Using Seasonal Time Series Models to Forecast Electrical Power Consumption in Fallujah City

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1 0 / Using esonl ime eries Models o Forecs Elecricl Power Consumpion in Fllujh Ciy.... ( ) / (00005). ARIMA (,, ) (0,, ) : 0. 0 Absrc his reserch del wih using sesonl ime series models o sudy nd nlysis he monhly d on consumpion of elecriciy in Fllujh ciy for he period (005 00), wheres his models re disinc wih high ccurcy nd flexible in nlysis ime series. he resuls of pplicion show h he proper nd efficiency model for represening ime series d re he muliplicive sesonl model of order : ARIMA(,, ) (0,, ) According o esimion resuls of his model done forecsing o monhly consumpion of elecrics cpciy for wo yers hed from he period Jn. 0 o Dec. 0, hese vlues show hrmonic direcion wih he sme originl ime series...

2 0 /. ) (. :. 0 0 : : (00005).. :. ( ).. Minib P Ver.7 : (Box & Jenkins (BJ)) : (BJ) esonl ime eries : (Brock Well & Dvis, 99 :53)

3 0 / (s) (s) (Fixed inervls) f( + s) = f(). (Mkridkis & McGee, 983 : 63) Anderson, ) ( ). (976 :3 : Auocorrelion (AC) : : () ˆ ρ = Cov(, Vr( ) Vr( + ) + = ) n = ( n = )( ( + ) ) : (Wei, 990 : )...( ). : : ρ, ρ, ρ3,..., ρ, ρ, ρ +,.... k=0, s, s,.. (ACF). Residul Auocorrelion Funcion (RACF) (ACF).. ( ) Pril Auocorrelion (PAC) : : 3 + (PACF). +,., + 3

4 0 / (ACF) Wei, 990 : ) ( ) ). (3 esonl ime eries Models : : 4 : :4 esonl Auoregressive Model (AR) (p) = Φ + Φ + Λ Λ + Φ P P + Φ ( Β ) = Φ B = 0...( ) :(3098: 985 : i=0,,,, p : i. : i=,,, p : Φ i ~ NID ( 0, σ ). : p : ( ) uni circle < Φ < Β = : (Wei, 990 : 6) : Bck shif operor B s=,,κκ AR() (ACF) ρ K = Φ 0 k = 0 k = s k=,, ΚΚ, s : AR(p). ( Mkridkis & McGee, 983 :64 ) p : : 4 esonl Moving Averge Model (MA) : (B) ( ) 4

5 0 / = Θ ( Β = ( Θ ) B Θ B ΛΛ Θ Q Β Q ) ) (Q) = Θ Θ ΛΛ Θ Q Q : (3098 : ( 3) :. : Θ i <Θ < i=,, ΛΛ, Q. : Q. ( Q ) Qs (MA). (PACF) : ( ) : 34 esonl Mixed (Auoregressive Moving Averge) Model (ARMA) : (B) Φ ( Β ) = Θ ( Β ) P Q ( Φ Β Φ Β Λ Λ Φ Β ) = ( Β Β ΛΛ ) P Θ Θ Θ Β Q :(3 :996 ) (P,Q) = Φ + Φ + Λ Λ + Φ + Θ p p ΛΛ Θ Θ Q Q...( 4) ARMA(P, Q) ( ). D D = ( Β ) D ) ( esonl Auoregressive Inegred Moving Averge Model Φ P Θ D ( Β ) = Q ( Β ) : ( Box nd Jenkins, 976 :54...( 5) ARIMA(P, D, Q) ( P, D, Q ) : : 44 Muliplicive esonl Model (ARIMA) ) 5

6 0 / ) ( p, d, q )*( P, D, Q ) Θ d D φ p ( Β) Φ P ( Β ) =θ q ( Β) Q ( Β ) (0,, )*(0, = Β : ( Anderson, 976 : 54 ; 5 : ( 6) :. : p. : d. : q. :φ p ( Β) d :. = Β d. :θ q ( Β). : P. : D. : Q. :Φ P ( Β D D :.. :Θ Q ( Β : ( Box nd Jenkins, 976 :73 ), ) Θ = ( θβ)( Β )...( 7) 3 ( Β)( Β ) = ( ΘΒ θβ+ θθβ ) = θ Θ + θθ3 < θ, Θ < + ) ). (,,, 3) (C) (y : =,,., NC).(3) esing ionriy of ime eries : : 5 6

7 0 / (ACF). (PACF) :( 64 : 000 ) *. *. *. : (53 : 003 ) :. : :. derending : y = = y = = = + = ( Β) : :. (d) : ( ) 3 esonl differencing : 000 ) : (63 y = 4 y = y. F =. W = F F F F ges of Building esonl Model : : 6 7

8 0 / Box & Jenkins ) ARIMA : ( Box nd Jenkins, 976 :43 Idenificion :. (PACF) (ACF) (PACF) (ACF) AR(P) MA(Q) ARMA (P, Q). : () (ACF) ( ) Q (Decys Exponenilly) (Cus off) ( ) (Decys Exponenilly) P. (8: 996 ) (PACF) (Cus off) ( ) (Decys Exponenilly) ( ) (Decys Exponenilly) Esimion : : ( Lwrence & Pul, 978 : 6964). ( ) Exc Mximum Likelihood Mehod (EML). Non Liner Les qure Mehod (NL) Dignosic Checking of Model : 3 : ( Wegmn, 000 : 443 ) 8

9 0 /. MA AR () r () o = n Pr.96 n : Residul nlysis Confidence Inervl Checking : ( r ()) s. (0.95) (µ.96/ r () +.96 = 0.95 n n )...( 8) ( ) ( ) r () ~ NID ( 0, ) n n : Pormneu ( Pierce & Box ) Q Box & Price, 970 : L Q = n r () ~ χ ((L m ), α ) k = ) k...( 9). : m H o χ : ( : L Q. : LJung nd Box Q * L () = n(n+ )...( 0) n k k = r k χ ((L m ), α ) * توجد صيغ أخرى للاختبار تعتمد على الارتباط الذاتي الجزي ي للبواقي. 9

10 0 / : (Akike, 974 : 7673).. Akike (p+ q) AIC= Log ( σ ) + n...( ) :. : σ. : (p+q) : : 998 ) AIC MAIC= n p+ q BIC= Log( σ ) + Log(n) n...( )...( 3) : ( 73 : chwrz Forecsing : 4 () (L=,,..). ˆ (L) = + L Differences Equion Form ) ( ) = ˆ (L) = Φˆ + L ΛΛ Q + L + L + Φ + L + + Φ p + Lp + Θˆ ˆ Θˆ + LQ ΛΛ ˆ. ( Box nd Jenkins, 976 :89 + L Θˆ + L...( 4) + L = + = E( + L ) ; + L E( L ) : 30

11 0 / : : : (7) ) / (... (005) () (66.300) (005) (95.000) (33.44) (00). (%0.76) (00). (3.74). (00005) : () yer Monh Jn Feb Mr Apr My Jun Jul Aug ep Oc Nov Dec : : : : : (). (). : : (Ln). (3 ) 3

12 0 / ( ). (4) (8) LJung & Box H * ( Q.s = LB Q= 387.> χ (8, 0.05) = r + 0.3) k : ρ = ρ = ρ = Λ Λ = ρ 0 3 k = 0. : : 3 : (5) ( ) =. LJung & Box * Q.s = LB Q= 8.97> χ (7, 0.05) = : (4 ) (6) ( ). () C4= = ( ) (C4) (7) ( 9 8 ) ( 0.3 r + 0.3) (8) k. () 3

13 0 / : : 3 MA AR (Conelogrmme) (9 8) (PACF) (ACF) () ( ) (k) ARIMA (,, ) ( 0,, ) or ( φ Β)( Β)( Β ) = ( θβ)( ΘΒ ) ARIMA (,, ) ( 0, : : 4. AKIAKE, ) (NL) : P Ver.7 Minib Finl Esimes of Prmeers ype Coeff. Dev. AR MA MA Differencing : regulr, sesonl of order Number of observion : originl series 7, fer differencing 59 Anlysis of Vrince : DF Adj.sum of squres Residuls Vrince Residuls ndrd error = Log Likelihood = AIC =.35 MAIC = BIC = ( ) : : 5 : : 33

14 0 / ( ) (0). ( 0.55 r () ˆ ) k Pormneu : NID ( 0, σ ) (Whie Noise) ~ (LJung ( ) rk () ˆ N & Box )Q * * Q.s = LB Q= 0.6< χ ( 9, 0.05) = 6.9. ( ) : () ().. : : 6 (4) (3) 0 0 (). (3). 0 0 yer 0 0 Monh Jn Feb Mr Apr My Jun Jul Aug ep Oc Nov Dec

15 0 / : 3 : : : ().() ) 4 ( BIC AIC :. 5. ARIMA (,, ) ( 0,, ) (4) : : 3 :.. 35

16 0 / " (985) ". (3098) " (996). " " (003) " Box Jenkins. (53). " " (998) " " (000).. " " (000) 6 7 Akike, H. (974), "A new look he sisicl model Idenificion ", IEEE rnscions on Auomic Conrol, Vol.9, No.6, PP Anderson, O.D. (976), " ime series nlysis nd forecsing ", Buer worhs, London nd Boson. 9 Box G., E., P., nd Jenkins G., M.,. (976), " ime eries Anlysis Forecsing nd Conrol ", n Frncisco, HoldenDy, U..A. 0 Box, G. E. nd Price, D. A. (970), " Disribuion of Residul Auocorrelions in Auoregressive Inegred Moving Averge ime eries Models ", JAA, Vol.55, No.33, PP Brock Well, P.J. nd Dvis, R.A. (99), " ime eries heory nd Mehods ", nd ed, pringer Verlg New York Inc, New York. Lwrence, A.K. nd Pul, I.N. (978), " on Condiionl Les qure Esimion for ochsic Processes ", Ann. of., Vol.6, No.3, PP Mkridkis,., Wheel Wrigh., C., nd McGee (983), " Forecsing Mehod nd Applicion ", nd ed, John Wily nd ons. Inc., U..A.. 4 Wegmn, E.J. (000), " ime eries Anlysis heory, D Anlysis nd Compuion ", AddisonWesley Publishing Compny. 5 Wei, W.. (990), " ime eries Anlysis : Univrie nd Mulivrie Mehods ", Addison Wesley Publishing Compny Inc., U..A

17 0 / : (). : () 37

0 / : (3) Auocorrelion Funcion for C Auocorrelion.0 0.8 0.6 0.4 0. 0.0 0. 0.4 0.6 0.8.0 7 7 Corr LBQ Corr LBQ Corr LBQ 0.86 0.77 7.8 4.4 55.7 00.3 8 9 0.53 0.50.65.48 76.54 97.9 5 6 0.6 0.4 0.69 0.

18 0 / : (3) Auocorrelion Funcion for C Auocorrelion Corr LBQ Corr LBQ Corr LBQ

0 / Pril Auocorrelion Funcion for C Pril Auocorrelion.0 0.8 0.6 0.4 0. 0.0 0. 0.4 0.6 0.8.0 7 7 PAC PAC PAC 0.86 7.8 8 0.0 0.9 5 0.04 0.3 3 0. 0.3.00.

19 0 / Pril Auocorrelion Funcion for C Pril Auocorrelion PAC PAC PAC : (4). : (5) 39

0 / Auocorrelion Funcion for C3 Auocorrelion.0 0.8 0.6 0.4 0. 0.0 0. 0.4 0.6 0.8.0 7 7 Corr LBQ Corr LBQ Corr LBQ 3 0.3 0.3 0.0.06.90 0..6 5. 5.3 8 9 0 0.07 0.0 0.4 0.58 0.9.06 7.68 7.

20 0 / Auocorrelion Funcion for C3 Auocorrelion Corr LBQ Corr LBQ Corr LBQ : (6). : (7) 40

21 0 / Auocorrelion Funcion for C4 Auocorrelion Corr LBQ Corr LBQ : (8) Pril Auocorrelion Funcion for C4 Pril Auocorrelion PAC PAC : (9). 4

22 0 / ACF of Residuls for C (wih 95% confidence limis for he uocorrelions) Auocorrelion PACF of Residuls for C (wih 95% confidence limis for he pril uocorrelions) Pril Auocorrelion : (0) 4

23 0 / Hisogrm of he Residuls (response is C) 0 Frequency Residul : () : () 43

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