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...
0 /. ) (. :. 0 0 : : (00005).. :. ( ).. Minib P Ver.7 : (Box & Jenkins (BJ)) : (BJ) esonl ime eries : (Brock Well & Dvis, 99 :53)
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
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
0 / = Θ ( Β = ( Θ ) B Θ B ΛΛ Θ Q Β Q ) ) (Q) = Θ Θ ΛΛ Θ Q Q : (3098 : 985...( 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
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 : 996...( 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 ) = + 3 + θ Θ + θθ3 < θ, Θ < + ) ). (,,, 3) (C) (y : =,,., NC).(3) esing ionriy of ime eries : : 5 6
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
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
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 : ( 50955 : 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
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
0 / : : : (7) ) / (... (005) () (66.300) (005) (95.000) (33.44) (00). (%0.76) (00). (3.74). (00005) : () yer Monh 005 006 007 008 009 00 Jn. 97.950 99.90 44.000 60.000 60.500 6.785 Feb. 97.330 98.650 39.500 46.000 47.000 48.730 Mr. 95.000.300 38.000 4.000 44.000 4.60 Apr. 95.00.000 38.000 45.000 46.000 59.50 My 95.000.000 38.500 46.500 47.750 6.07 Jun. 96.000 4.000 39.000 46.000 47.000 65.75 Jul. 97.000 30.50 40.000 50.000 5.000 6.30 Aug. 97.300 30.500 44.500 55.000 56.000 65.540 ep. 96.70 96.000 44.000 55.500 56.500 57.600 Oc. 96.000 96.000 45.000 58.000 59.000 4.800 Nov. 96.000 96.00 8.750 30.000 30.500 7.500 Dec. 97.000 98.450 40.600 45.000 47.000 66.300. : : : : : (). (). : : (Ln). (3 ) 3
0 / ( ). (4) (8) LJung & Box H * ( Q.s = LB Q= 387.> χ (8, 0.05) = 8.87 0.3 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) = 7.59. : (4 ) (6) ( ). () C4= = ( ) (C4) (7) ( 9 8 ) ( 0.3 r + 0.3) (8) k. () 3
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 0.840 0.0860 9.47 MA 0.978 0.09 33.40 MA 0.707 0.608 4.37 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 56 0.334307 0.0059697 ndrd error = 0.07764 Log Likelihood = 6.706898 AIC =.35 MAIC = 0.03597 BIC =.34004. ( ) : : 5 : : 33
0 / ( ) (0). ( 0.55 r () ˆ + 0.55 ) 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. 86.757 90.844 Feb. 7.604 77.734 Mr. 70.088 76.4 Apr. 75.398 8.664 My 75.568 83.598 Jun. 76.9 84.869 Jul. 77.506 86.759 Aug. 80.534 90.366 ep. 7.935 8.69 Oc. 67.487 77.9 Nov. 47.04 55.85 Dec. 68.35 78.493 34
0 / : 3 : : : 3.. 3 ().() ) 4 ( BIC AIC :. 5. ARIMA (,, ) ( 0,, ) (4) 6. 0 0. : : 3 :.. 35
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. 7673. 8 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.50955. 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.6964. 3 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. 3 4 5 36
0 / 00005 : (). : () 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 0.63 37.87 378. 3 0.7 3.8 39.6 0 0.46.33 35.49 7 0. 0.59 383.07 4 5 0.65 0.60.55.8 7.60 0.68 0.45 0.43.8.9 333.4 349.70 8 0.0 0.53 387. 6 7 0.57 0.55.93.79 7.8 5.85 3 4 0.33 0.8 0.89 0.76 359.4 366.68 38
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. 0.3.00.07 9 0 0.05 0.0 0.39 0.9 6 7 0.08 0.00 0.67 0.0 4 5 0.03 0.06 0.5 0.53 0.08 0.00 0.7 0.00 8 0.04 0.34 6 0.03 0.6 3 0.3.7 7 0.0 0.8 4 0.04 0.33. : (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 7.73 9.33 5 6 7 0.0 0.03 0.05 0. 0.3 0.33 8.60 8.7 8.97 4 5 6 7 0.3 0.03 0.0 0.0.00 0.3 0.76 0.5 6.37 6.44 7.9 7. 3 4 0.0 0.4 0. 0.08 0.09 3.0.4 0.50 9.35 4.0 8.06 8.58. : (6). : (7) 40
0 / Auocorrelion Funcion for C4 Auocorrelion.0 0.8 0.6 0.4 0. 0.0 0. 0.4 0.6 0.8.0 4 9 4 Corr LBQ Corr LBQ 0.0 0.05 0.00 8 0.4.60 4.8 0..67 3.00 9 0.0 0.05 4.8 3 0.00 0.03 3.00 0 0.3.50 8.7 4 0.8.06 8.6 0.03 0.8 8.4 5 0.08 0.5 8.54 0.37.7 8.5 6 0.5.00 9.99 3 0.06 0.33 8.78 7 0.05 0.35 0.8 4 0.04 0. 8.89. : (8) Pril Auocorrelion Funcion for C4 Pril Auocorrelion.0 0.8 0.6 0.4 0. 0.0 0. 0.4 0.6 0.8.0 4 9 4 PAC PAC 3 4 5 0.0 0. 0.00 0.34 0.09 0.05.67 0.0.65 0.66 8 9 0 0.04 0.07 0.0 0.0 0.3 0.33 0.5.5 0..74 6 7 0.37 0.0.8 0.7 3 4 0.05 0.06 0.38 0.47 : (9). 4
0 / ACF of Residuls for C (wih 95% confidence limis for he uocorrelions).0 0.8 0.6 Auocorrelion 0.4 0. 0.0 0. 0.4 0.6 0.8.0 3 4 5 6 7 8 9 0 3 4 5 PACF of Residuls for C (wih 95% confidence limis for he pril uocorrelions).0 0.8 Pril Auocorrelion 0.6 0.4 0. 0.0 0. 0.4 0.6 0.8.0 3 4 5 6 7 8 9 0 3 4 5. : (0) 4
0 / Hisogrm of he Residuls (response is C) 0 Frequency 0 0 0.0 0.5 0.0 0.05 0.00 0.05 0.0 0.5 0.0 0.5 0.30 Residul : () : () 43