The Comparison of Spline Estimators in the Smoothing Spline Nonparametric Regression Model Based on Weighted...

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1 he Coparson o Splne Esaors n he Soohng Splne Nonparaerc Regresson Model Based on Weghed... he Coparson o Splne Esaors n he Soohng Splne Nonparaerc Regresson Model Based on Weghed Leas Square (WLS and Penalzed Weghed Leas Square (PWLS n Longudnal Daa (A Sud on he Bab Growh n Indonesa Adj Achad Rnaldo Fernandes *, Luhaul Aalana, Sangun, Solun Sascs Deparen, Facul o Maheacs and Naural Scences, Unvers o Brawjaa, Indonesa *Correspondng Auhor: ernandes@ub.ac.d Absrac: he objecves o hs research are; ( o oban he esaon o soohng splne nonparaerc regresson based on PWLS and WLS, ( o oban he error varance-covarance arx based weghed esaon, (3 o exane he ecenc o Splne Esaor Curve n Soohng Splne Nonparaerc Regresson Model Based on Weghed Leas Square (WLS and Penalzed Weghed Leas Square (PWLS on he Longudnal Daa n he daa o bab growh. One o he was o easure he bab growh s b recordng he age and wegh o he bab onhl and havng wren n a card called as Karu Menuju Seha (KMS o deec he alnuron n oddlers. he esaon o soohng splne nonparaerc regresson PWLS (wh penal as ollow: A, A U U VU [ I U U ] Whou penal usng WLS as ollow: A, A. he esaon o error varance-covarance arx s as ollow: ( ( ' wh j j, j 0 0 NN he esaon o he curve o PWLS based nonparaerc regresson n he daa o bab growh s ore ecen han he esaon o he curve o WLS based nonparaerc regresson. I can be seen ro he ecenc o WLS based curve ha s onl 48.4% or less han 50%, copared o he ecenc o he PWLS based curve. In oher word, he use o esaon o he curve o penal (PWLS based soohng splne nonparaerc regresson has a beer ecenc level han he WLS (whou penal based. Kewords: Soohng Splne, Longudnal Daa, WLS, PWLS, Growh Curve 57 Inernaonal Journal o Conrol heor and Applcaons

2 . INRODUCION Adj Achad Rnaldo Fernandes, Luhaul Aalana, Sangun, Solun Regresson Analss s one o he sascal ehods used o nd he paern o he relaonshp beween response on predcor x. I he paern s saed n a graphcal or, he lnear relaonshp beween x and wll or be n he or o sragh lne, quadrac, cubc, exponen, ec. here are wo approaches ha can be used o esae he or o curve o regresson n descrbng he paern o he relaonshp beween he response and predcor, he are paraerc and nonparaerc regressons. Paraerc regresson s assued as he curve o regresson s dened or s or. here s an assupon ha should be e n paraerc regresson, ha s he assupon o he or o lnear relaonshp beween he response and he predcor. I he lnear assupon s no e and he nonlnear or s no or has no been dened, hen he nonparaerc regresson s used as an alernave wh an assupon ha he curve o regresson has no been dened or s or. here are hree pes o daa n order o conduc regresson analss, he are; cross-secon, e-seres, and longudnal. he cobnaon beween cross-secon and e-seres daa s called as longudnal daa. Longudnal daa s he daa resuled ro he observaon on N subjecs ha are ndependen each oher and each subjec s observed or several es n perod. A research conduced b Verbekke and Molenberghs [] suded he paraerc regresson or longudnal daa wh Generalzed Lnear Mxed Model approach. he approach accoodaed he correlaon o he observaons on slar subjecs b addng he eec o each subjec, he rando eec, ro he xed eec. he cobnaon beween rando eec and xed eec s called as xed eec. One o he approaches n nonparaerc regresson s soohng splne ha has a specc characersc b whch s able o adjus he changes o daa behavor ver well. In he soohng splne, does no need he kno selecon snce he esaon o he uncon s based on he crera o odel accurac and he soohness o he curve ha have been se b he soohng paraeer. A prevous research conduced b Fernandes [-5] suded he developen o b-response splne esaor speccall or longudnal daa b usng soohng splne wh he use o Reproducng Kernel Hlber Space (RKHS approach. Anoher research conduced b Budanara [6] suded on how o oban he esaor or o he curve o regresson or b-response longudnal daa usng Generalzed Penalzed Splne approach. In order o oban he esaon o nonparaerc regresson uncon or longudnal daa, can be conduced b usng he Penalzed Weghed Leas Square (PWLS opzaon, n whch, b addng varance covarance arx as weghed on he copleon o he leas quadrac opzaon. Lesar [7], Fernandes [- 5] had conduced a research b sudng he procedures n obanng he ul-responses esaor or o curve o nonparaerc regresson b usng PWLS. he addon o wegh s conduced snce n he longudnal daa, he slar subjecs are dependen, whle he deren subjecs are ndependen so ha he correlaon beween he observaons needs a wegh on he leas quadrac opzaon. In hs research, he nonparaerc regresson odel was used or longudnal daa b usng soohng splne approach and addng he wegh n esang he regresson uncon pleened n he daa o bab growh. he esaon on he paraeer, n whch consderng he penal conrollng he roughness/soohness o he curve, s called as PWLS ehod. In conras, does no use he penal, s called as WLS ehod. hereore, hs research s aed a esng he ecenc o he use o PWLS and WLS, whch one s beer. Based on he background above, he objecves o hs research are; ( o oban he esaon o soohng splne nonparaerc regresson based on PWLS and WLS, ( o oban he error varance-covarance arx based weghed esaon, (3 o exane he ecenc o Splne Esaor Curve n Soohng Splne Nonparaerc Regresson Model Based on Weghed Leas Square WLS and Penalzed Weghed Leas Square PWLS on he Longudnal Daa n he daa o bab growh. One o he was o easure he bab growh s b recordng he age and wegh o he bab onhl and havng wren n a card called as Karu Menuju Seha (KMS o deec he alnuron n oddlers. Inernaonal Journal o Conrol heor and Applcaons 58

3 he Coparson o Splne Esaors n he Soohng Splne Nonparaerc Regresson Model Based on Weghed.... SPLINE IN NONPARAMERIC REGRESSION FOR LONGIUDINAL DAA Nonparaerc regresson odel n longudnal daa, developed b N subjecs were observed repeaedl (repeaed easureen n he perod. Nonparaerc regresson odel or longudnal daa have a derence wh a cross-secon, whch s locaed on he observaon aong subjecs assued o be ndependen o each oher, bu beween observaons n he sae subjecs s dependen [9-]. Man researchers have developed a splne esaor n nonparaerc regresson odel or longudnal daa ([,3,6,]. he relaonshp beween predcors o response o longudnal daa nvolvng N subjec o he observaons o each subjec, ollowng he regresson odel as ollows: Inoraon: ( x ;,,..., N;,,...,. ( : Response on he subjec all he e observaon and all, x : Predcors on he subjec all he e observaon and all, : Regresson curve predcor relaonshp wh he response on he subjec, N : nuber o subjecs, : he nuber o observaons o each subjec, P : he nuber o predcors, : Rando error on he subjec all he e observaon and all, he regresson odel n equaon ( o nclude as a regresson curve ha accoodaes no labl observaons on he sae subjec. Rando error (,...,,,...,,..., N, N..., N s assued N-vara noral dsrbuon, wh ean E ( = (vecor easurng N and he varance-covarance arx Var ( = (arx easurng N N as ollows ([-7]: (, (, (, (, (, (, (, (, (, (, (, (, (, (, N N (, N (, N (, N N (, N N N ( N ( N. ( he arx can be spled no sub-arces and Inernaonal Journal o Conrol heor and Applcaons

4 Adj Achad Rnaldo Fernandes, Luhaul Aalana, Sangun, Solun N ( N ( N Sub-arx and 0 easurng, are presened as ollows: (, (, (, (, (, (, and Eleens ousde he dagonal (,, (,,..., (,, (,, nael he sub-arx s a rando error covarance beween observaons n he sae subjecs. hs covarance can be worh no 0, whch accoodaes he correlaon beween observaons n he sae subjecs. On he oher hand, he sub-arx ha 0 s he arx o all eleens o value 0 saes ha he covarance beween observaons n deren subjecs are uuall ndependen. Splne approach generall spec n equaon ( n he or o regresson curve shape s unknown, bu s assued sooh (sooh, n he sense o space s conaned n a parcular uncon, especall Sobolev spaces or wren ([3-5] W [ a, b ] where: a b x dx (3 b ( ( ( W [, ] :,,..., konnu absolu; [ ( ], a or a consan sang order polnoal splne. Copleon curve esaon regresson or longudnal daa n equaon ( usng Penalzed Weghed Leas Square PWLS nvolvng weghs n he or o nverse varancecovarance arx o rando errors sbolzed as has been descrbed n equaon (. o oban he esaes o he regresson curve usng he opzaon PWLS nael he copleon o opzaon as ollows [0,]: (4 N b ( Mn M ( ( ( (, [, ],,,..,. x dx W a b N a where (,,...,,,,...,,..., N, N,..., N ( ( x, ( x,..., ( x, ( x, ( x,..., ( x,..., ( x, ( x,..., ( x. N N N N N N, and Inernaonal Journal o Conrol heor and Applcaons 60

5 he Coparson o Splne Esaors n he Soohng Splne Nonparaerc Regresson Model Based on Weghed... PWLS opzaon n equaon (4 usng he soohng paraeer, as a conroller beween he goodness o (rs segen and a roughness penal (second segen 3. RESUL AND DISCUSSION here are hree research objecves ha wll be copleed n hs paper. Frs, o oban he esaon o soohng splne nonparaerc regresson wh penal, PWLS (wren n heore, and whou penal, WLS (wren n heore. Second, o oban he error varance-covarance arx based weghed esaon. hrd, o exane he ecenc o he Splne Esaor curve n he PWLS and WLS based Soohng Splne Nonparaerc Regresson Model n he bab growh. heore : Esaon o Curve Regresson wh Penal (PWLS When gven he daa pars ollowng he nonparaerc regresson odel nvolves a sngle predcor on longudnal daa ha ees he or o nonparaerc regresson uncons or longudnal daa as presened n equaon (, assung hen he splne esaor ha nzes PWLS s E( 0, Var(, (5 N b n M W [, ],,,..., x dx a b N a A, wh: A U U VU [ I U U ]. (6 M. U V Λ (7 Proo: Consderng he equaon ha s uncon d Vc,hen he nonparaerc regresson odel ( can be saed as [-7]: where s (N (N arx as ollow: d Vc , (8 0 0 N 6 Inernaonal Journal o Conrol heor and Applcaons

6 Adj Achad Rnaldo Fernandes, Luhaul Aalana, Sangun, Solun where,,,,,,,,,, (9 j x, j, wh,,..., ; j,,..., ( j! d s N-szed vecor, ro: d ( d', d ',, d ', where d' ( d, d,..., d, N V s (N (N-szed arx as ollow: V V 0 V, (0 0 0 V N where V, 0 0,, 0,,,, ( b ( x ( u xs u a (!, du,,,..., ; s,,..., s (4.8 c s N-szed vecor, ro: c ( c', c ',, c ', where c' ( c, c,..., c. N Nonparaerc regressonanalss s conduced o ge esaor o regresson curve. o ge he esaon, Reproducng Kernel Hlber Space (RKHS s used. he purpose s o oban he esaon o ha ees PWLS opzaon [0,]: n n (, (,,.., N,,.., N Inernaonal Journal o Conrol heor and Applcaons 6

7 he Coparson o Splne Esaors n he Soohng Splne Nonparaerc Regresson Model Based on Weghed... wh resrced:, 0. (3 hen, space uncon W [ a, b ] used s order- Sobolev space dened as ollow: ollow: b ( W [, ] : a b [ ( x ] dx, a Where a x b and =,,...,N. Based on he space, norn o ever W [ a, b ] s descrbed as b ( [ ( ] a x dx. Opzaon wh resrced n equaon ( can be saed as: Wh resrced n equaon (5 no: n n ( W [ a, b ] W [ a, b ],,.., N,,.., N, (4 b ( [ ( ], 0. a x dx (5 Weghng opzaon (4 wh equvalen resrced (5 b solvng Penalzed Weghed Leas Square (PWLS opzaon: where and penal : n W [ a, b ],,,..., N, (6 N b M x dx a M N and s soohng paraeer conrollng beween Goodness o : M N o solve opzaon n equaon (6 wh penal coponen: b, (7 x dx. (8 a 63 Inernaonal Journal o Conrol heor and Applcaons

8 Adj Achad Rnaldo Fernandes, Luhaul Aalana, Sangun, Solun N x dx c c b a ΛV, (9 where I I 0 Λ 0 0 I N. Usng d Vc as reerence, Goodness o n PWLS opzaon (5 can be saed as: V V M M d c d c. (0 Solvng PWLS opzaon b cobnng goodness o (0 and penal (9, can be descrbed as: n W [ a, b ],,,..., N N b M x dx a n N = c d N M d Vc d Vc c ΛVc n = N c N d d Vc d Vc c MΛVc M n ( = d Vc d d d N c N d V V V V V ΛV d c c c d c c c M c M n ( = d c V d d d Vc N c d N c V d c V V MΛV c M n Q( c, d. N = c ( N d Solvng opzaon ( s obaned b conducng dervave Q( c, d parall owards c and d, hen he resul equals o zero. he paral dervave s presened as ollow: Inernaonal Journal o Conrol heor and Applcaons 64

9 he Coparson o Splne Esaors n he Soohng Splne Nonparaerc Regresson Model Based on Weghed... Q( c, d 0 c, and he resul s: V V d V V MΛV c 0. V d [ V MΛI] c 0. d [ V MΛI] c 0. ( When arx U s presened as: equaon (3 can be saed as: U V MΛ. d Uc 0. Uc ( d Equaon (4 s doubled ro he le wh U and he ollowng equaon s onaned: (3 Furherore, paral dervave: c U ( d (4 resuls n: Q( c, d d 0, 0 d Vc Elaboraon o equaon (7 resuls n he ollowng equaons: Consderng equaons: d V{ U ( d } 0 d [ VU ] ( d 0. (5 U V MΛI, hen ( M V U ΛI, as he consequence, he resul s he ollowng VU ( U MΛI U VU ( I MΛU. Reduplcang he equaon above wh resulng n: 65 Inernaonal Journal o Conrol heor and Applcaons

10 Adj Achad Rnaldo Fernandes, Luhaul Aalana, Sangun, Solun VU I ΛU M. he equaon s subsued n equaon (4 resulng n: d [ I MΛU ] ( d 0 When he equaon above s elaboraed urher, he resul s: MΛ U MΛ U d 0. MΛ U d MΛ U. Boh segens o he equaon are reduplcaed wh ( MΛ and hen spled resulng n: - U U. d Equaon (4 s subsued no equaon (6 resulng n: (6 c U ( [ U U ] U [ I U U ]. (7 Based on equaon (6 and (7, esaor or nonparaerc regresson curve or longudnal daa nvolvng sngle predcor as ollows:,, d Vc N, N U U VU [ I U U ] { U U VU [ I U U ]} A, (8 where A U U VU [ I U U ]. Error varance-covarance arx wll be presened n he nex secon (heore 3, so heore uses as well as U V MΛ. resulng n: Inernaonal Journal o Conrol heor and Applcaons 66

11 he Coparson o Splne Esaors n he Soohng Splne Nonparaerc Regresson Model Based on Weghed... where - U U. d A, c U [ I U U ]. A U U VU [ I U U ]. Based on he he heore above, he equaon d Vc wh splne uncon esaon consderng he auocorrelaon (DM s A where A U U VU [ I U U ], And o esae he splne uncon whou consderng he auocorrelaon (M s equvalen I o A where A U U VU [ I U U ]. heore : Esaon o Curve Regresson Whou Penal (WLS When gven he daa pars ollowng he nonparaerc regresson odel nvolves a sngle predcor on longudnal daa ha ees he or o nonparaerc regresson uncons or longudnal daa as presened n equaon (, assung hen he splne esaor ha nzes WLS s E( 0, Var(, n M W [ a, b ],,,..., N 67 Inernaonal Journal o Conrol heor and Applcaons (5 A, wh: A. (6 Proo: Consderng he equaon ha s uncon d,hen he nonparaerc regresson odel ( can be saed as [-7]: d.

12 Adj Achad Rnaldo Fernandes, Luhaul Aalana, Sangun, Solun Or he purpose s o oban he esaon o ha ees WLS opzaon []: k n n (, (,,..., N,,..., N M. M d d (3 wh M=N. Solve WLS usng goodness o (3 whou penal, as ollow resul: n M d d N d = n N d d M d = n ( d d d d M N d = n ( d d d M N d = d N n Q(. d Solve he opaon o (4, cong ro derenal o Q( d b d, and equall o zero, as ollow: he resul: Q( d 0, d 0 d (5 he he solvng o curve esaon ro (5 as ollow: (4 wh d { } A, (6 Inernaonal Journal o Conrol heor and Applcaons 68

13 he Coparson o Splne Esaors n he Soohng Splne Nonparaerc Regresson Model Based on Weghed.... A. he second purpose o he sud us o esae error varance arx as weghng n PWLS or WLS. In sngle-response case, here s weghng ha accoodaes correlaon beween responses (Fernandes, [-3]. Esaon or error varance-covaran arx shown n heore 3. heore 3 he weghed usng Error varance-covarance arx or nonparaerc regresson longudnal daa odel usng axu lkelhood s as ollow: NN (7 Wh ( ( j j ' j, Proo: he sudes relaed o sngle-response nonparaerc regresson odel have been conduced exensvel. he researchers n general assued varance-covarance arx ro he rando error s unknown/ undened. As he eec, one should conduc esaon or he varance-covarance arx ro he rando error n sngle-response nonparaerc regresson odel. In order o do so, Maxu Lkelhood Esaor (MLE ehod s used. When s assued ha s he resul o norall dsrbued rando saple o M-vara (M = 3, and ean o E( = 0 (M-szed vecor and varance-covarance arx o Var( = (M M-szed arx, cobned dens uncon ro each observaon s obaned ro noral argnal dens. I s as ollow: Jon dens L(, exp ( ( / / o,, M 3 ( L ( (, exp ( ( M / / (5 ( ( n L(, can be elaboraed as ollow: r ( ( ( ( r ( ( 69 Inernaonal Journal o Conrol heor and Applcaons

14 Adj Achad Rnaldo Fernandes, Luhaul Aalana, Sangun, Solun ( ( r ( ( r ( ( (6 hus, equaon (5 s subsued usng equaon (6 and he resul s as ollow: L ( (, exp r ( ( M / / (7 Esaor or varance-covarance arx s obaned b axzng uncon o L(,, hrough L(, 0. As enoned n Fernandes [4-5], lkelhood uncon n equaon (6 wll ee he axu condon B, wh b b, and B ( (, or can be reorulaed as ollow: B b ( ( (8 Rando error varance-covarance arx n he sud s slar o equaon (8 or can be reorulaed as: Hence, equaon (7 can be reorulaed as: L(, exp{ {( ( ( ( M / / ( ( ( 3 3 ( ( ( ( ( 3 ( 3 3 ( 3 3 3( ( 3( 3 3 ( 3 3 3( } Inernaonal Journal o Conrol heor and Applcaons 70 NN

15 he Coparson o Splne Esaors n he Soohng Splne Nonparaerc Regresson Model Based on Weghed... / ( M L(, / exp ( ( + ( ( ( ( ( ( ( ( + ( 3 ( 3 3 ( 3 3 3( + ( 3 ( 3 3 ( 3 3 3( } exp ( / / ( (. ( ( ( ( ( ( ( / / / / / / / exp / ( (. exp / ( 3 33( exp / ( (. N exp / ( (. N exp / ( 3( N exp / ( 3 3 3( 3. N exp / ( 3( N exp ( / / 3 3 3( ( 3. N 7 Inernaonal Journal o Conrol heor and Applcaons

16 Adj Achad Rnaldo Fernandes, Luhaul Aalana, Sangun, Solun Esaor or varance-covarance arx s obaned b axzng uncon o L(, L(, j 0. j Elaboraon o each sub-arx o j s as ollow:, hrough For, s obaned ha: exp{ ( / / ( } (, L ( exp{ r / / ( ( '} ( r ( ( '} ln L(, 0, Based on he elaboraon o equaon (8, esaon o s as ollow: ( ( '. Usng he sae ehod,, 33,, 3, s: 3 ( ( '. ( ( ' ( ( '. ( ( ' Inernaonal Journal o Conrol heor and Applcaons 7

17 he Coparson o Splne Esaors n he Soohng Splne Nonparaerc Regresson Model Based on Weghed... ( ( ' Or can be orulaed ha: NN (9 wh ( ( j j ' j, Esaon o varance-covarance arx n equaon (9 can be used o predc regresson curve equaon (3. he heorecal ndngs above resuled n he esaon o he soohng splne nonparaerc regresson curve or PWLS based longudnal daa (wh penal n equaon ( and or he WLS based longudnal daa (whou penal n equaon (6. he applcaon on he daa used he weghed esaon n equaon (7. he daa used n hs research s he daa o he babes vsng Dnoo Coun Healh Cener o Malang C usng Karu Menuju Seha KMS. he daa onl nvolved N = 4 babes. he babes descrbe he bab growh aged 0-4 onhs. he observed response s he wegh o he bab ( n several onhlobservaon perods or 4 onhs. he predcor used n hs research s he age o he bab (x. able s he oupu o coecen esaon or PWLS (ncludng coecens c and d, and WLS (ncludng onl coecen c ehods. able he values o d and ĉ x x x3 x4 d 0,4364 0,499 0,888 0,37 d 0,944,0705 0, ,808 c -0,35-5,73-0,9994-0,08 c -0,37-5,904-0, -0,6378 c3-0, ,663-0,0685-0,3940 c4-0, ,6730-0,09-0,070 c5-0,096 0, ,049-0,044 he ollowng s he coparson beween he PWLS and WLS based soohng splne nonparaerc regresson odel. Fgure s he esaon o soohng splne nonparaerc regresson curve wh WLS (red lne and PWLS (green lne. 73 Inernaonal Journal o Conrol heor and Applcaons

18 Adj Achad Rnaldo Fernandes, Luhaul Aalana, Sangun, Solun Fgure : Esaon o he Curve o Soohng Splne Nonparaerc Regresson Usng WLS and PWLS he coparson beween he esaon resuls o he WLS and PWLS nonparaerc regresson shows ha he predcon values obaned n he PWLS nonparaerc regresson odel are near o he acual daa, copared o he predcon values n WLS approach. I can be proved ro he resul o coecen value o he deernaon on he splne nonparaerc regresson odel or 9,%. he esaon o WLS soohng splne nonparaerc regresson odel, whch equals o paraerc regresson, ha s global, has no e esaed ever observaon pon n deph. I s deren ro he esaon o PWLS based soohng splne nonparaerc regresson odel ha s ore local, so ha can esae ever observaon pon each oher n deal ver well. he esaor s sad o be ecen has a nu error varance. he ecenc o he esaor s he raon o nu error varance o he esaor. Meanwhle, he ecenc relave s he rao o he error varance o boh copared esaors. For exaple, g PWLS ( x and g WLS ( x are he wo esaors o he soohng splne nonparaerc regresson o g(. I boh esaors ollow he general condon o Craer-Rao, he Ecenc Relave (ER o g PWLS ( x and g WLS ( x s dened as he rao o he error varance as ollow; ER ( g, g PWLS WLS MSE MSE PWLS WLS ( ( I ER ( g, g <, hen g ( x s ore ecen han g ( x. PWLS WLS PWLS Inernaonal Journal o Conrol heor and Applcaons 74 WLS

19 he Coparson o Splne Esaors n he Soohng Splne Nonparaerc Regresson Model Based on Weghed... able Ecenc Relave o wo Mehods Subjec MSE(PWLS MSE(WLS ER Overall able above shows ha hose our PWLS based regresson curves are ore ecen han he WLS based regresson curves, o whch have onl 6,5% ecenc copared o he PWLS curves. he hghes slar s on he subjecs. In conras, n he ourh subjec, he PWLS and WLS esaons end o be slar, wh he Ecenc Relave near 00%. Overall, he esaon o he curve o PWLS based nonparaerc regresson s ore ecen han he esaon o he curve o WLS based nonparaerc regresson. I can be seen ro he ecenc o WLS based curve ha s onl 48.4% or less han 50%, copared o he ecenc o he PWLS based curve. In oher word, he use o esaon o he curve o penal (PWLS based soohng splne nonparaerc regresson has a beer ecenc level han he WLS (whou penal based. 4. CONCLUSIONS AND RECCOMENDAION Based on he analss resul and dscusson above, he concluson o hs research are ollow: ( he esaon o soohng splne nonparaerc regresson PWLS (wh penal as ollow: A A U U VU [ I U U ] Whou penal usng WLS as ollow: A A. ( he esaon o error varance-covarance arx s as ollow: NN wh ( ( j j ' j, (3 he esaon o he curve o PWLS based nonparaerc regresson n he daa o bab growh s ore ecen han he esaon o he curve o WLS based nonparaerc regresson. I can be seen ro he ecenc o WLS based curve ha s onl 48.4% or less han 50%, copared o he ecenc o he 75 Inernaonal Journal o Conrol heor and Applcaons

20 Adj Achad Rnaldo Fernandes, Luhaul Aalana, Sangun, Solun PWLS based curve. In oher word, he use o esaon o he curve o penal (PWLS based soohng splne nonparaerc regresson has a beer ecenc level han he WLS (whou penal based. he proble n hs research s ha has no e esaed he coecen o he auocorrelaon correcl, so ha here s a need o conduc urher research ha s able o accoodae he esaon o he coecen o auocorrelaon, as well as o prove he ecenc n he sulaon wh deren auocorrelaon levels. REFERENCES [] Verbekke, G., and Molenberghs, G. (000, Lnear Mxed Model or Longudnal Daa. Sprnger Seres n sascs. New York: Sprnger Verlag. [] Fernandes, A.A.R, Budanara, I.N, Ook, B.W., and Suharono. (05, Splne Esaor or B-Responses and Mul- Predcors Nonparaerc Regresson Model n Case o Longudnal Daa, Journal o Maheacs and Sascs, Vol, No, 05, pp [3] Fernandes, A.A.R, Budanara, I.N, Ook, B.W., and Suharono. (04, Splne esaor or b-responses nonparaerc regresson odel or longudnal daa. Appled Maheacal Scences, Vol. 8, no. 4, 04, [4] Fernandes, A.A.R, Budanara, I.N, Ook, B.W., and Suharono. (04, Reproducng Kernel Hlber Space and Penalzed Weghed Leas Square n Nonparaerc Regresson, Appled Maheacal Scence Vol 8, 04, No 46, pp [5] Fernandes, A.A.R, Budanara, I.N, Ook, B.W., and Suharono. (04, Reproducng Kernel Hlber Space or Penalzed Regresson ul-predcors: case n longudnal daa. Inernaonal Journal o Maheahcal Analss, Vol. 8, no. 40, 04, [6] Budanara, I.N, Ranasar, V., Rana, M., & Zan, I. (05, he Cobnaon o Splne and Kernel Esaor or Nonparaerc Regresson and s Properes, Appled Maheacal Scence, Vol 9, No, [7] Lesar, B., Budanara, I.N., Sunaro, S., and Mashur M. (00, Splne Esaor n Mulresponse Nonparaerc Regresson Model wh Unequal Correlaon o Errors. Journal o Maheacs and Sascs, 6(3: [9] Wu, H., & Zhang, J., (006, Nonparaerc Regresson Mehods or Longudnal Daa Analss. New Jerse: John Wle and Sons, Inc. [0] Wahba, G. (990, Splne Models or Observaonal Daa. Penslvana. SIAM. [] Wang, Y. (0, Soohng Splne Mehods Applcaons, CRC Press, New York. [] Wess, R.E. (005, Modellng Longudnal Daa. Sprnger exs n Sasc New York. Rereved, Januar, 3, 0. Webse: hp:// edu/books/ld. [3] Wang, J.L. (003, Nonparaerc Regresson Analss o Longudnal Daa. Calorna: Unvers o Calorna Press. [4] Howell, J.R. (007, Analss Usng Soohng Splnes As Ipleened In LME( In R. hess. Brgha Young Unvers. [5] K, Y.J., and Gu, C. (004, Soohng Splne Gaussan Regresson: More Scalable Copuaon va Ecen Approxaon. Roal Sascal Soce: Seres B, 66(, [6] Lee,.C.M. (004, Iproved Soohng Splne Regresson b Cobnng Esaes o Deren Soohness. Sascs and Probabl Leers, 67(, Inernaonal Journal o Conrol heor and Applcaons 76

THEORETICAL AUTOCORRELATIONS. ) if often denoted by γ. Note that

THEORETICAL AUTOCORRELATIONS. ) if often denoted by γ. Note that THEORETICAL AUTOCORRELATIONS Cov( y, y ) E( y E( y))( y E( y)) ρ = = Var( y) E( y E( y)) =,, L ρ = and Cov( y, y ) s ofen denoed by whle Var( y ) f ofen denoed by γ. Noe ha γ = γ and ρ = ρ and because