Performance of combined double seasonal univariate time series models for forecasting water demand

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1 1 Performance of combined double seasonal univariae ime series models for forecasing waer demand Jorge Caiado a a Cener for Applied Mahemaics and Economics (CEMAPRE), Insiuo Superior de Economia e Gesão, Technical Universiy of Lisbon, Rua do Quelhas 6, Lisboa, Porugal. Tel.: jcaiado@iseg.ul.p Absrac: In his aricle, we examine he daily waer demand forecasing performance of double seasonal univariae ime series models (Hol-Winers, ARIMA and GARCH) based on muli-sep ahead forecas mean squared errors. A wihin-week seasonal cycle and a wihin-year seasonal cycle are accommodaed in he various model speci caions o capure boh seasonaliies. We invesigae wheher combining forecass from di eren mehods for di eren origins and horizons could improve forecas accuracy. The analysis is made wih daily daa for waer consumpion in Granada, Spain. Keywords: ARIMA; Combined forecass; Double seasonaliy; Exponenial Smoohing; Forecasing; GARCH; Waer demand. 1. Inroducion Waer demand forecasing is of grea economic and environmenal imporance. Many facors can in uence direcly or indirecly waer consumpion. These include rainfall, emperaure, demography, land use, pricing and regulaion. Weaher condiions have been widely used as inpus of mulivariae saisical models for hydrological ime series modelling and forecasing. Maidmen and Miaou (1986), Fildes, Randall and Subbs (1997), Zhou, McMahou, Walon and Lewis (2000), Jain, Varshney and Joshi (2001) and Bougadis, Adamowski and Diduch (2005) adoped regression and ime series models for waer demand forecasing by using climae e ecs as explanaory variables for heir models. Wong, Ip, Zhang and Xia (2007) used a non-parameric approach based on he ransfer funcion model o forecas a ime series of river ow. Jain and Kumar (2007) and Coulibary and Baldwin (2005) employed ari cial neural neworks mehods for hydrological ime series forecasing. Such mehods are useful for assessing waer demand under some sabiliy condiions. However, heir abiliy o projec demand ino he fuure may be limied as a resul of weaher condiions variabiliy and changes in consumer behavior and echnology. Waer demand is highly dominaed by daily, weekly and yearly seasonal cycles. The univariae ime series models based on he hisorical daa series can be quie useful for shor-erm demand forecasing as we accommodae he various periodic and seasonal cycles in he model speci - caions and forecass. To avoid heir sensibiliy o changes in weaher condiions and oher seasonal paerns, we may combine forecass derived from he mos accurae forecasing mehods for di eren forecas origins and horizons. Combining forecass can reduce errors by averaging of individual forecass (Clemen, 1989, Armsrong 2001) and is paricularly useful when we are uncerain abou which forecasing mehod is beer for fuure predicion. Some relevan works on combined forecass of univariae ime series models are by Makridakis and Winkler (1983), Sanders and Rizman (1989), Lobo (1992) and Makridakis, Cha eld, Hibon, Lawrence, Mills, Ord and Simons (1993). In his paper, we examine he waer demand forecasing performance of double seasonal univariae ime series models based on muli-sep ahead forecas mean squared errors. We invesigae wheher combining forecass from di eren

2 2 mehods and from di eren origins and horizons could improve forecas accuracy. Our ineres in his problem arose from he ime series compeiion organized by Spanish IEEE Compuaional Inelligence Sociey a he SICO 2007 Conference. The remainder of he paper is organized as follows. Secion 2 describes he daase used in he sudy. Secion 3 discusses he mehodology used in ime series modelling and forecasing. Secion 4 presens he empirical resuls. Secion 5 o ers some concluding remarks. 2. Daa We analyze he daily waer consumpion series in Spain from 1 January 2001 o 30 June 2006 (2006 observaions). We have drop February 29 in he leap year 2004 in order o mainain days in each year. This series is ploed in Figure 1. The daase was obained from he Spanish IEEE Compuaional Inelligence Sociey (hp:// We use he rs 1976 observaions from 1 January 2001 o 31 May 2006 as raining sample for model esimaion, and he remaining 30 observaions from 1 June 2006 o 30 June 2006 as possample for forecas evaluaion. The series exhibis periodic behavior wih a wihin-week seasonal cycle of 7 periods and a wihin-year cycle of periods. The observed increases (decreases) in demand in he summer (winer) days seem o be caused by good (bad) weaher. The analysis of weekly seasonaliy shows a consumpion drop in demand on Saurdays and Sundays as a resul of he shudown of indusry. Figure 2 shows he sample auocorrelaions (ACF) and he sample parial auocorrelaions (PACF) for he raining sample. The ACF decays very slowly a regular lags and a muliples of seasonal periods 7 and. The PACF has a large spike a lag 1 and cu o o zero afer lag 2. This suggess boh a weekly seasonal difference (1 B 7 ) and a yearly seasonal di erence (1 B ) o achieve saionariy. Figures 3 and 4 presen he double seasonal di erenced series (1 B 7 )(1 B )Y and heir esimaed ACF and PACF funcions. 3. Mehodology 3.1. Forecas evaluaion Denoe he acual observaion for ime period by Y and he forecased value for he same period by F. The mean squared error (MSE) saisic for he pos-sample period = m+1; m+2; :::; m+ h is de ned as follows: MSE = 1 h m+h X =m+1 (Y F ) 2. (1) This saisic is used o evaluae he ou-ofsample forecas accuracy using a raining sample of observaions of size m < n (where n is he sample size) o esimae he model, and hen compuing recursively he one-sep ahead forecass for ime periods m + 1, m + 2,... by increasing he raining sample by one. For k-sep ahead forecass, we begin a he sar of he raining sample and we compue he forecas errors for ime periods = m + k, m + k + 1,... using he same recursive procedure Random walk The naïve version of he random walk model is de ned as F +1 = Y. (2) This purely deerminisic mehod uses he mos recen observaion as a forecas, and is used as a basis for evaluaing of ime series models described below Exponenial smoohing Exponenial smoohing is a simple bu very useful echnique of adapive ime series forecasing. Sandard seasonal mehods of exponenial smoohing includes he Hol-Winers addiive rend, muliplicaive rend, damped addiive rend and damped muliplicaive rend (see Gardner, 2006). We implemened he double seasonal versions of he Hol-Winers exponenial smoohing (Taylor, 2003) in order o ake ino accoun he wo seasonal cycle periods in he daily waer consumpion: a wihin-week cycle of 7 days and a wihin-year cycle of days. In an applicaion o half-hourly elecriciy demand, Taylor (2003) used a wihin-day seasonal cycle of 48

3 Jan 01 Jan 02 Jan 03 Jan 04 Jan 05 Jan 06 Figure 1. Daily waer demand in Spain for he period 1 January 2001 o 30 June ACF PACF Figure 2. ACF and PACF of he waer demand series

4 Figure 3. Waer demand series afer yearly seasonal di erencing and weekly seasonal di erencing ACF PACF Figure 4. ACF and PACF of he di erenced waer demand series

5 5 half-hours and a wihin-week seasonal cycle of 336 half-hours. The double seasonal addiive mehods ouperformed he double seasonal muliplicaive mehods. Wihin he double seasonal addiive mehods, he addiive rend was found o be he bes for one-sep ahead forecasing. The forecass for Taylor s exponenial smoohing for double seasonal addiive mehod wih addiive rend are deermined by he following expressions: L = (Y S 7 D ) + (1 )(L 1 + T 1 ) (3) T = (L L 1 ) + (1 )T 1 (4) S = (Y L D ) + (1 )S 7 (5) D = (Y L S 7 ) + (1 )D (6) F +h = L + T h + S +h 7 + D +h + h [Y (L 1 T 1 S 7 D )] (7) where L is he smoohed level of he series; T is he smoohed addiive rend; S is he smoohed seasonal index for weekly period (s 1 = 7); D is he smoohed seasonal index for yearly period (s 2 = ); and are he smoohing parameers for he level and rend; and are he seasonal smoohing parameers; is an adjusmen for rs-order auocorrelaion; and F +h is he forecas for h periods ahead, wih h 7. We iniialize he values for he level, rend and seasonal periods as follows: L = T = 1 X 1 2 =1 Y X730 =366 Y! X Y =1 S 1 = Y 1 L 7 ; S 2 = Y 2 L 7 ; :::; S 7 = Y 7 D 1 = L 7 Y 1 L ; D 2 = Y 2 L ; :::; D = Y L The smoohing parameers,,, and are chosen by minimizing he MSE saisic for one-sep-ahead in-sample forecasing using a linear opimizaion algorihm ARIMA model We implemened a double seasonal muliplicaive ARIMA model (see Box, Jenkins and Reinsel, 1994) of he form: p (B) P1 (B s1 ) P2 (B s2 )(1 B) d (1 B s1 ) D1 (1 B s2 ) D2 (Y c) = q (B) Q1 (B s1 ) Q 2 (B s2 )" (8) where c is a consan erm; B is he lag operaor such ha B k Y = Y k ; p (B) and q (B) are regular auoregressive and moving average polynomials of orders p and q; P1 (B s1 ), P2 (B s2 ), Q1 (B s1 ) and Q 2 (B s2 ) are seasonal auoregressive and moving average polynomials of orders P 1, P 2, Q 1 and Q 2 ; s 1 and s 2 are he seasonal periods; d, D 1 and D 2 are he orders of inegraion; and " is a whie noise process wih zero mean and consan variance. The roos of he polynomials p (B) = 0, P1 (B s1 ) = 0, P2 (B s2 ) = 0, q (B) = 0, Q1 (B s1 ) = 0 and Q 2 (B s2 ) = 0 should lie ouside he uni circle. This model is ofen denoed as ARIMA(p,d,q)(P 1,D 1,Q 1 ) s1 (P 2,D 2,Q 2 ) s2. We examine he sample auocorrelaions and he parial auocorrelaions of he di erenced series in order o idenify he ineger values p, q, P 1, Q 1, P 2 and Q 2. Afer idenifying a enaive ARIMA model, we esimae he parameers by Marquard nonlinear leas squares algorihm (for deails, see Davison and MacKinnon, 1993). We check he adequacy of he model by using suiable ed residuals ess. We use he Schwarz Bayesian Crierion (SBC) for model selecion GARCH model In many pracical applicaions o ime series modelling and forecasing, he assumpion of nonconsan variance may be no reliable. The models wih nonconsan variance are referred o as condiional heeroscedasiciy or volailiy models. To deal wih he problem of heeroscedasiciy in he errors, Engle (1982) and Bollerslev (1986) proposed he auoregressive condiional heeroskedasiciy (ARCH) and he generalized ARCH (or GARCH) o model and forecas he condiional variance (or volailiy). The

6 6 GARCH(p,q) model assumes he form: 2 =! + px j 2 j + j=1 qx i " 2 i, (9) i=1 where p is he order of he GARCH erms and q is he order of he ARCH erms. The necessary condiions for he model (9) o be variance and covariance saionary are:! > 0; P j 0, j = 1; :::; p; i 0, j = 1; :::; q; and p j=1 j + P q i=1 i < 1. Las summaion quani es he shock persisence o volailiy. A higher persisence indicaes ha periods of high (slow) volailiy in he process will las longer. In mos economical and nancial applicaions, he simple GARCH(1,1) model has been found o provide a good represenaion of a wide variey of volailiy processes as discussed in Bollerslev, Chou and Kroner (1992). In order o capure seasonal and cyclical componens in he volailiy dynamics, we implemened a seasonal-periodic GARCH model of he form: 2 =! " " " 2 MX 2kS 2kS + k cos + ' 7 k sin 7 k=1 2kD 2kD +u k cos + v k sin + 0 k" 2 2kS 7 cos 7 +' 0 k" 2 2kS 7 sin 7 +u 0 k" 2 2kD cos +vk" 0 2 2kD sin, (10) where S and D are repeaing sep funcions wih he days numeraed from 1 o 7 wihin each week, and from 1 o wihin each year, respecively. A similar approach was used by Campbell and Diebold (2005) o model condiional variance in daily average emperaure daa, and by Taylor (2006) o forecas elecriciy consumpion. In he empirical sudy, we se M = 3 for he Fourier series. We esimae he model by he mehod of maximum likelihood, assuming a generalized error disribuion (GED) for he innovaions series (see Nelson, 1991) Combining forecass We examine wheher combining forecass from he various univariae mehods for di eren forecas origins and horizons could provide more accurae forecass han he individual mehods being combined. The forecass can be combined by using simple and opimal weighs Simple combinaion We consider all possible combinaions of he forecas mehods Hol-Winers (HW), ARIMA (A) and GARCH (G), and we compue he simple (unweighed) average of he forecass for one o seven days ahead, F S = F (HW ) + F (A) 3 + F (G) ; (11) where F () is he forecased value of mehod () in ime period. This approach is simple and useful when we have no evidence abou which forecasing mehod is more accurae. We drop he random walk (he wors mehod) of he combinaion Opimal combinaion We consider wo approaches for compuing opimal weighs. Firsly, we compue he opimal combinaion of he forecass using weighs by he inverse of he MSE of each of he individual mehods (see Makridakis and Winkler, 1983), as follows: F MSE = h (MSE MSE (HW ) )F (HW ) +(MSE +(MSE MSE (A) )F (A) MSE (G) )F (G) 2MSE, (12) where MSE = MSE (HW ) + MSE (A) + MSE (G) is he sum of he pos-sample forecas mean squared errors of he hree mehods. Secondly, we compue opimal combinaion of he pos-sample forecass using weighs by he inverse of each of he forecas squared errors (SE) i

7 7 of each of he individual mehods, as follows: h F SE = (SE SE (HW ) )F (HW ) +(SE +(SE SE (A) SE (G) )F (A) )F (G) 2SE, (13) where SE = SE (HW ) + SE (A) + SE (G) is he sum of he pos-sample forecas squared errors of he hree mehods in ime period. 4. Empirical sudy 4.1. Esimaion resuls The implemenaion of he double seasonal Hol-Winers mehod o he waer demand series Y gives he values: = 0:000, = 0:755, = 0:303, = 0:294 and = 0:607. Afer evaluaing di eren ARIMA formulaions, we apply he following muliplicaive double seasonal ARIMA model: (1 1 B 2 B 2 4 B 4 )(1 1 B 7 2 B 14 ) (1 B 7 )(1 B )(Y c) = (1 9 B 9 )(1 3 B 21 )(1 1B )" This model can be represened as ARIMA(4; 0; 9) (2; 1; 3) 7 (0; 1; 1), wih 3 = 0, 1 = = 8 = 0, and 1 = 2 = 0. The esimaed resuls and diagnosic checks are shown in Table 1. All he parameer esimaes are signi can a he 5% signi cance level. The residual auocorrelaion funcion (ACF) and parial auocorrelaion funcion (PACF) exhibi no paerns up o order 7. The Ljung-Box saisic, Q = 18:31, based on 20 residual auocorrelaions is no signi can a he convenional levels. These resuls sugges ha he model is appropriae for modeling he waer demand series. We hen ed a signi can parameer ARIMA- GARCH model of he form: (1 1 B 2 B 2 4 B 4 )(1 1 B 7 2 B 14 ) (1 B 7 )(1 B )(Y c) = (1 9 B 9 )(1 3 B 21 )(1 1B )" i and 2 =! " " 2 2D +' 1 sin + ' 0 3" 2 sin 6D The model esimaes and diagnosic checks are given in Table 2. The Ljung-Box es saisics show evidence of no serial correlaion in he residuals (mean equaion) and no serial correlaion in he squared residuals (variance equaion) up o order 20. Thus, we conclude ha his model is also adequae for he daa Forecas evaluaion resuls The performance of he esimaed univariae mehods were evaluaed by compuing MSE saisics for muli-sep forecass from 1 o 7 days ahead. Table 3 and Figure 5 give he forecass resuls for he pos-sample period from 1 June 2006 o 30 June An iniial inerpreaion of he resuls suggess ha he abiliy o forecas waer demand did no decrease as he forecas horizon increased, excep from 1 o 2 days ahead. The ARIMA and GARCH models appear o have he same forecas performance especially for shor-erm forecass (one o wo days ahead). For one o four days ahead forecass, he sochasic models ARIMA and GARCH performed beer han he Hol-Winers mehod. In conras, he Hol-Winers ouperformed he ARIMA and GARCH models in long horizons. The random walk model ranked las for any of he forecas horizons considered. The opimal combinaion of Hol-Winers, ARIMA and GARCH weighed by inverse squared errors is more accurae han he various simple combinaions, excep for 7-sep ahead forecasing in which he Hol-Winers ouperformed he opimal combined forecasing. For one day ahead, he average MSE for he individual forecasing mehods (HW, ARIMA and GARCH) was 0.36 while i was 0.33 for he opimal combined forecass a error reducion of 8.33%. For wo and hree days ahead forecass, combining reduced he MSE by 12.77% and 10.64%, respecively. Table 4 and Figure 6 give he forecas resuls.

8 8 for each of he 7 days of he week in he same period. The resuls suggess ha he Thursdays exhibi irregular demand paerns in he possample period used in his sudy. From he daa, we found ha he waer consumpion decreased 10.37% on he rs Thursday of he pos-sample period (1 June 2006), whereas i increased 4.22% and 18.44% on he following Thursdays (8 June 2006 and 15 June 2006, respecively). Possible reasons for his unusual paern are weaher changes and any resricions on waer demand. In erms of he day of he week e ec on forecasing performance, he opimal combinaion HW-A-G (SE) appears o be mos useful for Monday, Tuesday and Wednesday forecass combining reduced he MSE of muli-sep ahead averaged forecass by 12.15%, 45.45% and 14.60%, respecively, when compared wih he average of he individual mehods. The Hol-Winers appears o be he mos appropriae mehod for Thursday, Friday and Saurday forecass and he GARCH model appears o be he bes mehod for Sunday forecass. 5. Conclusions In his aricle we compared he forecas accuracy of individual and combined univariae ime series models for muli-sep ahead daily waer demand forecasing. We implemened double seasonal versions of he Hol-Winers, ARIMA and GARCH models in order o accommodae he wo seasonal e ecs (wihin-week cycle of 7 days and wihin-year cycle of days) on he variabiliy of he daa. The empirical resuls sugges ha he opimal combined forecass can be quie useful especially for shor-erm forecasing. However, he forecasing performance of his approach is no consisen over he seven days of he week. The deerminisic mehod Hol-Winers and he sochasic mehod GARCH can be used independenly o improve forecas accuracy on Thursdays o Saurdays and Sundays, respecively. Acknowledgmen: The auhor graefully acknowledges he helpful commens of he paricipans in he Spanish IEEE Compuaional Inelligence Sociey a he SICO 2007 Conference. This research was suppored by a gran from he Fundação para a Ciência e a Tecnologia (FEDER/POCI 2010). REFERENCES 1. Armsrong, J. (2001). "Combining forecass", in Principles of Forecasing: A Handbook for Researchers and Praciioners, J. S. Armsrong (ed.), Kluwer Academic Publishers. 2. Bollerslev, T. (1986). "Generalized auoregressive condiional heeroskedasiciy", Journal of Economerics, 31, Bollerslev, T., Chou, R. and Kroner, K. (1992). "ARCH modeling in Finance", Journal of Economerics, 52, Bougadis, J., Adamowski, K. and Diduch, R. (2005). "Shor-erm municipal waer deamand forecasing", Hydrological Processes, 19, Box, G., Jenkins, G. and Reinsel, G. (1994). Time Series Analysis: Forecasing and Conrol, 3rd ed., Prenice-Hall, New Jersey. 6. Campbell, S. and Diebold, F. (2005). "Weaher forecasing for weaher derivaives", Journal of he American Saisical Associaion, 100, Clemen, R. (1989). "Combining forecass: a review and annoed bibliography", Inernaional Journal of Forecasing, 5, Coulibaly, P. and Baldwin, C. (2005): "Nonsaionary hydrological ime series forecasing using nonlinear dynamic mehods", Journal of Hydrology, 307, Davison, R. and MacKinnon, J. (1993). Esimaion and Inference in Economerics, Oxford Universiy Press, Oxford. 10. Engle, R. (1982). "Auoregressive condiional heeroscedasiciy wih esimaes of he variance of Unied Kingdom in aion", Economerica, 50, Fildes, R., Randall, A. and Subbs, P. (1997). "One-day ahead demand forecasing in he uiliy indusries: Two case sudies", Journal of he Operaional Research Sociey, 48, Gardner Jr., E. (2006). "Exponenial smooh-

9 9 0,55 0,5 0,45 MSE 0,4 0,35 0,3 1 sep 2 sep 3 sep 4 sep 5 sep 6 sep 7 sep Forecas horizon (days) HW ARIMA GARCH HW A HW G A G HW A G HW A G (MSE) HW A G (SE) Figure 5. Comparison of muli-sep ahead forecass for pos-sample period 4,50 4,00 3,50 3,00 MSE 2,50 2,00 1,50 1,00 0,50 0,00 Monday Tuesday Wednesday Thursday Friday Saurday Sunday HW ARIMA GARCH HW A HW G A G HW A G HW A G (MSE) HW A G (SE) Figure 6. Comparison of muli-sep ahead averaged forecass for each of he seven days of he week

10 10 ing: The sae of he ar - Par II", Inernaional Journal of Forecasing, 22, Jain, A., Varshney, A. and Joshi, U. (2001). "Shor-erm waer demand forecas modeling a IIT Kanpur using ari cial neural neworks", Waer Resources Managemen, 15, Jain, A. and Kumar, A.M. (2007). "Hybrid neural nework models for hydrologic ime series forecasing", Applied Sof Compuing, 7, Lobo, G. (1992). "Analysis and comparison of nancial analyss, ime series, and combining forecass of annual earnings", Journal of Business Research, 24, Maidmen, D. and Miaou, S. (1986). "Daily waer use in nine ciies", Waer Resources Research, 22, Madridakis, S. and Winkler, R. (1983). "Average of forecass: Some empirical resuls", Managemen Science, 29, Madridakis, S., Cha eld, C., Hibon, M., Lawrence, M., Mills, T., Ord. K. and Simons, L. (1993). "The M2-compeiion: A real-ime judgmenally based forecasing sudy", Inernaional Journal of Forecasing, 9, Nelson, D. (1991). "Condiional heeroskedasiciy in asse reurns: a new approach", Economerica, 59, Sanders, N. and Rizman, L. (1989). "Some empirical ndings on shor-erm forecasing: Technique complexiy and combinaions", Decision Sciences, 20, Taylor, J. (2003). "Shor-erm elecriciy demand forecasing using double seasonal exponenial smoohing", Journal of he Operaional Research Sociey, 54, Taylor, J. (2006). "Densiy forecasing for he e cien balancing of he generaion and consumpion of elecriciy", Inernaional Journal of Forecasing, 22, Wong, H., Ip, W., Zhang, R. and Xia, J. (2007). "Non-parameric ime series models for hydrological forecasing", Journal of Hidrology, 332, Zhou, S., McMahon, T., Walon, A. and Lewis, J. (2000). "Forecasing daily urban waer demand: a case sudy of Melbourne", Journal of Hidrology, 236,

11 11 Table 1 Seasonal ARIMA model esimaes for waer demand series Model: ARIMA(4,0,9)(2,1,3) 7 (0,1,1) Residual ACF Residual PACF Parameer Lag Esimae Sandard error Lag Esimae Lag Esimae c R 2 adjused = 0.662; Q(20) = 18:31 (0.11). Noes: Q(20) is he Ljung-Box saisic for serial correlaion in he residuals up o order 20; p-value in parenheses. Table 2 Seasonal-periodic GARCH model esimaes for waer demand series Model: ARIMA(4,0,9)(2,1,3) 7 (0,1,1) GARCH(1,1)(0,1) Mean equaion Residual ACF Residual PACF Parameer Lag Esimae Sandard error Lag Esimae Lag Esimae c Variance equaion Sq. residual ACF Sq. residual PACF Parameer Lag Esimae Sandard error Lag Esimae Lag Esimae! ' ' GED R 2 adjused = 0.657; Q(20)=19.20 (0.08); Q 2 (20)=13.61 (0.33). Noes: Q(20) (Q 2 (20)) is he Ljung-Box saisic for serial correlaion in he residuals (squared residuals) up o order 20; p-value in parenheses.

12 12 Table 3 MSE for muli-sep-ahead forecass for pos-sample period Forecas Simple combinaion Opimal combin. horizon RW HW ARIMA GARCH HW-A HW-G A-G HW-A-G MSE SE 1-sep sep sep sep sep sep sep Average Table 4 MSE for muli-sep ahead forecass for each day of he week in pos-sample period Forecas Day of he Simple combinaion Opimal combin. horizon week RW HW ARIMA GARCH HW-A HW-G A-G HW-A-G MSE SE 1-sep Monday Tuesday Wednesday Thursday Friday Saurday Sunday sep Monday Tuesday Wednesday Thursday Friday Saurday Sunday sep Monday Tuesday Wednesday Thursday Friday Saurday Sunday Average Monday Tuesday Wednesday Thursday Friday Saurday Sunday