Pemilihan Model Terbaik pada Mars Respon Kontinu

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1 Sttstk, Vol. 8 No., 9 9 e 008 Pelh odel erk pd rs Respo Kotu Bg Wdjrko Otok eg Pegjr d Jurus Sttstk, IS, Sury e-l: [email protected]; [email protected] Astrk ultvrte dptve regresso sple (ARS) dlh slh stu odel regres opretrk, ytu odel yg egsusk etuk huug fugsol tr vrel respo d predktor tdk dkethu. ARS dlh kos yg koplek tr etode sple deg rekursf prts utuk eghslk ests fugs regres yg kotu.hsl peelt eujukk hw ests preter odel ARS utuk vrel respo kotu dlkuk deg pelzed lest squre (PLS). Pelh odel ARS terk dlkuk deg prosedur forwrd d ckwrd stepwse ddsrk pd l GCV. Prosedur forwrd dlh thp utuk edptk fugs ss ksu yg eckup pegruh efek ut, terks, d kot. Sedgk prosedur ckwrd dlh thp utuk egels fugs ss yg kotrusy tdk sgfk. Hsl kj jug eujukk hw GCV deg potog regres ler dpt terukt ekerj deg k dl eetuk pelh odel terk pd ARS respo kotu. Kt-kt Kuc: ARS, Pelzed Lest Squre, GCV.. Pedhulu Dl erg dspl lu, perslh peelt uuy dlh edekt seuh fugs eerp vrel predktor pd erg ttk dl rug vrel respo. slh terjd dl dg sttstk (regres opretrk ultvrt), d dg lu koputs d rekys ( eurl etwork). Hl dktk keju yg cukup pest pd tekktekk oder utuk pegupul dt, yg telh eghslk dt dl julh esr k dl ukur upu des. Utuk eyelesk perslh terseut dperluk sutu odel pd vrel respo ()Y dl stu tu leh vrel predktor (,...,) yg eerk huug pd dt. Secr uu, lss yg dguk utuk egeksplors huug tr ()Y d (,...,) dkel deg regres. Regres ler dlh slh stu pedekt pretrk yg plg populer dl peodel dt. Dl prktek, pedekt pretrk sergkl tdk fleksel dl eodelk pol oler yg terseuy pd dt erdes tgg. Hl eotvs peggu regres opretrk, yg egutk flekslts deg egsusk sooth (ulus) dl rt fugs regres yg tdk dkethu dlh kotu d dferesel. ultvrte dptve regresso sple (ARS) dlh slh stu odel regres opretrk, ytu odel yg egsusk etuk huug fugsol tr vrel respo d predktor tdk dkethu, d epuy etuk fugsol yg fleksel. ARS dlh etode yg dperkelk oleh Fred (99). etode dlh pleets tekk-tekk utuk eyelesk slh regres, deg tuju utuk epredks vrel respo yg erl kotu erdsr eerp vrel predktor. odel ARS dsusu pd pegtur eerp koefse fugs ss d secr keseluruh dkedlk pd dt regres. Hste et l. (00) ey tk hw odel ARS ergu utuk egts perslh dt des tgg yg dkel deg curse of desolty d eghslk predks vrel respo yg kurt, sert utuk 9

2 0 Bg Wdjrko Otok egts keleh regres prts rekursf (RPR) ytu egh slk odel yg kotu pd kot, yg ddsrk pd l geerlzed cross vldto (GCV) u. Oleh kre tu peelt egkj secr teorts pelh odel terk pd ARS deg krter GCV. Sel tu, dlkuk suls d kj stud ksus utuk pleets dr kj teor.. ju Pustk Fred (99) eytk hw odel ARS dlh kos yg koplek tr etode sple deg rekursf prts utuk eghslk ests fugs regres yg kotu. Sple dlh slh stu jes potog polol, ytu polol yg elk sft tersege. Sft tersege eerk flekslts leh drpd polol s, sehgg eugkk utuk eyesuk dr secr leh efektf terhdp krkterstk lokl pd sutu fugs tu dt. Wh (990) eujukk hw sple elk sft - sft sttstk yg ergu utuk eglss huug dl regres. Sple dl regres opretrk terus erkeg sp pd odel dptve (Bler d Fhrer, 00) d ultvrt respo (Holes d llck, 003). Sel tu, He d Sh (998) egegk pedekt ootocty utuk egests fugs ss sple, sedg-k Hll d Opsoer (005) egguk pedekt pelt kudrt terkecl. Regres prts rekursf (RPR) erupk pedekt fugs yg tdk dkethu deg egguk pegeg fugs ss. Sel tu, RPR dlh sutu kosep geoetr yg eg derh deg dsr rtetk, ytu pejulh d perkl. org d Soqust (963) eperkelk R PR dl rset utotc tercto detecto (AID), seljuty org d esseger (973) eeuk het AID (HAID), yg dguk utuk eproduks poho-poho klsfks. Veles d Rpley (994) egguk tur pegklsfks pd etode HAID utuk epredks sutu oyek. odel ARS dguk utuk egts keleh RPR ytu eghslk odel yg kotu pd kot. Beerp perk yg dlkuk utuk egts keterts RPR, tr l eghslk fugs ss ejd: K () q ()() k v( k,) k k B s t q f () Ests dr kurv regres secr uu ddptk ellu pelzed lest squre (PLS) yk euk pers erkut: y f f d Setelh dlkuk odfks odel RPR d dkosk deg sple, esttor odel ARS dpt dtuls seg erkut: deg K fˆ() [()] s t 0 K s 0(,) k v k k k = kostt (ss duk) = koefse dr fugs ss ke- = yky fugs ss (ocostt ss fucto) = derjt terks k = ly v ( k,) = vrel depede t k = l kot dr vrel predktor v ( k,) Deg egguk esttor ARS, k odel ARS dlh : () Sttstk, Vol. 8, No., e 008

3 Pelh odel erk pd rs Respo Kotu deg K y [ s.()]() t B 0(,) k 0 v k k k K B () [ s.()] t k v( k,) k k Dl etuk trks dpt dtuls ejd: Y B ε deg, Y Y, 0 Y (,,) (,,) ε, (,,) K K ()() s t s t K K ()() s t s t B K K ()() s t s t (,) (,) k k (,) (,) k k (,) (,) k k () (3) 3. Ests D GCV odel rs Respo Kotu odel ARS dl pers ( 3), ˆ erupk preter yg k dests dr dt, ellu pelzed lest squre (PLS) yg telh dodfks. Hl djrk dl ukt teore dwh. eore. Deg egguk esttor ARS dl Pers (), d 0 sgulr d preter soothg k pedug dr ˆ dlh deg ˆ () B B D B Y B [,() ] K v,( k,) tk B trk o (4) Bukt: Perhtk pers: y f f d Deg eperhtk, d, B f (,)() ˆ 0 B ' ' f (,)() ˆ 0 B '' '' f (,)() ˆ 0 (5) (6) (7) Sttstk, Vol. 8, No., e 008

4 Bg Wdjrko Otok '' slk, k, deg, f d B () d '' '' (())(())() S 0 [(()) B ] B B '' '' d '' '' [(()) B ]()() d f d d S d J j B () j () B d Sehgg pers d ts ejd tu ASR RP (( y,))() ˆ f J dl etuk trks ASR()()() Y B Y B D D ASR() ASR RP slk, k koefse fugs ss ˆ dperoleh deg euk pers (3) tu (4). Sehgg, ASR() u k ASR() ˆ 0 ˆ Y B D ASR() ˆ [()() Y B ] 0 ˆ ˆ (5) D B Y B Y BY B B ˆ ˆ 0 B Y B B ˆ ˆ 0 B B ˆ () B B D ˆ ˆ ˆ() B B D B Y Peyeles yg derk dl Pers ( 6) dlh euk erupk esttor g ˆ. Jd eore terukt. ASR() (8) (9) (0) () () (3) (4) (6) d Sttstk, Vol. 8, No., e 008

5 Pelh odel erk pd rs Respo Kotu 3 eore. Deg egguk esttor ARS dl pers (), d B trk o sgulr d preter soothg 0 k pedug dr ˆ dlh ˆ () B B B Y (7) K deg B [,() ] v,( k,) tk Bukt: Perhtk pers: y f f d deg 0, k pers d ts ejd: y f Dr Pers (3), k ˆ() f B, sehgg Pers (8) ejd: Y B Y B Y B ()() Z (8) Utuk eperoleh esttor ˆ dguk etode kudrt terkecl, yg pd prspy euuk Z, dytk seg erkut: Y B Z ()() Y B ˆ ˆ B Y Y B B B Z () Y Y ˆ ˆ ˆ ˆ B Y B B Z ( Y Y ) ˆ ˆ ˆ Utuk eperoleh pers orl dlkuk deg euruk secr prsl terhdp ˆ deg hsl seg erkut: Z B Y B B ˆ 0 ˆ B Y B B ˆ 0 B Bˆ B Y (9) kre B trks o sgulr, k - ˆ ( B B) B Y (0) sedgk turu kedu terhdp ˆ dlh Z B B > 0. ˆ Jd eore terukt. Pd peodel ARS, pelh odel dguk etode stepwse. Forwrd stepwse dlkuk utuk edptk fugs deg julh fugs ss ksu. Krter pelh fugs ss pd forwrd stepwse dlh deg euk ASR. Utuk eeuh kosep prseo (odel sederh) dlkuk ckwrd stepwse ytu elh fugs ss yg dhslk dr forwrd stepwse deg euk l geerlzed cross-vldto (GCV). Sttstk, Vol. 8, No., e 008

6 4 Bg Wdjrko Otok Utuk euktk GCV pd odel ARS dperluk eore erkut. eore 3. Jk rk dr P dlh (+) k P elk (+) ut l ege yg tdk erl ol d seyk [N-(+)] ut l ege yg erl ol. Bukt: slk dlh l ege d vektor ege, k: P () 0 jk dklk ejd P P P, sehgg () tu () 0 d P dlh trk depote ( P P ) k. Hl eujukk hw l ege dr P erl seyk (+) d erl 0 seyk [N -(+)], sehgg rk dr trk P s deg (+). eore 4. slk R dlh trks kudrtk deg R () R X X, B dlh fktor Cholesky dr X X. slk U d Q trks dgol sedek hgg UQ U RDR. seljuty, Z X() R U k Z U RX d slk ˆ U()() R R U ˆ k peyeles ˆ dlh: () I + Q ˆ Z Y =(U B)X Y Seljuty Xˆ Zˆ d trks Ht, d S()() Z I + Q Z I + Q Z Z I + Q tr[()] S {() tr } {() }() tr Q Q dlh trk dgol ke- dr Q. deg derjt es, Bukt: d jug, X X D R ()()() R D R R R R DR R ()()()() R I RDR R R U I Q U R U R() X X D R U I Q errt, U R()(())() X X D R U U R ˆ U R X Y tu, () I R ˆ Z Y jug sehgg X ZU () R ˆ () ˆ ˆ. Jd eore 4 terukt. X ZU R Z eore 5. slk dguk odel ARS pd Pers (), k optl dperoleh deg krter GCV seg erkut: (( y,))ˆ f GCV () C() () C ()() C. d deg,, C() =rce(()) B B B B, d 4. Sttstk, Vol. 8, No., e 008

7 Pelh odel erk pd rs Respo Kotu 5 Bukt: Perhtk Pers (3) d esttor fugs ARS kotu dlh ˆ () B B D B Y, k f (,)() ˆ ˆ H Y deg, H()() ˆ B B B D B. Seljuty deg eperhtk pers erkut, ()((,)) CV y f ˆ () d GCV() (( y,)) f [(())] tr I A k yky preter odel ARS sel ss duk, dpt dperoleh deg esusttusk pers (0) ke dl pers (3) seg erkut: Y B[()()]() B B B Y B B B B Y PY () Deg peukt eore 3, k P = () B B B B ˆ erukur ()(). Kre P dlh trk setrs d depote k rce dr trks P s deg rk dr P yg erupk yky preter fugs ss sel kostt d yky preter yg dests, dotsk seg C() =rce(()) B B B Seljuty erdsrk eore 4 d pers (), k peyeut pd GCV, dlh deg, Y B B B D B Y A Y ˆ [()()]() B A()() B B B + D B. Sehgg, tr[()] I A [() tr ] I B B B + D B tr[() B B B ] + D B tr[() B B B ] (())( tr peh pd duk ( 0 B B B B tr BD B + B D B C() (()). tr B B B B d C() B ) tr(() ), sedgk d dsrk erl d 4. Jd eore 5 terukt. B B B B B kre dl odel ARS sellu eltk ss. Berkut dsjk prosedur forwrd d ckwrd:. Prosedur Stepwse Forwrd ARS. Algort. Forwrd Stepwse ( ) B() ; ( ) Loop utl > : lof* ~ ( 3) For = to (-) do: ( 4) For v {v(k,) k K} ( 5) For t {vj B(j) > 0} ( 6) g B ( ) B ( )[ ( v t t)] B ( )[ ( v t)] ( 7) lof,,+ LOF(g) Sttstk, Vol. 8, No., e 008

8 6 Bg Wdjrko Otok ( 8) f (lof < lof*) the (lof* < lof); * ; v* v; t* t; ( 9) ed f (0) ed for () ed for () ed for (3) B ( ) B* ( )[ ( v* t*)] (4) B ( ) B* ( )[ ( v* t*)] (5) (+) (6) ed loop (7) ed lgort Algort () egpleetsk g forwrd pd ARS deg eodfks lgort regres prts rekurs (RPR). Pgkt (q=) pd fugs ss dsusttusk utuk step fucto pd rs 6,, 3. Pret ss fucto dsukk dl eodfks odel pd rs 6 d ssy d updte ellu logk pd rs -4. Perkl fugs ss dpks utuk eperoleh fktor elput vrel yg ered deg kotrol loop ts vrel dl rs 4. Algort k eghslk q= tructed power. Fugs ss erupk suset dr tesor yg legkp, ss deg loks kot pd seu rgl dt yg ered ly. Sepert pd lgort regres prts rekurs (RPR) pd set ss fucto dhlgk deg ckwrd stepwse utuk eghslk set ss fucto khr. Loks kot dhuugk deg pedekt, dguk turu ss pecewse cuc yg kotu pd turu pert, sehgg eghslk odel khr deg turu yg kotu.. Prosedur els ckwrd ARS. Algort (). Bckwrd Stepwse ( ) J*={,,, -; K* K ( ) lof { jj *} LOF[()] jbj j jj * ( 3) For = to do: ~; L K* ( 4) For = to do: K L-{}~; L K* ( 5) lof { } [()] k k K LOF k B k kk ( 6) f (lof < ) the ( lof); K* K; ed f ( 7) f (lof < lof*) the (lof* lof); J* K; ed f ( 8) ed for ( 9) ed for (0) ed lgort hp ckwrd stepwse pd ARS eghslk rego yg overlp. Peug fugs ss pd ARS tdk eghslk lug pd rug vrel predktor (sel ss kost tdk dhlgk). Pd lgort (), rs pert eujukk sl dr fugs ss J* yg dhslk oleh lgort (), setp ters d lur for loop eyek stu fugs ss dug d ddl for loop utuk elh fugs ss. Sel elkuk ckwrd stepwse tdk perh eug kostt ss B()=. Algort k eetuk rs odel seyk (-), setp rs epuy stu fugs ss leh sedkt ddgk deg rs seeluy. odel terk rs erd pd kupul J* st seles seleks. Hl dlh g pd ARS yg eyeleks suodel dl euk predks ests resdul deg krter geerlzed cross vldto (GCV) pd pers (). 4. Hsl Eprs Kj eprs dl peelt elput du hs. Pert dlkuk utuk eujukk hw GCV dpt ekerj deg k dl proses peetu odel ARS Sttstk, Vol. 8, No., e 008

9 Pelh odel erk pd rs Respo Kotu 7 terk. Bg dlkuk deg egguk sutu dt suls utuk fugs yg oler. Kedu dfokusk pd peodel ARS respo kotu pd dt rel ytu es gs ug kedr erh ks solr (opsts) Ekspere suls dl rgk eujukk g odel ARS yg dperkelk dpt ekerj deg k. Krter odel ARS dpt ekerj leh k dlht dr l geerlzed cross vldto ( GCV) terkecl. Kj eprk egguk dt y () suls dlh egu odel kurv regres, f, =,,..., deg fugs: SIULASI : 5 5 f () e deg 50, 00, 50; 0. 05, 0. 5, ; ~( N,) 0 ; X ~( U,) 0 SIULASI : f () s() deg 50, 00, 50; 0. 05, 0. 5, ; ~( N,) 0 ; X ~( U,) 0 Nl GCV deg erg vrs pd fugs, pegt () secr legkp tersj pd el erkut. el. Nl GCV Pd Sple ructed d ARS GCV Fugs Sple ARS 0, , , SIULASI 00 0, , , , , , SIULASI 00 0, , , Berdsrk el, pd odel sple, teryt l GCV sek esr pd odel suls, deg pegt yg sek esr d vrs kost. Sedg pd odel suls, deg pegt yg sek esr d vrs kost eerk l GCV yg sek kecl. Sedgk pd ARS, l GCV sek kecl pd odel suls, deg pegt yg sek esr d vrs kost. Sedg pd odel suls, deg pegt yg sek esr deg vrs = 0,05 eerk l GCV yg sek kecl, tetp pd vrs = 0,5 d l GCV ervrs. Secr keseluruh fugs optl terjd pd vrs kecl deg serg. Sedgk utuk peodel es gs ug erh kr solr (0psts) dpegruh oleh vrel erk (X), odel (X), jes kedr (X3), tpe es (X4), jes turo (X5), thu peut (X6), s slder (cc) (X7) d jrk tepuh (k) (X8). Dl peodel ARS, utuk eperoleh odel terk dsrk julh fugs ss ksu tr ( sp 4) kl julh vrel depede. odel ARS deg vrs ksu oservs d terks tersj dl el. Berdsrk el, odel terk dlh odel 3. Hl dpt dtujukk pd l GCV yg plg kecl d R yg esr. Sehgg odel ARS utuk opsts dytk dl pers erkut: Sttstk, Vol. 8, No., e 008

10 8 Bg Wdjrko Otok f ˆ() BF3.060 BF5 deg, BF = (0, AHUN_P ); BF3 = (ODEL = OR ODEL = 7 OR ODEL = 8 OR ODEL = ); BF5 = (ODEL = 4 OR ODEL = 6 OR ODEL = 7 OR ODEL = 8 OR ODEL = 0 OR ODEL = OR ODEL = ) * BF; Pers d ts dpt dterpretsk seg erkut: () Koefse BF3 Setp peh BF3 seesr stu-stu k ekk Opsts (Y) seesr,0 perse deg egggp BF5 kost. Leh ljut dpt dktk hw peruh kek opsts (Y) terjd pd odel ke dr (ft(), Pther(7), Prego(8) d Kud()) () Koefse BF5 Setp peh BF5 seesr stu-stu k euruk Opsts (Y) seesr,060 perse deg egggp BF3 kost. Leh ljut dpt dktk hw peruh peuru opsts (Y ) terjd pd odel kedr (Everest(4), Elf(6), Pther(7), Prego(8), Colt(0), Kud(), Dy()) deg thu peut d ts thu 995. Vrel yg epegruh opsts dlh odel kedr d thu peut. odel kedr erupk vrel yg petg pert deg kotrus 00 perse dl epegruh es opsts, seljuty thu peut erupk vrel yg petg kedu deg kotrus 59,05 perse. el. Peetu odel ARS terk pd Opsts odel BF I O # BF Ler GCV Kuk R SE. Regresso odel ,7 83, ,859 odel ,53 76,554 0,96 5,6 odel ,7 76,9 0,96 5,6 odel ,785 77,84 0,959 5,9 Keterg: BF= fugs ss, I= ksu terks, O= u oservs pd setp surego 5. Kespul Ests preter pd kurv regres dlkuk deg euk pelzed lestsqures (PLS) yg dodfks ertuju eeuk sutu peyeles 0 d,,...,. Seg telh djrk dl teore d teore dperoleh ests utuk odel ARS respo kotu. Hsl kj eprs eujukk hw GCV dpt ekerj deg k dl eetuk pelh odel terk yg dterpk pd odel ARS respo kotu. Pd khry, hsl kj tetg GCV dktk deg dt rel eerk teu ru, ytu dperolehy pelh odel terk ddsrk pd GCV deg potog regres ler. Dftr Pustk []. Bller, C., d Fhrer, L. 00. Byes vryg-coeffcet odel usg dptve regresso sple. Sttstcl odelg. []. Fred, J. H. 99. ultvrte Adptve Regresso Sples (wth Dscusso). he Als of Sttstcs, 9:-4. [3]. Hll, P., d Opsoer, J. D heory for pelzed sple regresso. Boetrk, 9:- 05. [4]. Hste,., shr, R., d Fred, J. H. 00. he Eleet of Sttstcl Lerg: Dt g, Iferece, d Predcto. Sprger seres Sttstcs, New York. [5]. He, X., d Sh, P oote B-Sple Soothg, Jourl of Aerc Sttstcl Assocto, 93. [6]. Holes, C. C., d llck, B. K Geerlzed Noler odelg wth ultvrte Free- Kot Regresso Sple, Jourl of Aerc Sttstcl Assocto, 98. Sttstk, Vol. 8, No., e 008

11 Pelh odel erk pd rs Respo Kotu 9 [7]. org, J. N., d Soqust, J. S Prole the lyss of survey dt d proposl. Jourl of Aerc Sttstcl Assocto, 58: [8]. org, J., d esseger, R HAID: A sequetl serch progr for the lyss of ol scle depedet vrles. (echcl Report). Isttute for socl Reserch, Uversty of chg, A Aror, chg. [9]. Veles, W. N. d Rpley, B. D oder Appled Sttstc wth S Plus. Sprger Verlg, New York. [0]. Wh, G Sple ethod for Oservtol Dt: Socety for dustrl d Appled thetcs, Phldelph, Peyylvc. Sttstk, Vol. 8, No., e 008