Kajian Metode Estimasi Parameter dalam Regresi Semiparametrik Spline

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1 W. Wowo, S. Hrytm, I N. Budtr, Kj Metode Estms Prmeter... Kj Metode Estms Prmeter dlm egres Semrmetrk Sle Whyu Wowo, Sr Hrytm, I Nyom Budtr whyu.stk@gml.com Jurus Mtemtk, Uversts Gdjh Md Yogykrt Jurus Sttstk, Isttut ekolog Seuluh Noemer Sury Itsr Pd regres semrmetrk sle, estms kurv regres dt dselesk deg metode kudrt terkecl, kudrt terkecl terelt, d reroducg kerel Hlert sce. Msg-msg edekt memlk krkterstk yg ered. Pd edekt kudrt terkecl, mslh utm dlh memlh yk kot d loksy. Pd kudrt terkecl terelt d reroducg kerel memuy ermslh yg sm, ytu meetuk l rmeter eghlus yg otml. Nmu, secr mtemts, metode reroducg kerel memer hsl yg leh umum ddgk deg kudrt terkecl terelt kre erlku utuk semu derjt oloml sle yg k derguk. Mklh k mejelsk eggu metode kudrt terkecl, kudrt terkecl terelt, d reroducg kerel hlert sce sert ered msg-msg dlm estms kurv regres semrmetrk sle. Kt kuc : egres semrmetrk, lest squre, elzed lest squre, reroducg kerel Hlert sce, sle Astrct Curve estmto or sle semrmetrc regresso c e doe y usg lest squre, elzed lest squre d reroducg kerel Hlert sce method. Ech methods hs deret chrcterstcs. Lest squre method hs rolem out how to choose the umer o kots d ther locto. Pelzed lest squre d reroducg kerel hs the sme rolem out how to choose the otml smoothg rmeter. However,reroducg kerel roch s more geerl mthemtclly th elzed lest squre due to e vld or y degree o olyoml sle tht to e used. hs er wll exle out lest squre, elzed lest squre, reroducg kerel Hlert sce s weel s derece o ech method curve estmto or sle semrmetrc regresso. Keyword : semrmetrc regresso, lest squre, elzed lest squre, reroducg kerel Hlert sce, sle. Pedhulu egres semrmetrk meruk koms tr regres rmetrk d regres ormetrk. Koms dlm hl dmksudk hw dlm regres semrmetrk memut seklgus model regres rmetrk d model regres ormetrk. egres semrmetrk mucul kre dy ksus-ksus emodel yg huug tr vrely sel d yg ler jug d yg tdk dkethu etuky. Keerd du komoe yg ered dlm regres semrmetrk mejdk emk model mejd lus d secr teor erkemg sgt est. Perkemg sel kre lks jug kre erkemgy ergkt kers tekolog komuts yg memermudh d memercet komuts. 0

2 Berkl MIPA, 3(), Jur 03 Estms kurv regres semrmetrk sle dt dselesk deg metode kudrt terkecl, kudrt terkecl terelt, d reroducg kerel hlert sce. Msg-msg edekt memlk krkterstk yg ered. Pd edekt kudrt terkecl, mslh utm dlh memlh yk kot d loksy. Pd kudrt terkecl terelt d reroducg kerel memuy ermslh sm, ytu meetuk l rmeter eghlus yg otml. Nmu, secr mtemts, metode reroducg kerel memer hsl yg leh umum ddgk deg kudrt terkecl terelt kre erlku utuk semu derjt oloml sle yg k derguk Mklh k mejelsk eggu metode kudrt terkecl, kudrt terkecl terelt, d reroducg kerel hlert sce sert ered msg-msg dlm estms kurv regres semrmetrk sle. Pejels k dmul deg kostruks model regres semrmetrk sle, metode kudrt terkecl, kudrt terkecl terelt, d reroducg kerel Hlert sce. Seg eutu k derk kesmul erkt deg metode-metode terseut.. Model Pdg smel rdom deg vrel d msg-msg smel dlh ( y, x, t ),,,...,. Dlm hl dsumsk y seg vrel reso, x seg vrel redktor yg dkethu eregruh ler, d t seg vrel redktor yg tdk dkethu etuk egruhy terhd reso. Seljuty detuk model regres semrmetrk y 0 x ( t ),,,..., () ( 0, ) rmeter utuk komoe rmetrk, d komoe ormetrk, dlm hl meruk ugs yg tdk dkethu.,,..., dlh error rdom yg slg deede deg me ol d vrs. Model () dt dytk dlm ots mtrk mejd dm YXβ +ε () y Y, y x X, x 0 β, ( t), ( t) ε Mslh estms d regres semrmetrk mucul kre dy komoe ormetrk eru ugs yg tdk dkethu etuky. Oleh kre tu, hmr terhd etuk ugs terseut dt dlkuk deg leh dr stu etuk ugs. Beer dtry dlh sle, kerel, ourer, wvelet, d oloml lokl. Secr lks, hmr-hmr memlk keleh yg ered. Seg cotoh, hmr sle sgt cocok utuk dt yg memlk erlku eruh-uh dlm su-su tervl tertetu. Hmr ourer sgt cocok utuk dt yg memlk ol erulg tu musm. Pejels mege mcm-mcm hmr dt dlht d Eggermot d Lrcc (009). 3. Metode Kudrt erkecl Metode kudrt terkecl meruk metode yg sgt lzm derguk dlm regres ler. Prs metode dlh memmumk kudrt resdul. Metode jug s derguk dlm regres semrmetrk sle. Wowo, dkk (009, 00) telh megguk metode utuk estms rmeter d regres semrmetrk d jug 03

3 W. Wowo, S. Hrytm, I N. Budtr, Kj Metode Estms Prmeter... st-st sttstk estmtor yg deroleh. Peggu metode mesyrtk etuk sesk ugs (t) dlm model (). Betuk ugs sle yg s derguk dlh ugs ss sle oloml tructed. Fugs sle oloml tructed derjt deg k ttk kots,,..., k dsjk dlm etuk, deg ( t ) 0 ( t ) t t... t t k t k ( )... ( ) ( t ), t 0, t Sehgg model () dt dytk mejd Al dytk dlm mtrks, deroleh deg y x t t... t 0 0 ( t )... ( t ), k k,,..., Y = X β + Z + ε (5) x x ( t ) ( t k ) Z= x x ( t ) ( t ) ( k ) (3) (4) 0 k x( k) Seljuty, dtetuk C x z,, sehgg (5) dt dytk deg α YC ε Seljuty ddesk kudrt resdul seg erkut, ε ε ( Y C ( Y C = Y Y- C Y+ C C Estmtor deroleh deg cr memmumk (6) terhd. Seljuty (6) dturuk terhd d dsmdegk ol, sehgg deroleh : C Y + C C (7) C CC Y yg meruk ersm orml. Peyeles ersm k meruk estmtor. Sesu ljr mtrk, kre mtrk C memuy rk (3 + + k) d C C mtrk ostve-dete, mk C C k meruk mtrk osgulr. Sehgg ersm (6) k memuy eyeles tuggl, ˆ ( ) - C C C Y (8) Dlm hl, ˆ meruk estmtor kudrt terkecl. Meggt ersm (), estmtor (8) erlku hy utuk derjt oloml d yk kots k yg tertetu. Sehgg leh tet klu estmtor dytk deg ˆ ( ) -, C C C Y (6) 04

4 Berkl MIPA, 3(), Jur 03 ˆ ˆ ( ) - Y C, C C C C Y Y ˆ H ( ;,..., ) Y k deg H( ;,..., ) C( C C) C k - Permslh seljuty dlh gm meetuk derjt oloml d yk kots k yg k dguk dlm estmtor. Utuk keerlu k derguk krter Geerlzed Cross Vldto (GCV), yg ddesk : ( I H( ;,,..., k ) y k trce( I H( ;,,..., k ) GCV ( ;,,..., ) Nl d k dlh deg meyelesk otms GCV Metode estms kudrt terkecl secr GCV ( ;,,..., ) m ( ;,,..., ) ot ot ot kot + k k mtemts dt dselesk deg lgkh-lgkh yg sederh d meghslk model sttstk yg mudh dterretsk. Kesult metode dlh meetuk derjt oloml, yk kot d loks kot-kot terseut. 4. Metode Kudrt erkecl erelt Metode kudrt terkecl terelt meruk erlus metode kudrt terkecl deg memhk rmeter eghlus d elt d ugs yg k derguk. Fugs yg k derguk meruk kelurg ugs yg terderesl d tervl [,] d kotu solut d turu ertm. Betuky derk seg erkut, S(, ) { y ( x ) ( t )} 0 { ( )} ( m) t dt Dlm hl meruk rmeter eghlus, sedgk elt derk oleh { ( )} ( m) t dt Jk 0, mk hsl estms medekt hsl metode kudrt terkecl. Selky, jk, mk estms k megterols ttk-ttk dt. Estmtor terk meruk komrom tr l jumlh kudrt resdul d rmeter eghlus yg s ddtk deg memmumk l GCV. Dlm Gre d Slverm (994) dytk hw ugs yg memmumk (9) meruk ugs turl cuc sle, yg derk seg erkut : 3 ( t) ( t t ) c ( t t ) d ( t t ), t t t, =,,, t, t,, t meruk kot yg memeuh t t t. Seljuty, elt dlm (9) k dytk dlm etuk l turu kedu deg lgkh seg erkut.. Ddesk ( t ) d "( t ),,,,.. Pdg = (,, ), γ = (,, ), h t t,,,, 3. Susu mtrk, mk Q d, deg ketetu seg erkut. (9) 05

5 W. Wowo, S. Hrytm, I N. Budtr, Kj Metode Estms Prmeter... Q dlh mtrk erukur ( ) deg elemet q j,,,, d j,3,,. q, qj hj hj, qj, j h j j,3,,, q 0, jk j. j, j h j Mtrks Q s ddeks mul j, sehgg eleme terts Q dlh q. Seljuty dlh mtrks symmetrc erukur ( ) ( ) deg elemets r j,,j=,3,, ( ) seg erkut 4. Desk mtrk K deg j ( ) rj h 3 h, r, j r, h 6, d rj 0 utuk j K Q Q Seljuty huug tr Q,,,γ d K terdt dlm d Gree d Slverm (994), yg dytk dlm teorem erkut. - heorem : Vektor d γ meetuk turl cuc sle jk d hy jk memeuh Q γ Seljuty elt k dt dytk mejd { "( t)} dt γ γ K Bukt dt dlht d Gree d Slverm (994). Sehgg (9) dt dytk dlm S β, ( Y Xβ ) ( Y Xβ ) K ots mtrk mejd ; Seg hsl teorem, jumlh kudrt terelt (9) dt dytk, ( ) S β Y Xβ ( Y Xβ ) K (0) Memmumk (9) equvlet deg memmumk, Sβ, ( Y Xβ ) ( Y Xβ ) K () Seljuty, deg meuruk (0) terhd β d, d meymdegk ol, deroleh ˆ β ( X X) X ( Y ˆ) ˆ = ( I K ) (Y Xβˆ) () Dlm hl terhd du smoother, Sβ X( X X) X meghslk dug Xβ ˆ d smoother ly dlh rmetrk derk oleh ormetrk dlh meghslk S = I K meghslk dug ˆ. Seljuty dug ( ) S Y X X X X Y Xβ ˆ sedgk dug utuk β( ) ( ) ( ) ˆ = S (Y Xβ ˆ). Susttus dug ormetrk ke g ertm (), ˆ XXβX ( Y S (Y Xβ)) X ( I S ) Xβ X ( I S ) Y Persm dlh ersm orml utuk geerlzed lest squre orml, deg eleme o-dgol eru mtrk emoot ( I - S ). Sehgg, rmeter β d dt dselesk deg, 06

6 Berkl MIPA, 3(), Jur 03 ˆ ( ( ) ) ( ) β X I S X X I S Y ˆ = S (Y Xβˆ ) S Xβˆ S Y Estmtor elzed lest squre msh tergtug d rmeter smoothg, oleh kre tu hrus dlh yg otmum deg memmumk GCV. GCV ddesk dlm metode dormulk deg Y Yˆ GCV ( ) ( tr( A)) A meruk ht mtrk yg memeuh ˆ ˆ β ˆ AY Xβ X I Y ˆ ˆ deg X X X X A X I X ( I K) I Betuk kurv regres hsl metode kudrt terkecl terelt kudrt terkecl ergtug d rmeter eghlus. Nmu metode mejd leh komlek jk terdt leh dr stu redktor ormetrk. Sel tu, terrets egruh redktor ormetrk terhd reso tdk derk secr esklst mellu model sttstk, k tet mellu kurv regres. 5. eroducg Kerel Hlert Sce Peggu metode reroducg kerel Hlert sce dlm regres semrmetrk d dsry meruk erlus metode kudrt terkecl terelt deg megguk ugs yg terderesl d tervl [,] d turuy yg ke-m kotu solut d tervl terseut. Deg kt l, ugs terseut termut d dlm rug Soolev m ( m) ( m) W deg m [, ] W [, ] { :,, kot. s., ( t) dt } eroducg Kerel Hlert Sce (KHS) H dlh sutu rug Hlert dr ugs erl rel d [0,] deg st hw utuk set t [0,], ugsol Lt ( ) ( t) meruk ugsol ler terts, dlm rt hw terdt M sedemk hgg erlku L () t M t H dt ddekomoss mejd H H0 H deg H 0 rug Null, d H dlh rug yg tegk lurus deg rug Null. eroducg kerel dr H dlh ugs yg ddesk d [0,] [0,], sedemk hgg utuk set ttk t [0,] erlku t H deg t ( s) ( t, s) d Lt t,, H Utuk megguk metode reroducg kerel dlm estms rmeter regres semrmetrk, model () dytk mejd y 0 x Lt,,,..., (3) deg L, ( t ) t t Seljuty, estms rmeter deroleh deg memmumk elzed lest squre 07

7 W. Wowo, S. Hrytm, I N. Budtr, Kj Metode Estms Prmeter... deg ( m) { ( t)} dt P, W m S(, ) { y ( x ) L } P P royeks orthogol d H dlm 0 t (4) Estmtor yg k deroleh terdr dr estmtor sle rsl seg komoe ormetrk (), d estmtor rmetrk (). Estmtor sle rsl deroleh erdsrk teorem erkut. H. eorem : Al H H0 H d,,, m meruk ss d rug H 0, sert m meruk mtrk ull rk erorde m yg derk oleh m { L v},,,, ; v,,, m mk ugs yg memumk S(, ) { y ( x ) L } P dlh deg P 0 m ˆ d c (5) v v v d ( d, d,, d m )' ('M ) M ' ( c M,, ;, j,,, j ( c, c,, c )' M (I - ('M )) 'M ( Sedgk estmtor rmetrk yg ersesu dlh ˆ β ( X'(I - A( ))X) X'( I - A( ))' Y (6) Bukt derk d khr mklh. Sm d s, telh deroleh estmtor utuk regres semrmetrk sle deg metode (KHS). Nl tksr utuk vrel reso dt dytk mejd Y ˆ = Xβ + (7) λ λ Keleh estmtor yg deroleh deg metode KHS dlh keumumy yg erlku utuk serg oloml sle sm derjt-m. Iterrets egruh redktor ormetrk terhd reso dt deg mudh dlkuk mellu kurv regres. Jk terdt leh dr stu redktor ormetrk, mk k mucul kesult secr mtemts d grs. 6. Kesmul d Sr Mslh utm d estms rmeter regres semrmetrk dlh dy komoe ormetrk eru ugs yg tdk dkethu etuky. Peggu metode kudrt terkecl megsumsk etuk ugs sle oloml tructed d memerk kemudh terrets mellu model sttstk. Peggu kudrt terkecl terelt meghslk ugs turl cuc sle seg komoe ormetrk d etuk kurv regresy tergtug d rmeter eghlus. Estmtor yg deroleh deg reroducg kerel Hlert sce jug tergtug d rmeter eghlus, mu 08

8 Berkl MIPA, 3(), Jur 03 estmtory erlku utuk serg derjt oloml sle. Sehgg estmtor yg deroleh deg edekt reroducg kerel Hlert sce erst leh umum ddgk deg estmtor yg deroleh deg metode kudrt terkecl muu kudrt terkecl terelt. Dtr Pustk Gree, P.J. d Slverm, B.W., 994, Normetrc egresso d Geerlzed Ler Model, Chm & Hll, Lodo Eggermot, P.P.B., d Lrcc, V.N., 009, Mxmum Pelzed Lkelhood Estmto, Volume II : egresso, Srger Seres Sttstcs Wh, G., 990, Sle Model or Oservtol Dt, SIAM, XII, Phldelh Wowo, W., Hrytm, S., Budtr, I.N., 009, Lest Squre Methods or Estmtg Curve o Sle Semrmetrc egresso, Proceedg o Ntol Semr o Mthemtc d Mthemtc Educto, Yogykrt Stte Uversty, Decemer 5 th 009, , ISBN : Wowo, W., Hrytm, S., Budtr, I.N., 00, Ierece Ad Codece Itervl For egresso Curve I Sle Semrmetrc Model, Proceedg o Ntol Semr o Mthemtc d Mthemtc Educto, Uversty o Muhmmdyh Mlg, Jury 30 th, 00 Bukt eorem : Utuk ukt estmtor ormetrk, dt dlht d Wh (990), sedgk utuk estmtor rmetrk derk secr sgkt. Utuk medtk estmtor rmetrk, ersm (5) dytk dlm etuk mtrk mejd d c Seljuty, deg meggt huugy deg model (), mk (5) dt dytk jug mejd Y - Xβ - Mc c Dlm hl, dt dlht seg hsl smoothg terhd umum mejd A( )( = Y - Xβ - ( + ) c = Y - Xβ - c c = Y - Xβ - Y - Xβ - A( )( = (I - A( ))( D l hk, c = d = A( )( - d c' c = c'a( )( = (I - A( ))( A( )( S β, ( Y Xβ ) ( Y Xβ ) c' c Y Xβ, sehgg secr 09

9 W. Wowo, S. Hrytm, I N. Budtr, Kj Metode Estms Prmeter... ( Y Xβ) ( I A( )) ( I A( ))( Y Xβ) Y Xβ I A I A Y Xβ c c (I - A( ))( A( )( ( ) ( ( )) ( ( ))( ) ' S β, β 0 β ( X'(I - A( ))X) X'( I - A( ))' Y 0