An Expansion of the Derivation of the Spline Smoothing Theory Alan Kaylor Cline

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1 A Epaso of the Derato of the Sple Smoothg heory Ala Kaylor Cle he classc paper "Smoothg by Sple Fctos", Nmersche Mathematk 0, ) by Chrsta Resch showed that atral cbc sples were the soltos to a oel formlato of the data smoothg problem Resch employed what he called "stadard methods of the calcls of aratos" to obta hs reslts ad ths clded the se of Eler-Lagrage eqatos It s my tet here to epad that derato ad to aod ay referece to the calcls of aratos he materal here smply replaces the frst paragraph of Secto 3 of Resch's paper I hae tred to preset t so that oly calcls ad lear algebra are reqred for derstadg he Problem Cosder that we are ge a set of trples {,, )} y ad a oegate ale S sch that < < < ad δ > 0 for =,,, We seek to fd a fcto f to mmze y ) d oer all fctos g wth two cotos derates o [, ] sch that g ) S I the ery terse frst paragraph of Secto 3, Resch showed: Ay solto to the problem mst be a atral cbc sple wth kots { }, = For S > 0, the gaps the thrd derates of a solto are proportoal to the weghted resdals the appromatos hat s for some o-egate ale of p f ) y f ) f ) + = p for =,,,, where the sbscrpts + ad - dcate rght- ad left-sded derates, respectely, ad the terpretato at the edpots s that f ) = 0 ad f ) + = 0 he Epaso We beg wth a lemma that relates tegrals of atral cbc sples to ther thrd derate gaps Lemma : If g s a atral cbc sple wth kots < < < ad l s a fcto wth two cotos derates o [, the ] where ) = 0 ad ) + = 0 ) l ) d = l ) ) ) ), +

2 Proof: Althogh s ot cotos o [, ], t s cotos o each teral, + ), for =,,,, Itegratg by parts o each of the smaller terals we fd + ) l ) d = ) l + + ) ) l ) ) l ) d Howeer the terms wth the smmato ca be separated ad all of the prodcts ) l ) ecept the frst ad last) wll cacel each other resltg + = ) l ) ) l ) ) l ) d Bt from the atral ed-codtos of g ad the fact that s pece-wse costat o each of the terals, + ), we hae Fally by otcg that = 0 ) l ) l )) + + = ) l ) + ) l ) = ) l ) + ) l ) ) = ) + + +, we obta = ) l ) + ) l ) + = ) ) ) l ) + Resch's frst proposto s a easy coseqece of Lemma = Ay solto to the problem mst be a atral cbc sple wth kots { } Proof: If f s a solto to the problem ad g s a atral cbc sple that terpolates f o the set { } = e, g ) = f ) for =,,, ), the g certaly satsfys g ) S sce f does Frthermore, epressg f as f g) + g, we hae f ) d = f ) )) d+ f ) )) ) d+ ) d Bt from Lemma ad the terpolato codtos, we hae

3 ) )) ) )) ) + ) ) ) = f g d+ f g g g + g d f ) )) d 0 ) d = + + he frst term s clearly o-egate, ad ths ) ) f d d Frthermore, the eqalty s strct ad ths the solto mst be a atral cbc sple) less f ) g )) d = 0 Howeer for ths to happe f 0 o [, ] Itegratg twce, we hae that f g mst be a lear polyomal o [, ], yet from the terpolato codtos, that lear mst be detcally zero hs f caot be a solto to the problem wthot beg a atral cbc sple wth kots { } = he et lemma smply says that less two ectors hae eactly the same drecto the we ca fd a ector wth a poste compoet drecto of oe bt a egate compoet wth respect to the other Lemma Let ad be o-zero -ectors If for o o-egate scalar p does = p, the there ests a -ector w so that w> 0 ad w< 0 Proof: Let z deote the ecldea orm ad hs, ad Smlarly Let are ot the same drecto, w =, the z z) < / < From the Cachy-Schwartz eqalty, sce 0 < 0 <

4 Yet w w = = > 0 = = < 0 he ector w wll be sed to show that less a partclar fcto s a solto to or problem, slght pertrbatos ca be made that redce the objecte fcto costrats g ) Resch's secod proposto s S ) d wthot olatg the If S > 0, for some o-egate ale of p the for =,,,, f ) y f ) f ) + = p Proof: Sppose for some atral cbc sple g wth kots { } = there s o o-egate ale of p for whch g ) y ) ) + = p for =,,, I partclar, ths mples that the qattes ) ) + are ot all zero Ether g ) the qattes y are also ot all zero or they are all zero Sppose tally that they are ot all zero Lemma the apples ad we ca fd ales ad w for w ) ) ) > 0 + g ) y w < 0 =,,, so that

5 If we let l be ay fcto wth two cotos derates o [, ] that terpolates the data pars {, w ) } e, l ) = w, for =,,, ), the ad Now for ay scalar λ l ) ) ) ) > 0 + g ) y l ) < 0 ) + λl )) d = ) d + λ ) l ) d + λ l ) d λ + λ = ) d + l ) ) ) ) + l ) d ) λ ) ) ) + ) λ = d l + l ) d Notce that for all poste ales of λ the mddle term o the rght-had sde s egate ad, for sffcetly small poste ales of λ, the sm of the last two terms s egate hs for those ales of λ Howeer, at the same tme, ) + λl )) d< ) d g + λl g g l l ) = λ + λ + δy δy δy δy ) ) ) ) ) Aga for all poste ales of λ the mddle term o the rght-had sde s egate ad, for sffcetly small poste ales of λ, the sm of the last two terms s egate If g ) the for sffcetly small poste ales of λ g ) + λl ) S < he coclso s that, f g does ot satsfy the proportoalty relato of the hypothess, the a fcto g+ λl satsfes the costrats ad redces the objecte fcto hs g cold ot hae bee a solto S, g ) Alterately, we assme that all of the qattes y w so that are zero It s stll possble to fd a ector

6 As aboe, a fcto l cold be fod for whch w ) ) ) > 0 + ) + λl )) d< ) d for sffcetly small poste ales of λ Howeer sce So oce aga, g g ) + λl ) l ) λ δy δy S > 0, for small ales of λ = S cold ot be the solto he fal part of ths epaso of Resch s derato shows that, f at the solto f ) y < S, the f ) f ) + = 0, for =,,,, e, the parameter p = 0 ), sce otherwse the argmet cold be repeated to prodce a better solto Of corse, these codtos reslt o gaps the thrd derates A solto mst the be a sgle cbc Howeer the atral ed codtos the mply that the cbc solto s actally a lear he reslt s that ether a solto rests at the edge of the costrats or s a straght le f ) y = S Addtoal Note o Resch s Paper: At the ed of Secto 4, Resch s presets a terate algorthm ad the a fal paragraph where he commets that oe cold also apply a Newto s method to fd the recprocal of p Wth that, he clams, Whe tested wth few eamples, coergece was always reached wth a slghtly redced mber of teratos Althogh he does ot state t eplctly, the ALGOL code Secto 5, actally mplemets the terato to fd the recprocal of p As Resch states the secod paper, "Smoothg by Sple Fctos II", Nmersche Mathematk ), the terato to fd the recprocal of p may ot coerge here he sggests that stead of sg Newto s method o the eqato F p), that faster ad garateed) coergece s obtaed wth the eqato / F p ) = S / he oly chage that ths wold reqre from the algorthm Secto 4 of the frst paper, s the replacemet of the Newto pdate / p p+ e Se) )/ f p g) by / p p+ e e/ S) ) / f p g) = S /