Jurl Eksoesil Volume, Nomor, Noember ISSN 85-789 Pegesimsi Prmeer Model Auoregresif Movig Averge (ARMA deg Meode Ucodiiol Mximum Likelihood Esimio The Esimio of Prmeer Auoregressive Movig Averge - Model Wih Ucodiiol Mximum Likelihood Esimio Suyio Progrm Sudi Sisik FMIPA Uiversis Mulwrm Absrc A ime series is ordered seuece of observios. The orderig is usully hrough ime or riculrly i erms some eully ime iervls, d i my lso be ke hrough oher dimesios, such s sce. There re vrious objecives for sudyig ime series. These iclude he udersdig d descriio of geerig mechism, he forecsig of fuure vlues d oiml corol of sysem. The irisic ure of ime series is h is observios re deede or correled, d he order of he observio is ideiclly o he sme imes mesure. The rocedure o hd ime series re model ideificio, rmeer esimio, digosic checkig & model selecio, d forecsig. I his ricle discussed he secod se h is rmeers esimio he uoregressive movig verge (ARMA models by usig Ucodiiol Mximum Likelihood Esimio. Uder he ssumio of kow order d of he ARMA rocess, i rmeers c be esimed by usig ucodiiol mximum likelihood esimio d hrough he simulio ARMA(, d yield he sme vlue of he esimor rmeer. Keywords : Auoregressive movig verge models, he esimio rmeer, ucodiiol mximum likelihood, forecsig, bck-forecsig, sum sure error I. PENDAHULUAN Model umum d lisis dere wku dimk model Auoregressive Iegreg Movig Avrge (ARIMA yg elh dieljri secr medlm oleh George Box d Gwilym Jekis (976, d m merek serig disioimk deg roses ARIMA. Pd model ARIMA erdiri dri du bgi yiu bgi uoregressive d movig verge. Secr umum model ARIMA ii diulisk deg osi ARIMA(,d,, dim meyk orde roses uoregressive (AR, meyk orde roses movig verge (MA d d meyk orde rsormsi embed (differecig. Pd model ARIMA(,d, jik hrg d = mk model mejdi ARIMA(,, u dimk model ARMA(,, jelsy model ARMA(, dlh model ARIMA uuk d dere wku yg ssioer yg idk meglmi rsormsi embed. Jik d = d =, mk model dimk ARIMA(,, u model ARMA(, u lebih umum dimk model AR( yki model uoregressive orde, d jik = d d = mk model ARIMA mejdi model ARMA(, u dimk model Movig Averge orde d diosik deg MA(. Berdsrk edek Box-Jekis, dlm melkuk lisis dere wku erd em h yiu: ( ideifiksi model yg erdiri dri merumusk model umum d ee model semer; ( eksir (esimio rmeer; (3 emeriks digosik model (digosic checkig d (4 erml (forecsig. Pd embhs sebelum (Suyio, elh membhs egesimsi rmeer model AR deg meode mome, meode kudr erkecil d meode mksimum Likelihood bersyr, deg hsil eelii bhw jik orde roses AR dikehui mk egesimsi d dilkuk deg megguk ig meode yiu meode mome, ordiry les sure (OLS d meode mksimum likelihood (ML, d keig meode ersebu memberik hsil eksir rmeer yg sm, d jik orde AR idk dikehui mk rosedur egesimsi rmeer megikui h Box-Jekis yiu: ( ideifiksi model semer; ( egesimsi rmeer uuk beber orde d h erm; (3 memilih orde yg memberik ili iformio crieri miimum. D sebgi keljuy d rikel ii dibhs egesimsi model ARMA deg meode Mksimum Likelihood k bersyr (Ucodiiol Mximum Likelihood, dim r eelii msih msih relif sediki yg membhs eori ii. Pd rikel ii beruru-uru k diurik model umum ARMA(,, fugsi uokorelsi d fugsi uokorelsi rsil, egesimsi rmeer ARMA d liksiy d d dere wku Progrm Sudi Sisik FMIPA Uiversis Mulwrm 3
Jurl Eksoesil Volume, Nomor, Noember ISSN 85-789 II. TINJAUAN PUSTAKA Model ARMA Seeri yg elh diurik d edhulu bhw model Auoregressive Movig Averge u ARMA(, dlh model khusus dri model ARIMA. Model ARMA meruk model cmur yiu cmur model Auorgressive (AR d Movig Averge (MA. Beuk umum model cmur ARMA(, dlh ( B Z ( B, ( Z Z ; E Z deg ( dlh me roses {Z }; { } dlh roses whie oise d ( B B B B ; ( B B B B ser B dlh oeror bckshif yg didefiisik j oleh B Z Z j. Persm ( d diulis Z Z Z Z u Z Z Z Z ( Deg (,(Wei,999. Proses ARMA(, ssioer jik semu kr-kr ersm ( B erlek di lur ligkr su d iveribel jik semu kr-kr ( B erlek di lur ligkr su. Jik kedu rus ersm ( diklik Z k d kemudi dihiug ili hr u ekseksiy did fugsi uokovrisi roses ARMA(, : k k k k E( Z k E( Z k E( Z k E( Z uuk k j. (3 Kre k j mk fugsi uokovrisi roses ARMA(, d diulis k k k k ; k, (4 sehigg fugsi uokorelsi (fk roses ARMA(, uuk k dlh k k k k (5 Uuk hrg = d =, mk model d ersm ( dimk roses ARMA(, yg model umumy dlh ( B Z ( B u Z Z. (6 Proses ARMA(, d ersm (6 ssioer jik d iveribel jik. Berdsrk ersm (3 mk fugsi uokovrisi roses ARMA(, dlh k k E( Z k E( Z k (7 Uuk k =, mk ( ( (, d uuk k = did, sehigg fugsi kovrisi roses ARMA(, dlh ( ; ( ( (, ( k k uuk k, (8 d fugsi uokorelsiy (fk dlh uuk k ( ( k, uuk k k uuk k (9 Berdsrk ersm (9, fugsi uokorelsi roses ARMA(, uru secr eksoesil (siusoid meuju deg bermbhy lg (k u dies dow uru secr ekoesil (dies dow. Nili fk d lg erm ergug d rmeer d d fk d lg kedu d seerusy megikui ol fk AR(. Sedg rumus umum fugsi uokorelsi rsil (fk roses ARMA(, dlh komleks d idk derluk. Tei erlu dic bhw kre ARMA( memu roses MA( mk fky jug dies dow semki besr k yg beuky ergug d rmeer d, (Aswi-Sukr, 6. Progrm Sudi Sisik FMIPA Uiversis Mulwrm 4
Jurl Eksoesil Volume, Nomor, Noember ISSN 85-789 Jik fk smel dlh k ( Z Z ( Z k Z ˆ ˆ k k, ˆ ( Z Z k =,,,..., mk deg meode mome mellui ersm (9 d deg memerhik syr kessioer ser iveribel roses ARMA(, mk esimsi rmeer d dieuk d kemudi d dieuk deg moode ierif, (Mkridkis 998. Uuk seljuy, d rikel dibhs bgim egesimsi rmeer ARMA(, deg megguk meode Ucodiiol Mximum Lilkelihood Esimio (ML k bersyr. Pegesimsi Prmeer Proses ARMA deg Meode Ucodiiol ML Seeri yg dielh diurik sebelumy bhw { } d ersm ( dlh roses whie oise yg slig bebs d berdisribusi ideik N(,. Kemudi eryy dlh kh bis dilkuk erml mudur (bck -foreccs ili-ili yg idk dikehui Z* ( Z,, Z, Z d * (,,,. Teu sj ii d dilkuk kre sebrg model ARMA d diyk dlm model mju ( B B Z u model mudur ( B B, ( ( F F Z ( F F e, ( j deg F Z Z j. Kre sif ssioer mk ( d ( memuyi srukur kovri yg sm, sehigg { e } jug meruk roses whie oise deg me sm deg ol d vrisiy e. Kre { } dlh iid. N(,, mk memuyi fugsi ked elug (FKP / f (, ex, ( d fugsi ked elug bersm dri (,, dlh P(, = L(, = f (,. f (,.. f (, / =.ex. (3 Box d Jekis (976 memberi eujuk uuk fugsi log-likelihood k bersyr yiu S( l L(, l (4 deg S( dlh fugsi jumlh kudr yg diberik oleh S(, E( Z, (5 deg E( Z dlh ekseksi bersyr dri jik dikehui ˆ, ˆ, d Z. Hrg-hrg d ˆ yg memiimlk S( d ersm (5 dimk Ucodiiol Mximum Likelihood Esimors (ML k bersyr. Kre ili l ( d ersm (4 L ergug d d egm d suku S( sehigg eksir ML k bersyr dlh ekuivle deg Ucodiiol Les Sures Esimor deg memiimlk fugsi S (. Dlm rkeky ejumlh (5 diesimsi oleh sebuh ejumlh berhigg S(, E( Z, (6 M deg M dlh bilg bul yg cuku besr sedemiki sehigg E ( Z E( Z ; ( M uuk bilg osii yg cuku kecil yg dieuk. Berdsrk ersm (6 mk berimliksi E( Z d oleh kre iu E( d dibik uuk ( M. D seelh edug ˆ, ˆ d ˆ dieuk, mk edug yiu ˆ d dihiug S( ˆ, ˆ, ˆ ˆ. (7 Progrm Sudi Sisik FMIPA Uiversis Mulwrm 5
Jurl Eksoesil Volume, Nomor, Noember ISSN 85-789 Perhiug jumlh kudr k bersyr (6 dilkuk secr rekursif megguk ersm (, deg rodur sebgi beriku : ( melih kessioer d, jik d idk ssioer d me mk dilkuk rsormsi embed, ( berdsrk d yg ssioer, e uuk dihiug ili deg megguk model bckcsig ( deg ili wl e d Z Z ser ili wl rmeer deg memerhik syr kessioer d iveribel, (3 meghiug ili Z ; ( M deg megguk form bckcsig deg ili e d meeuk hrg M yg memeuhi ersm (6; (4 meghiug ili uuk M d meghiug jumlh kudr ersm (6; d (5 lgkh 4 diulgi smi memeroleh ili jumlh kudr miimum. III.METODE PENELITIAN Peelii ii dlh eelii ekserime, embuki liksi dri suu eori mellui sudi lierur. D eelii dlh d simulsi (ekserime yiu d bgki model ARMA(,, egesimsi rmeer model megguk meode ucodiiol ML d erhiug dilkuk deg membu suu rogrm liksi d rogrm MATLAB. IV.HASIL DAN PEMBAHASAN D dere wku d bel, meruk d dere wku deg ukur = hsil simulsi model ARMA(, deg ersm (,8B Z (,5B. Deg melih ol d d gmbr mk bhw ol d sejjr deg sumbu medr ( yg meujukk d dere wku ssioer. Hl ii dierku oleh FAK d FAKP d gmbr d 3 yg msig-msig bersif dies dow. Seelh h ideifiksi deg kesimul bhw d dere wku ssioer mk h berikuy dlh egesimsi rmeer deg meode ucodiiol ML. Deg megikui h dlm egesimsi rmeer d yg elh diurik di s d deg megmbil ili ˆ d ˆ did ili miimum S( 74 E( (,9, Z,(,5 dlh 7477,67 uuk M = 74 (uuk 75 egm deg Z 74 Z 75,433,5 d ili edug rmeer ˆ, 9, ˆ, 5 ˆ 3, ˆ,7886 d ˆ, 5 ser Z 99. Sedgk uuk meghsilk jumlh kudr error S = 448,4 deg M = 8. Sebgi erbdig, erhiug deg megguk rogrm MINITAB did ˆ,7886 d ˆ, 5, deg jumlh kudr gl S = 89, d egm (bckforecs dikelurk. V. KESIMPULAN DAN SARAN Berdsrk egesimsi rmeer deg meode Ucodiiol ML meghsilk model yg sm seeri model d d bgki. Agr egesimsi rmeer lebih kur deg jumlh digi yg lebih besr uuk edug rmeer yg meghsilk jumlh error yg lebih kecil lgi dierluk egembg embu bhs emrogrm yg bis mejwb ersol ii, sehigg dierluk eelii lju. DAFTAR PUSTAKA Aswi & Sukr, 6. Alisis Dere Wku, Mksr: Adir Publisher. Box, G.E.P & Jekis, GM., 976. Time Series Alysis Forecsig d Corol, d Ediio, S Frcisco : Holde-Dy. Hmilo, J.D., 994. TimeSeries Alysis, New Jersey : Priceo Uiversiy Press. Judge, G.G., Griffihs, W.E., L u keol, H., Hill, R.C., Lee, T.C., 985. The Theory d Prcice of Ecoomerics, d Ediio, USA: Joh Wiley & Sos, Ic. Kirchgsser, G., & Wolers, J., 7. Iroducio o Moder Time Series Alysis, Berli: Sriger-Verlg. Kousoyiis, A., 977. Theory Of Ecoomerics: A Iroducory Exosiio of Ecoomeric Mehods, d Ediio, USA: Hrer & Row Publishers, Ic. Mkridkis, S., Wheelwrigh, S.C., & MicGee, V.E., 998. Forecsig d Alicios, d, Joh Wiley & Sos, Ic. (lih bhs: Hri. Soejoei, Z., 987. Alisis Ruu Wku, Jkr: Kuri Uiversis Terbuk. Tsy, R.S.,. Alysis of Ficil Time Series: Ficil Ecoomerics, New York:Joh Wiley & Sos. Ic. Wei, W.W.S., 994. Time Series Alysis: Uivrie d Mulivrie Mehods, Clifori: Addiso-Wesley Publishig Comy. Progrm Sudi Sisik FMIPA Uiversis Mulwrm 6
Jurl Eksoesil Volume, Nomor, Noember ISSN 85-789 LAMPIRAN Tbel : D simulsi dere wku (,9B Z (,5B 97.84 88.64 86.4 89.37 85.4 89.97 99.89.69 3.96 5.6 4.3 6.97 5.5 4.8 7.45 5.93 9.96.9 8.5.5.4 3.43 3.3.89. 3.54.6 3.7.5.9 99.69.7 7.75.58 9.55 4.6 9.37 3.4 7.96 5.33 5.38. 97.7.98.9.79 4.3 99.57 86.45 8.9 Sumber : Simulsi 78.68 8.35 87.9 98.3 5.65 3.35 3.3 98.88 96.37 96.45 96.68 95.4.56 93.87 9.94 98.3 4.36 8.64 9.4.97 5.3 3.9. 6.58 97.8 93.7 93.86 96.34 4.7 6. 7.7.5 7.96 3.55.78.4 5.83 99..67 4.46 5.63 6.48.88 98. 99. 95.64 97.6.64 99.69 96.34 Z( 3 9 8 Time Series Plo of ARMA(, 3 4 Gmbr : Grfik dere wku simulsi ARMA(,: (,9B Z (,5B Auocorrelio..8.6.4.. -. -.4 -.6 -.8 -. 4 6 8 5 4 Lg 6 6 8 7 AuocorrelioFuciofor ARMA(, (wih5%sigificcelimisfor heuocorrelios Gmbr : FAK dere wku (,9B Z (,5B 8 9 4 Pril AuocorrelioFuciofor ARMA(, (wih5%sigificce limis for he ril uocorrelios Pril Auocorrelio..8.6.4.. -. -.4 -.6 -.8 -. 4 6 8 4 6 8 4 Lg Gmbr 3 : Grfik FAKP dere wku (,9B Z (,5B Progrm Sudi Sisik FMIPA Uiversis Mulwrm 7
Jurl Eksoesil Volume, Nomor, Noember ISSN 85-789 Tbel : Tbel ili bckcsig forecsig e d ili ˆ, uuk ˆ, 8 d 5 Z e -74-73 -7-98 99,388.434.4793 84.99558 94.43953 97.837,645 99,694 96,343.77478 7.56893.694.75757.38839 -.58.47475.37355.374836 6.6539967 9.55899.55775 5.3395474 Tbel 3. Nili S( ˆ, ˆ, ˆ uuk beber sg ˆ d ˆ ˆ ˆ M S( ˆ, ˆ, ˆ.7886,9 -,5 -,5 8 74 448,4 7477,67 Progrm Sudi Sisik FMIPA Uiversis Mulwrm 8