Sisi Vol 9 No 09 3 Nopember 009 Perbningn Meoe Permln ARIMA n ARFIMA p D Long Memory GUMGUM DARMAWAN Sf Pengjr Jurusn Sisi FMIPA UNPAD e-mil : gums@yoocom ABSTRAK P ml ini n i bningn u meoe permln ri long memory Meoe perm menggunn meoe permln ARIMA imn sebelumny ilun pemben (ifferencing) engn nili pembe yng el ienun Meoe eu menggunn meoe permln ARFIMA lngsung Moel ARFIMA yng iji l Moel ARFIMA(0) Moel ARFIMA(0) n Moel ARFIMA() Perben ri eu meoe ini ienun bersrn nili ri MSE (Men Sure Error) Sofwre yng igunn p peneliin ini l Sofwre R (OSSR) K Kunci : ARFIMA MSE OSSR long memory ABSTRACT Tis pper compres wo forecsing meos from ARFIMA moel Te firs meo uses ARIMA forecsing meo ime series re firsly ifference by e vlue of ifferencing prmeer Te secon meo uses ARFIMA forecsing meo irecly Tis pper uses ree ARFIMA moels ie ARFIMA(0) ARFIMA(0) n ARFIMA() moels Te ifference from ese wo meos is eremine bse on e vlue of MSE (Men Sure Error) Keywors : ARFIMA MSE (Men Sure Error) OSSR long memory PENDAHULUAN Mermln suu ejin merupn suu proses penenun suu nili yng i ieui yng mungin erji p ms yng n ng P nlisis ere wu unu membnu penenun nili yng mungin erji iperlun sebelumny Unu eperlun permln m ere wu ms llu rus isesuin engn eperlun penelii Ji ingin memperirn nili ms ng lm rin m erlebi ulu ibu lm inervl rin Seel iperole m lng selnjuny l memoeln bersrn ienifisi seperi p Meoe Box-Jenins menggunn plo ACF n PACF Alny plo ACF n PACF menunjun pol long memory ini erli ri nili-nili uoorelsi p plo ACF u PACF urun secr lmb unu lg yng semin mening Ienifisi ini menginisin bw nili ri (oefisien pembe ifferencing) bernili pecn seingg moel yng pling coco l Moel ARFIMA ( Auoregressive Frcionlly Inegre Moving Averge) Pemoeln ARFIMA perm li iembngn ole Grnger n Joyeux (980) yng merupn pengembngn ri moel ARIMA (Auoregressive Frcionlly Inegre Moving Averge) Hosing (98) mengji sif-sif long memory ri moel ARFIMA ssioner n nonssioner Sowell (99) mengembngn pensirn prmeer pembe mellui Meo Exc Mximum Lielioo Bern (995) mengembn sebu penen Mximum Lielioo unu prmeer pembe mellui Meoe Nonliner Les Sure (NLS) Wlupun moel ARFIMA lebi pliif n ur lm memoeln ibningn engn Moel ARIMA n epi msi erp beberp esulin lm permlnny Proses permln moel ARFIMA i semu moel ARIMA bi secr memi mupun secr ompusi Unu iu p peneliin ini n iji meoe permln ARFIMA Meoe perm lun pemben ri bersrn nili yng el iienifisi 09
0 Gumgum Drmwn seingg mengiui moel ARIMA(p0) n permln mengiui meoe permln ARIMA Meoe eu permln ilun mellui meoe ARFIMA(p) secr lngsung MODEL ARFIMA Moel ARFIMA(p) yng iembngn Grnger n Joyeux (980) l sebgi beriu φ ( B )( ( Z μ) = θ ( () engn: = ines ri pengmn = prmeer pembe (bilngn pecn) μ = r-r ri pengmn IIDN(0 σ ) φ( = φ B φ B φ B p p l polinomil AR(p) θ( = θ B θ B θ B l polinomil MA() ( ) ( B = = )( ) B operor pembe pecn = 0 Unu suu bernili pecn operor ifferencing frsionl ( Γ ( + ) ( = + B = Γ ( )! Γ ( + ) Ji persmn λ ( ) = Γ ( )! m: unu = iperole unu = iperole unu = 3 iperole iefinisin sebgi () p persmn () ijbrn unu berbgi nili Γ ( + ) ( )! = = Γ ( )! ( )!! Γ ( + ) ( + )! ( ) = = Γ ( )! ( )!! Γ ( + 3) ( + )! ( )( ) = = Γ ( )3! ( )!3! 6 n seerusny Persmn () p iulis embli menji ( ) B = + λ ( ) B = engn λ 0 ( ) = λ ( ) = ( ) ( ) λ = λ 3 ( ) = ( )( ) n seerusny 6 Seingg persmn () i s p iulis menji ( = B ( ) B 6 3 ( )( ) B (3) Sisi Vol 9 No Nopember 009
Perbningn Meoe Permln ARIMA n ARFIMA Permln Moel ARIMA Permln p moel ARIMA p ml ini mengiui persmn p (Cryer 986) engn persmn permln AR() MA() n ARMA() msing-msing sebgi beriu: Z( ) = μ+ φ ( Z μ) Z ( ) = μ θ (4) Z = μ + φ Z μ φ θ ( ) ( ) engn l perioe yng n irmln µ l r-r ere wu φ l prmeer Auoregresi θ l prmeer Moving Averge l resiul e- Permln Moel ARFIMA Permln p moel ARFIMA p srny sm engn moel ARIMA p Persmn (4) p ibenu menji persmn ( p )( ) ( φ φ φ θ θ θ ) ( θb θb θb ) B B B B Z = B B B Z φ Z φ Z φ Z = p ( p p Menuru Persmn () p ibenu persmn sebgi beriu ( ) ( + ) ( )! ( λ ( )) Γ B = B = B = 0 Γ = 0 seingg Persmn (5) i s menji ( θb θb θb ) Z = ( φz + φz + + φpz p) + ( λ ( ) ) B = 0 Dengn menglin seip suu ri persmn i s engn menji engn θ θ Z = φz + φz + + φpz p + f( ) f( ) f( ) (5) m persmnny (6) Sisi Vol 9 No Nopember 009
Gumgum Drmwn f() = λ ( ) B = 0 f( ) = λ ( ) B = 0 f( ) = λ ( ) B = 0 Tsirn lng e epn iperole engn menggni ines menji Nili 0 n T θ Ẑ = φz + + φpz p+ f(t + ) f(t + ) + = unu permln Z Z 0 E( Z Z TZ T ) = f > 0 0 E( Z TZ T ) = 0 > 0 3 KAJIAN SIMULASI (7) Simulsi menggunn Sofwre R versi 7 engn bnyny T = 300 600 engn perulngn 000 li Unu mengifn fsilis pembe pecn (frcionl Difference) p Sowre R Sebelumny i insll Pcge frciff Moel ARFIMA yng ibngin mengiui Moel ARFIMA(0) n Moel ARFIMA(0) engn nili = 0 n 04 Prmeer φ n θ msing-msing 05 mengiui Disribusi Norml engn r-r nol n vrins Aursi pensirn prmeer ienun engn mengiung r-r n snr evisi ri 000 nili unu Moel ARFIMA Lng-lng lm melun simulsi Bngin ARFIMA engn T = 300 600 n perulngn sebny 00 li engn = 0 n 04 engn r-r nol n vrin Dengn Moel AR() n MA() msing-msing prmeerny 05 P Moel ARMA() prmeerny φ n θ = -0 Bgi menji u yiu rining sebny T-0 perm n 0 erir sebgi esing 3 Unu permln Meoe ARIMA lun pemben p rining sebesr = 0 n 04 emuin lun permln unu 0 perioe e epn engn persmn 4 4 Unu permln Meoe ARFIMA lun permln unu 0 perioe e epn engn menggunn persmn 7 5 Tenun nili MSE n snr evisi ri MSE ri eu meoe ersebu unu 0 perioe e epn Sisi Vol 9 No Nopember 009
Perbningn Meoe Permln ARIMA n ARFIMA 3 Tbel Nili MSE n Snr Devisi ri permln Moel ARFIMA T 300 600 Moel ARFIMA = 0 = 04 MSE SD(MSE) MSE SD(MSE) ARFIMA(0) 0048 007 0044 0073 ARFIMA(0) 005 0035 003 007 ARFIMA() 0086 03 0096 04 ARFIMA(0) 0045 0066 0049 0073 ARFIMA(0) 005 0036 008 004 ARFIMA() 05 005 0089 06 Bersrn sil simulsi p bel Moel ARFIMA(0) n ARFIMA(0) lebi ur ibningn engn permln p Moel ARFIMA() Dismping nili MSE nili snr evisi ri moel ARFIMA() relif lebi besr ibningn engn u moel linny 4 KESIMPULAN Bersrn sil perbningn u meoe permln secr simulsi mp bw Nili MSE secr eselurun memberin sil yng cuup bi Meoe pensirn mellui pemben erlebi ulu ri long memory llu ilun permln engn Meoe ARIMA relif sm engn meoe pensirn mellui Meoe ARFIMA secr lngsung DAFTAR PUSTAKA [] Bern J (994) Mximum Lielioo Esimion of e Differencing Prmeer for Inverible Sor n Long Memory Auoregressive Inegre Moving Averge Moels Journl of e Royl Sisicl Sociey Vol 57 l 659-67 [] CryerJD (986) Time Series Anlysis PWS-KENT Publising Compny BosonUSA [3] Grnger C W J n JoyeuxR (980) An Inroucion o Long-Memory Time Series Moels n Frcionl Differencing Journl of Time Series Anlysis Vol l 5-9 [4] Hosing JRM (98) Frcionl Differencing Biomei Vol 68 l 65-76 [5] Sowell F (99) Mximum Lielioo Esimion of Sionry Univrie Frcionlly Inegre Time Series Moels Journl of economerics Vol53 l65 88 Sisi Vol 9 No Nopember 009