Estimasi Spline Dan Mars Menggunakan Kuadrat Terkecil

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1 ol. 4 o. - Jl 007 Ess Sple D rs egg dr Terecl bg Wdro Oo. Shd Abr Rpog Absr T peel dlh edp ess dr esor sple d ARS fgs op egg rv rced sple d lvre Adpve Regresso Sples ARS ell d sls dr fgs ere deg berbg r spel. Sel pd d sls dg boosrp elh esbl vrs dr ed rv erseb. rer SE elh fgs g op. Hsl peel e bhw sr dr esor: Trced Sple : ~ T T β [ X X ] X ~ ARS : deg [ { v } ] d â dperoleh dr Deg ASR ˆ 0 ASR c D Pd epr bss fgs opl pecewse ler berdsr SE erd pd vrs g ecl sebrg peg. rv rced sple pd pecewse ler lebh b dbdg deg ARS re h elb d des redh. Sedg peboosrp pd rv sple p ARS eber hsl g lebh b dbdg rv sl. c: oosrp ARS SE regres oprer Trced Sple. Pedhl Alss regres eperlh hbg d pegrh vrbel predor erhdp vrbel respo. sl dlh vrbel respo d dlh vrbel predor bh peg secr hbg r d dp dls sebg ber : g ;... deg dlh ses rdo d g erp rv regres. hssw S3 e UG ogr Dose Jrs Ss ITS Srb oo_bw@hoo.co Dose Jrs Ss ITS Srb shd_lbr@hoo.co Sf pegr pd Jrs e F.IPA Uverss Hsdd ssr rpog@hoo.co

2 bg Wdro Oo. Shd Abr Rpog U egess rv preer regres d d pede prer d oprer. Pede prer dg d fors sebel eg be rv g dperoleh berdsr eor pegl s ll sehgg dp d bhw egess rv evle deg egess preer d hsl ess eg odel ere. Sedg pede oprer dg d d fors eg be rv g d ergg pd ss be rv ere sehgg eber flesbls g lebh besr Eb 988; Hrdle 990; Eod d Seve 997 Whb 990 e bhw sple el sf-sf ss g berg eglss hbg dl regres. Sple dl regres oprer ers berebg sp pd odel dpve ler d Fhrer 00 d lvre respo Holes d llc 003. U egess bss fgs sple elh debg beberp eode seper oooc He d Sh 998 d pelsed Hll d Opsoer 005. Fred 99 eodfs eerbs g dl eode recrsve prog regresso g del deg ARS. Sple ep eleh pd s orde sple gg os g b d o g erll de ebe r dl perhg g hpr sglr sehgg pers orl d dp dseles. ss l g dp egs eleh dlh bss -Sple. oosrp dlh eode g berbss oper g dg pegr rs dr sr ss pedg besr ss d selg eperc. oosrp erp slh s eode esf perhg g erbr eperb ess/lsfs g d sbl. Secr esr berg slh rg d berdes gg. Sple dl regres oprer e preer peghls g dp dplh deg berbg eode. Peel el eor sls ebdg l SE pd rv sple rced d ARS sebel d sesdh dl peboosrp ee fgs g opl d erp.. Sple Trced d ARS sl dber odel regres g ;... deg erp resdl d g rv regres. Pd pers d d fors sebel ege be rv regres g dg regres oprer deg g rv regres g d deh dess resdl rdo depede deg e ol d vrs. Dl regres opreer h dss be lf dr g o d dferesbel. Pd fgs sooh pbl secr geoer grde berbh d erll cep dp egg s dser erseb sebg ess o. erdsr osep d s sl g fgs sooh dl r g er dl s rg d fgs hss rg Sobolev W [ b] { f f f f... f } o bsol pd [b] f d d L[ b] deg f dlh fgs r e-d. L [ b] dlh hp fgs g dr eregrl pd ervl [b] b d d L [ b] { f [ f ] d } Scher 98

3 bg Wdro Oo. Shd Abr Rpog 3 Apbl dg pede rv sple rced rv regres g dp dls ed : Eb 988 g d... deg... dlh o d 0 eger o egf. l e der sple rced. Recrsve Prog Regresso RPR erp pede dr fgs f g d deh deg: S f ˆ c 3 d I[ R ] I[.] e fgs dor g ep l s per ber R d 0 ol slh c erp oefse os g de dl sbrego. Pee os pd regres d dl secr l re el des d g redh d hl d egl esl sedg d g berdes gg erdp esl. U egs hl erseb dg odel Recrsve Pro Regresso re pee os ergg oos dr d. Tep odel sh erdp eleh odel g dhsl d o pd os Pd odel ARS sel pee os g dl secr oos dr d g eghsl odel g o pd os. odel ARS dp dls sebg ber: d: 0 S v fˆ 0 [ s. v ] = bss fgs d = oefse dr bss fgs e- = s bss fgs ocos bss fgs = der ers = l = vrbel depede = l os dr vrbel depede v Pebr dr Pers 4 dp ds sebg ber: 4 fˆ 0... [ s [ s [ s v v v ] ][ s ][ s v v ] ][ s 3 v3 3 ] 5 d secr Pers 4 dp dls sebg ber: f ˆ 0 f f f

4 bg Wdro Oo. Shd Abr Rpog 4 Pers 6 e bhw pelh per elp se bss fgs s vrbel pelh ed elp se bss fgs ers r d vrbel pelh eg elp se bss fgs ers r g vrbel d seers. sl { v } dlh hp dr vrbel g dhbg deg bss fgs e- sep pelh per pd Pers 6 dp d sebg: f 7 Jd f erp pelh se bss fgs s vrbel d erp sple deg der q= g erepreses fgs vr. Sep fgs bvr pd Pers 5 dp dls sebg: f 8 Pers erseb erepreses pelh se bss fgs d vrbel d. Pebh eghbg orbs vr g dls sebg ber: * f f f f 9 Fgs rvr pd pelh g eg dperoleh deg elh se bss fgs g vrbel g dls sebg ber: f 0 3 Pebh fgs vre d bvre ep orbs dl be: * f f f f f f f f Pelh odel ARS dp dl deg eode sepwse forwrd d bcwrd. Forwrd edp fgs deg lh bss fgs s. rer pelh bss fgs pd forwrd dlh deg e Averge S Sqre Resdl ASR e Sqre Error SE. cwrd elh bss fgs g dhsl dr forwrd deg e l Geerlzed Cross-ldo GC. Fred d Slver Sf-Sf Aso Esor sl... dlh spel rdo deg fgs epd probbls f. sf osses brs esor ddefs sebg ber: Defs : Rohg975 rs esor T T dseb osse dl probbls bl sep 0 d sep l P T T bl p

5 bg Wdro Oo. Shd Abr Rpog 5 Defs : Rohg975 J { X } brs vrbel rdo { X } d overge e X dl probbls dls deg. Defs 3: Rohg975 X p X sep 0 { X X } 0 P s.. rs { X } d overge hpr ps e X d d sebg X X P{ l X X}. Defs 4: Rohg975 rs X } d overge dl dsrbs e X d d sebg X d { X F F se d F X o. X X Dr defs overges dperoleh hbg eoverge sebg ber: - J s brs overge hpr ps brs erseb g overge dl probbls. - J s brs overge dl probbls brs erseb g overge dl dsrbs. Fergsso996 Teore : Teore L Serl lvr Prb.se993 J { X } erp veor rdo..d d p R deg e X 0 deg l. d d r ovr Defs 5: Hrdle990 rs { } d { b } d sebg O b erdp blg rel sede hgg l. b Defs 6: Hrdle990 rs { } d { b } d sebg o b l 0. b Sel O ep sf-sf sebg ber: Rohg 975 J O b O b O b b. O b O b. O b O b O bb. J 0 dlh os J Defs 7. Hrdle990 rs { } d { b } d sebg O b erdp blg rel d sede hgg P b. p

6 bg Wdro Oo. Shd Abr Rpog 6 Defs 8. Hrdle990 rs { } d { b } d sebg o b 0 sede hgg l P 0. b p Teore : sl Z ~ d A s r veor U AZ ~ A AA. : U AZ A Z A r U [ U U U U ] [ AZ A AZ A ] [ A Z Z A] A [ Z Z ] A A r Z A A A re U fgs ler dr Z ~ d g pers..6 d..7 erb U AZ ~ A AA. Defs 9: Rode968 S fgs d o d sep 0 d 0 sehgg f f deg Defs 0. rle976 S fgs f. d eeh ods Lpschz order d os A > 0 sehgg f f A. 4. Hsl Peel Peel secr grs besr erbg dl d eor ber deg preer pd esor sple d ARS d eprs ell d sls deg fgs ere d erp ber deg plees hsl eor. 4.. Ess Sple d ARS Sple les sqres erp geerlss regres polol Eb988 d ess rv regres f dperoleh ell fgs g ddefs sebg ber: g dl pers g erp sple orde- deg os.... Hp dr se fgs S... dlh s rg veor berdes + g erdr dr poog polol orde- g el - r g o d dso pd r e-- d. Deg elh... } {

7 bg Wdro Oo. Shd Abr Rpog 7 f dp dess deg egess oefse-oefse dr pers..34. Slh s eode eeles hl erseb dlh deg egg les-sqres. Ddefs: ~ T deg β 3 Esor sple les-sqre dr f dlh: d f β 4 ~ T β... dlh s zer dr SE ; 5 erhdp ~. Lebh els ddefs: X { }... ; ~ dlh s peeles pers orl: T ~ T X X X 7 d =... T. J X ep r + : ~ T T β [ X X ] X ~ 8 Dr Pers 4 d 8 erlh bhw deg {... } } f dlh esor ler dr f. odel ARS dp d dl pers ber: ε 9 deg = vrbel respo = oefse dr bss fco ε = error. [ { v } ] = bss fco Ess preer deg eode dr erecl g pd prsp e lh dr error d sebg ber:

8 8 bg Wdro Oo. Shd Abr Rpog 0 Pers orl dperoleh deg er 0 secr prsl erhdp deg hsl sebg ber: Jlh preer ARS sel bss fgs d dp dperoleh deg esbss Pers e dl Pers 9 sebg ber: I l Trce dr rs I dlh lh preer bss fgs sel os d lh preer g dess dlh Trce C ] [ ~. Peeles Pers deg lss erc d efse sehgg dl deg QR decopoo Whb 990 dl Scho 997. J bss fgs dps e l r-r ol proporsol deg ovrs rs bss fgs ber: c 3 ] [ 4 c 5 d d dlh r-r dr selrh d. Pers 3 dseles sep los o sep vrbel v se bss fgs d se ers. U q= forl deg ess les sqre dlh: 0 v v v c c v v v 7 s s v v v v / d v v s d dlh elee dr rs d bss fgs dlh elee dr rs d sl d dlh d respo. Pers orl dr peodel ARS dp dseles deg egg Choles Decopoo Dogrr oler ch d Sewr 979 Sco 997 deg odfs ber c D 8

9 bg Wdro Oo. Shd Abr Rpog 9 d D dlh rs dgol berr ++ dr elee rs dgol rs. oefse dr bss fgs dperoleh deg er forl ber ASR c D Eprs Seelh ebg odel g lgh per dde deg rv sple rced d ARS. l SE deg berbg vrs pd fgs peg secr legp ers pd Tbel ber. Tbel. l SE Pd Sple Trced d ARS Fgs SE SE oosrp Sple ARS Sple ARS 5 5e s Sber: D Dolh erdsr Tbel er l SE deg b peg g se besr d vrs 5 os eber hsl g se besr pd fgs g 5e. Sedg pd fgs g s l SE deg b peg g se besr d vrs os eber hsl g se ecl. Secr eselrh fgs opl erd pd vrs ecl deg sebrg. Sedg pd ARS l SE deg b peg g se besr d vrs os eber hsl 5 g se ecl pd fgs g 5e. Sedg pd fgs g s l SE deg b peg g se besr pd vrs = 005 eber hsl g se ecl ep pd vrs = 05 d l SE bervrs. Secr eselrh fgs opl erd pd vrs ecl deg sebrg.

10 bg Wdro Oo. Shd Abr Rpog 0 5. espl Trced Sple: erdsr hsl peel dp dspl sebg ber: T X X β X ARS : deg [ { v } ] T d â dperoleh dr ASR 0 deg ASR ˆ c D. Fgs opl erd pd vrs g ecl sebrg peg. l SE pd rv rced sple ler lebh ecl dbdg deg ARS ler pd se fgs. Hl dp dr bhw rv rced sple lebh b dbdg deg ARS re sh egg d des redh. Seer pd rv ARS ee pd ers vrbel predor. Oleh re perl lebh l pd d des gg elb b predor d ers. Dfr Ps [] C. ler d L. Fhrer00 es vrg-coeffce odels sg dpve regresso sple Sscl odellg. [] I.. dr S. Fred.W. Oo d S. Gro 006 Peodel Sple d ARS pd l s erhdp IP hssw Jrs Ds os sl U- PETRA Srb Jrl TeIdsr U PETRA SURAAA [3]. Efro d R.J. Tbshr 993 A Irodco o he oosrp Chp d Hll Ic. [4] R.L. Eb 988 Sple Soog d oprerc Regresso rcel Deer: ew or. [5] J.H. Fred d.w. Slver 989 Fleble prso soohg d ddve odelg Techoercs [6] J.H. Fred 99 lvre Adpve Regresso Sples Wh Dscsso The Al of Sscs [7] P. Hll d J.D. Opsoer 005 Theor for pelsed sple regresso oer 9. pge.05 [8] W. Hrdle 990 Appled oprerc Regresso Cbrge Uvers Press: ew or. [9] X. He d P. Sh 998 oooe -Sple Soohg Jorl of he Aerc Sscl Assoco; 9344 [0] C.C. Holes d.. llc. 003 Geerlzed oler odellg Wh lvre Free-o Regresso Sple Jorl of he Aerc Sscl Assoco; 9846.

11 bg Wdro Oo. Shd Abr Rpog [].W. Oo S. Gro d Sbr 004 sclssfed Wh Approch oprerc UISA dg. [].W. Oo S. Gro d Sbr 004 Alss Dsr d ARS lsfs Perb d Idoes Ser FS3I e II UG ogr. [3].W. Oo 005 lsfs Perb deg Pede CART d ARS Jrl Wd ee & As UW Srb. [4].W. Oo S. Gro d Sbr 006 Opze o d ss Fco Trced Sple d lvre Adpve Regresso Sples" ICOS UISA dg. [5] L.L. Scher 98 Sple Fos: sc Theor Joh Wle & Sos Ic: Cd. [6] J. Sho d D. T 995 The Jcfe d oosrp Sprger-erlg ew or Ic. [7] G. Whb 990 Sple odels for Observol D SIA CS-SF Regol Coferece Seres Appled hecs Phldelph.