variable for public holidays, o esimae he effec of hese days on he average consumpion of gas (his correcion considered rend and seasonaliy as deerminisic). The observed values were correced using he esimaed coefficiens for he public holidays.

13 Following he models proposed by De Livera e al (2011) presened in secion II, a TBATS 2 model is fied o he daa. The esimaed model is: 1, ˆ 0.971, p 4, q 5, m, k 7,1, m, k , 4, TBATS A ransformaion is no needed ( 1 ), 1indicaes ha he damping is slow and harmonics in he TBATS model significanly reduce he number of seed values o be esimaed. The decomposiion shows increasing level, a growh rae of almos zero and wo seasonal paerns, weekly and annual. Figure 9. Decomposiion of daily average gas consumpion per cusomer measured in m 3 in Las Rosas (Sana Fe, Argenina), for he model TBATS. (November 9, 2012 o November 12, 2012) Source: Own calculaions based on daa from a privae company. 2 The BATS model is no presened because a proper fi was no achieved.

14 Figure 10. Predicions of daily average gas consumpion per cusomer measured in m 3 in Las Rosas (Sana Fe, Argenina) wih he 95% predicion inervals for he model TBATS (November 9, 2012 o November 12, 2012). Source: Own calculaions based on daa from a privae company. Mos of he observed values fall wihin he 95% forecas inervals. Table 2. MAPE of he TBATS model for differen horizons (h). Days (h) TBATS If he values in Table 2 are compared wih hose in Acosa, P. (2013), he value obained here for 30 days is hree imes he oher and he value obained for 62 days is almos four imes he one obained by Acosa, showing he forecasing superioriy of he more complee model. IV. Concluding Remarks This paper discusses he benefis of forecasing using innovaions sae space models for series wih a single seasonal period and for a complex seasonaliy. TBATS and BATS models are used wih Argeninean ime series. The wo ime series of sales (axles and suspensions) of a mealworking firm in Argenina show a good performance, as i was expeced for his ype of series. The average percenage of error for five-monh-ahead ou-of-sample forecass is 9.4% for axles and 6.9% for suspensions, wih a conrollable level of uncerainy.

15 These resuls are consisen wih hose reached by De Livera e.al. (2011), who recommend his ype of models for uiliy demand series. TBATS and BATS models are used for series wih complex seasonal paerns. For he oal number of vehicles passing per day hrough he oll booh General Lagos on he Rosario-Buenos Aires highway, resuls show ha boh ypes of models are suiable for describing and predicing his series. The TBATS model has some advanages over he BATS model such as: i) beer goodness of fi (lower AIC), ii) lower percenage of error in heir ou-of-sample forecass for differen forecas horizons (measured wih MAPE); iii) reducion in compuaion ime o esimae he model, due o he lower number of seed values. However, for he daily average gas consumpion per cusomer measured in m 3 in Las Rosas (Sana Fe, Argenina) during he period March 1, 2008 o Augus 31, 2011, he performance of he proposed models is no as good. The BATS model does no provide good fi and in he case of he TBATS model, alhough i fis he daa well, he forecass have more error han he ones obained using an SSM wih Spline. One possible explanaion for he lower qualiy of forecass using TBATS model is ha no explanaory variables were included in hese models and in his applicaion climae variables have significan influence. Climae variables were included in he SSM approach. However, given he simpliciy of he use of TBATS, i canno be compleely discarded. Fuure research will coninue invesigaing he validiy of he models, boh heoreically and empirically, o improve he qualiy of forecass made in highly flucuaing series. IV. References ACOSTA, P. (2013) Time series models for daily consumpion of gas. Maser Thesis in Applied Saisics, Faculy of Economics and Saisics Sciences, Naional Universiy of Rosario. BOX, GEP, and JENKINS, GM (1970) Time Series Analysis: forecasing and conrol (1 s ed) San Francisco. Holden-Day. DE LIVERA, AM, Hyndman, RJ, and Snyder, RD (2011) Forecasing ime Series wih Complex Smoohing exponenial Using seasonal paerns. Journal of he American Saisical Associaion From 1513 o HARVEY, A. (1989) Forecasing Srucural Time Series Model and he Kalman Filer, New York: CambridgeUniversiy press. HARVEY, A. AND KOOPMAN, SJ (1993) Forecasing Hourly Elecriciy Demand Using Tie Varying Splines. Journal of he American Saisical Associaion

16 HYNDMAN, RJ, KOEHLER, AB, ORD, JK and SNYDER, RD (2008) Forecasing wih Exponenial Smoohing: The Sae Space Approach, Berlin, Springer- Verlag. HYNDMAN, RJ and KANDHAKAR, Y. (2008) Auomaic Time Series Forecasing: he Forecas Package for R. Journal of Saisical Sofware, 26 (3) R DEVELOPMENT CORE TEAM (2011). R: A language and environmen for saisical Foundaion for Saisical Compuing. R, Vienna, Ausria. ISBN , URL hp:// TAYLOR, JW (2003). Shor-Term Elecriciy Demand Forecasing Journal of he Operaional Research Sociey, 54, TAYLOR, JW (2010) Seasonal Triple Mehods for Shor-Term Elecriciy Demand Forecasing. European Journal of Operaional Research, 204, TAYLOR, JW and Snyder, RD (2009) Forecasing Inraday Time Series Wih Seasonal Cycles Muliple Seasonal Exponenial Smoohing Using Parsimonious. Technical Repor 09/09. Dep. of Economeric and Business Saisics, Monash Universiy. WINTERS, PR (1960) Forecasing sales by exponenially weighed moving averages, Managemen Science, 6, WEST, M. and HARRISON, J. (1997) Bayesian Forecasing and Dynamic Models (2 nd Ed.) New York: Springer-Verlag.