b denotes trend at time point t and it is sum of two

Transcription

1 Inernaional Conference on Innovaive Applicaions in Engineering and Inforaion echnology(iciaei207) Inernaional Journal of Advanced Scienific echnologies,engineering and Manageen Sciences (IJASEMSISSN: X) Volue.3,Special Issue.,March.207 MODIFIED HOL s LINEAR MODELS FOR INRADAY DAA B.SAROJAMMA, RAMAKRISHNA REDDY 2, S.V.SUBRAHMANYAM 3, S.VENKAARAMANA REDDY 4*,S.K.GOVINDARAJULU 5,R.ABBIAH 6,2,3,5: Deparen of saisics, s.v.universiy, irupai Deparen of physics, s.v.universiy, iorupai * Auhor for correspondence: drsvreddy23@gail.co ABSRAC:here are any ypes of ie series odels and soe of he are univariae ie series odels, ulivariae ie series odels, inerval ie series odels and inraday ie series odels ec.. in his paper wo inraday ie series odels are inroduced and hey are odified hol s odeli and odified hol s odelii are esiaed using Roo ean square error crieria y aking alpha rages fro 0. o 0.9 ane0.99 for all hree odels i.e. hol s linear odel, odified hol s linear odeli,odified hol s linear odelii epirical invesigaion are calculaed using inraday daa of eperaure of dalles KEYWORDS: Hol s linear odel, odified Hol s linear odeli, odified Hol s linear odelii, eperaure I.INRODUCION here are so any odels are developed and odified for inraday odels. Inraday odels possesses here won odels, i can no ake usual odels which saisfies year wise daa, onhly daa and quarer daa. General odels for ie series are single exponenial soohing odel, Hol linear odel, Hol winer odel, Adopive exponenial soohing odel, Auo regressive odel, oving average odels, Auo regressive oving average odels for differen orders of Auo regressive and oving averages, Auo regressive inegraed oving averages(p, q, r) for differen values of p, q, r, differen odels of Auo regressive condiionally heeroscedasiciy like generalized Auo regressive condiional heeroscedasiciy(garch), IGARCH,EGARCH ec. hese odels are univariae odels of ie series.mulivariae odels of ie series are vecor auo regression (VAR), vecor auo regressive oving average (VARMA), vecor auo regressive inegraed oving average (VARIMA), uliple Regression, Discoun weighed regression odels ec., In his paper we explain Hol s linear odel and inroduced wo odified Hol s odels for inraday daa. Modified Hol s odel is fied y odificaion of Hol odel i.e., in place of linear rend y susiuing exponenial curve. Modified Hol s odel2 is fied y odificaion of Hol odel i.e in place of linear rend y susiuing power curve. Which odel is es aong Hol s linear odel, Hol s odified odel, Hol s odified odel2 using Roo ean square error crieria. Generally Hol s odel is ased on seasonaliy and rend. In Hol s odel rend follows addiive odel. Hol s odel is a wo paraeer odel. Hol (957) exended single exponenial soohing o linear exponenial soohing o allow forecasing of daa wih rends. Equaions for rend, level and forecas are as follows L ˆ u ( )( L ) where L denoes level a ie and i oains su of wo coponens, firs coponen is produc of consan wih ie series value Y and second coponen is su of level L and esiaed value of rend ˆ and his su is uliplied wih ( ) L is level a ie u is ie series value a ie L is level a ie poin ˆ is rend a ie poin is a consan lies eween 0 and, generally is susiued for differen values like 0., 0.2, 0.3, 0.4,,0.9 and upon using ean square error crierion we conclude which is he es sui o daa, y iniu value of MSE. Equaion for rend ( L L ) ( ) Here denoes rend a ie poin and i is su of wo coponens, firs coponen is L is suraced fro L and ha difference is uliplied wih consan β and second coponen is produc of ( ) wih. Where is rend a ie poin Page 32

2 Inernaional Conference on Innovaive Applicaions in Engineering and Inforaion echnology(iciaei207) Inernaional Journal of Advanced Scienific echnologies,engineering and Manageen Sciences (IJASEMSISSN: X) Volue.3,Special Issue.,March.207 L is level a ie L is level a ie poin is rend a ie is consan lies eween 0 and. F L.for forecas. Hol odel assues linear for of forecas equaion and calculaion will e changed y adding one year daa. hus ˆ is an uniased esiaor of. Generally Hol s odel is used when ie series daa is in he for of year wise or onh wise where as daa is in he for of day wise or hour wise i can no sui well. herefore in his paper we are inroduced wo odified Hol s odels. Modified Hol s odel : Generally Hol s odel follows linear rend. odified rend Hol s odel follows exponenial curve of for Y L he exponenial curve used for forecas is Y L Modified Hol s odel is convered in o General Hol s odel y aking naural logarihs on oh sides. logy log L logy log L log F A B where F is forecas for odified Hol s odel. A is level for odified Hol s odel. B is rend for odified Hol s odel. he hree equaions for odified Hol s odel are as follows For level A Y ( )( A B ) For rend B ( A A ) ( ) B For Forecas Y A B where lies eween 0 and, lies eween 0 and. Modified Hol s ehod 2: In his odified odel, power curve is used in place of linear rend of Hol s odel. In general power curve is Y ax he odified Hol s ehodii is F L By aking log on oh sides, we ge log F log L log Y A B log F Page 33 Y A log L B log he hree equaions for level, rend and forecas using odified Hol s odel2 are For level A Y ( )( ) 2 A 2 For rend B 2( A A ) ( 2) B For forecas F A B Here oh 2 and 2 are lies eween 0 and. Generally linear odel ay flucuae i.e., i ay ake posiive or negaive values. By aking log o any daa i soohes daa and i decreases errors. Roo Mean Square Error: Posiive square roo of ean square error gives Roo ean square error ( RMSE ). RMSE n i ( Y F) n Mean square error and Roo ean square error crieria is especially used o choose which odel possesses es fi o daa copared wih soe oher odel. A odel which akes sall MSE or RMSE is he es odel copared o oher odels. Relaive Errors: Relaive or percenage errors are copued y aking difference of original value and forecas value which gives error, his error is divided wih original value and uliplies 00. Y F PE 00 Y Mean Percenage Error ( MPE ): Su of percenage errors divided y nuer of oservaions gives Mean Percenage Error ( MPE ). Mean Asolue Percenage Error ( MAPE ): Asolue percenage error divided y oal nuer of oservaions gives ean asolue percenage error ( MAPE ) n PE n MAPE. Generally all he aove errors of accuracy are used o sudy aou consans fied o daa and o choose es odel o daa. A odel which possesses iniu error is good for daa. Epirical invesigaions: Inraday daa plays an vial role in any fields like aospheric science for forecas of hour o hour eperaure in suer, in rainy season oisure in air ec., in usiness arkes forecas of rading hour o hour, loss or gain of 2

3 Inernaional Conference on Innovaive Applicaions in Engineering and Inforaion echnology(iciaei207) Inernaional Journal of Advanced Scienific echnologies,engineering and Manageen Sciences (IJASEMSISSN: X) Volue.3,Special Issue.,March.207 rading ec. in elephone exchange deparen forecas of call deand of hour o hour asis, in call cener o esiae aou cell services ec. In his we are paperfied hree odels of Hol s ehod o forecas eperaure of Dallas daa. Hol s ehod: For forecas of Dallas daa we are fiing Hol s ehod. Hol s ehod conains rend, level and forecas equaions separaely level rend L ˆ u ( )( L ) ( L L ) ( ) F L forecas Where L is level a ie Page 34 is rend a ie poin L is level a ie poin is rend a ie, are consans lies eween 0 and. By aking ie series value as Dallas eperaure (u ), we fi Hol linear odel for differen values of α and β as follows HOL,s MODEL CALCULAION PERIODS EMPERAURE alpha ea L alpha ea forecas when = Error Error square E E E FORECAS FOR 24 HOURS

4 Inernaional Conference on Innovaive Applicaions in Engineering and Inforaion echnology(iciaei207) Inernaional Journal of Advanced Scienific echnologies,engineering and Manageen Sciences (IJASEMSISSN: X) Volue.3,Special Issue.,March Modified Hol s odel: Year wise, quarer wise and onh wise daa can no saisfy univariae odels of inraday and inra week daas. In his odified Hol s odel we are aking curve of for exponenial in place of rend in Hol s ehod. By akeing logarihs o exponenial curve i cover o rend. Generally log soohes daa and i reduces he flucuaions. Y L ake log on oh sides logy log L logy log L log F A B he odified Hol s odel conains level, rend and forecas equaions are For level A Y ( )( A B ) For rend B ( A A ) ( ) B For Forecas Y A B where lies eween 0 and, lies eween 0 and. Exponenial curve calculaion: Periods eperaure alpha ea Log y L B as alpha ea forecas Error square Page 35

5 Inernaional Conference on Innovaive Applicaions in Engineering and Inforaion echnology(iciaei207) Inernaional Journal of Advanced Scienific echnologies,engineering and Manageen Sciences (IJASEMSISSN: X) Volue.3,Special Issue.,March Forecas for 24 hours: Page 36

6 alpha ea Inernaional Conference on Innovaive Applicaions in Engineering and Inforaion echnology(iciaei207) Inernaional Journal of Advanced Scienific echnologies,engineering and Manageen Sciences (IJASEMSISSN: X) Volue.3,Special Issue.,March.207 Modified Hol s odel 2: Hol s linear odel forecas follows linear odel. In his odified Hol s odel2 we are assuing ha forecas follows power curve of for Y ax he forecas equaion of odified Hol s odel 2 is F L By aking log on oh sides, we ge log F log L log Y A B log F, log WhereY A log, L B, he hree equaions for level, rend and forecas are as follows For level A Y ( )( ) 2 A 2 For rend B 2( A A ) ( 2) B For forecas Y A B α and β are consans lies eween 0 and. Power curve calculaion forecas when = ERROR error square Period alpha ea logy L B E E E E E E E E05 6.2E09 forecas for 24 hours E E06 6.8E E E05.E E E06 4.7E E E07.6E E E07.E3 Page 37

7 Inernaional Conference on Innovaive Applicaions in Engineering and Inforaion echnology(iciaei207) Inernaional Journal of Advanced Scienific echnologies,engineering and Manageen Sciences (IJASEMSISSN: X) Volue.3,Special Issue.,March E E08 3E E E08 3.3E E E08.4E E E09 5.9E E E0.7E E E0 5.5E E E09.E E E09.2E E E09.2E E E09.E E E09.E E E09.E E E09.E E E09.E E E09.E E E09.E E E09.E E E09.E E E09.E8 Y A B SUMMARY AND CONCLUSIONS In his paper we inroduced wo odified Hol s linear odels for inraday daa. Modified Hol s odel: Year wise, quarer wise and onh wise daa can no saisfy univariae odels of inraday and inra week daas. In his odified Hol s odel we are aking curve of for exponenial in place of rend in Hol s ehod. By akeing logarihs o exponenial curve i cover o rend. Generally log soohes daa and i reduces he flucuaions. Y L ake log on oh sides logy log L logy log L log F A B he odified Hol s odel conains level, rend and forecas equaions are For level A Y ( )( A B ) For rend B ( A A ) ( ) B For Forecas Y A B where lies eween 0 and, lies eween 0 and. Modified Hol s odel 2: Hol s linear odel forecas follows linear odel. In his odified Hol s odel2 we are assuing ha forecas follows power curve of for Y ax he forecas equaion of odified Hol s odel 2 is F L By aking log on oh sides, we ge log F log L log log F WhereY, A log L B, log Page 38, he hree equaions for level, rend and forecas are as follows For level A Y ( )( ) 2 A 2 For rend B 2( A A ) ( 2) B For forecas α and β are consans lies eween 0 and. hese odels are epirically calculaed using inraday daa of Dallas and 24 hours forecas is given. Biliography [] Gould, P.G., Koehler, A.B., synder, R.D., Hyndan, R.J& Vahid Araghi,(2008).Forecasing ie series wih uliple season paern European journal of operaional Research, 9, [2] Harvey, A. & koopan, s.j.(993). Forecasing hourly elecriciy deand using ievarying splines, journal of he Aerican saisical associaion, 88, [3] Jaes w aylor 200. Exponenially ehods for forecasing inraday ie series wih uliple seasonal cycle. Inernaional journal of forecasing 26, [4] Porier, D.J. (973). Piece wise regression using cuic splines, journal of he Aerican saisical Associaion, 68, [5] Spyros Makridakis, seven C. wheelwrighr, Ro J.hyndan. forecasing ehods and Applicaions, 3 rd Ed., John wiley & sons.