Least-squares Fit of a Continuous Piecewise Linear Function

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Albert Ralf Goodwin
7 years ago
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1 Les-squres F of Couous Pecewse Ler Fuco Nkol Golovcheko 3-Augus-4 Absrc The er descrbes lco of he les-squres mehod o fg couous ecewse ler fuco. I shows h he soluo s uque d he bes f c be foud whou resorg o erve omzo echques. Problem Gve se of rs of d os: x, y,.. x y deede vrble; deede vrble; dex; umber of os; d fxed bouds of he segmes of he couous ecewse ler fuco:,..m x coorde of segme ed o; ed o dex; m umber of segme ed os; m- umber of segmes; fd he y coordes of he segme ed os (b ) of couous ecewse ler fuco, whch mmze he sum of squres of he dsce bewee he fuco d corresodg d os: S Σ(f(x ) y ) f(x ) fed ecewse ler fuco. Noe h he erm couous s used he sese h he dce segmes of he fuco shre he sme ed o. See Fgure for grhcl exmle of he roblem.

2 Fgure. Exmle of fg couous ecewse ler fuco. Soluo Geerl Les-squres Mehod Frs, we wll oule some key ses used he les-squres mehod. Gve fuco f(x, b, b m ), where b, b m re ukow rmeers, d se of d os (x, y ), where.., we eed o mmze he followg obecve fuco: ( ) (.) s f y f f(x, b, b m ) vlue of he fed fuco x ; The mmum of he fuco c be foud by lyzg s rl dervves he ukow rmeers. The frs order dervve would dce he exremum o(s) whe s equl o zero. The secod order dervve would dce f he exremum o s cully he mmum d f s uque mmum. For exmle, f he secod dervve s osve cos, he frs dervve hs uque erseco wh zero d chges sg from egve o osve he erseco, whch corresods o he fuco mmum.

3 The obecve fuco dervves c be exressed erms of fuco f(x, b, b m ) d s dervves: ds ( f y ) (.) d s d f + ( f y ) (.3)..m dex of he ukow rmeers. The rmeers re foud by equg he frs dervve equos o zero d solvg he resulg sysem of equos: ( ) (.4) f y Noe h f he fed fuco s olyoml, he secod dervve of he obecve fuco s osve (uless x re zero) d smlfes o: d s Therefore, fg olyomls resuls uque soluo. (.5) Fg Segmeed Fucos If he fed fuco cosss of severl cosecuve segmes, he sme les-squres mehod c be used wh some ddol cosrs. The cosrs deed o how he segmes re coeced, e.g.: ) he fuco s couous he o os bewee segmes (he sme o s shred by dce segmes); ) he fuco s couous d smooh he o os (he sme o s shred by dce segmes d he frs dervve x s couous). The equos.-.4 c be exressed slghly fere wy for segmeed fucos: m (,, ) (.6) s f y m, ( f, y, ) (.7) ds

4 m d s d f,, ( f, y, ) + (.8) m, ( f, y, ) (.9) m umber of segme ed os; m- umber of segmes;..m- segme dex; umber of os -h segme;.. o dex -h segme; x, deede vrble; y, deede vrble; f, f (x,, b, b k ) fed fuco vlue; f (x, b, b k ) fed fuco used -h segme; k umber of ukow vrbles;..k dex of ukow vrble (corresods o equo for ech ukow vrble). I order o solve he sysem of equos (.9), he fucos f mus clude he cosrs for he segme o os. If he cosrs re formuled serely, usg ddol equos, he sysem becomes overdeermed d here s o soluo. Foruely, s ossble o fcor he couy d smoohess cosrs he fed fucos f rovded hey hve olyoml form: ( ) q f ( x) b x (.), q q q.. b,, b, olyoml order; olyoml coeffce dex; ukow rmeers (olyoml coeffces); fxed x coorde of he frs ed o of -h segme. The couy cosr mles h f ( + ) f + ( + ). Afer subsug hs codo (.), we fd: ( ) q b, q + b+, (.) q I oher words, oe of he rmeers fuco f + s deermed from he revous segme fuco f. The smoohess cosr c be hdled smlrly:

5 ( + ) + ( + ) (.) dx dx q ( ) q b, q + b+, (.3) q The smoohess cosr deermes oher rmeer fuco f + from fuco f, b +,. Fg Couous Pecewse Ler Fuco A couous ecewse ler fuco cosss of segmes defed by frs-order olyomls (.) wh he couy cosr (.): ( ) f ( x) b + b x (.4),, Afer cludg he couy cosr (.4), ll fed fucos c be exressed he sme form: f ( x) ( b b ) x + b b +,,, + +, + (.5)..m- segme dex; m- umber of segmes; (, b, ) x,y coordes of he frs ed o of -h segme; ( +, b +, ) x,y coordes of he secod ed o of -h segme. We c dro he secod dex he b rmeers becuse s lwys zero: ( b b ) x + b b f ( x) m- segme dex; m- umber of segmes; (, b ) x,y coordes of he frs ed o of -h segme; ( +, b + ) x,y coordes of he secod ed o of -h segme. + (.6) Le s lyze he dervves of he fuco resec o s rmeers. There re oly wo rmeers volved b d b + : x (.7)

6 x + + (.8) d f d f (.9) + (.) The secod dervves re equl o zero. Therefore, he secod dervves of he obecve fuco re osve (.8) d he soluo s uque. The sysem of equos c be formed by removg he frs dervves rmeers equl o zero from (.9):,, ( f, y, ) + ( f, y, ) (.)..m rmeer dex (corresods o equo er ech vlue); m umber of ukow rmeers; m- umber of segmes; umber of os -h segme; m umber of os udefed segmes (corresodg sums dser); o dex segme; There re wo exceos he sysem whe - or re equl o zero ( or m). They corresod o udefed fucos f d f m. Sce he sum of squres does deed o fucos f d f m, he corresodg sums (.) should be omed. The sysem of equos c be exded by subsug exressos from (.6-.8) o (.): ( b b ) x + b b x +,, y, ( b b ) x + b b x + +, + +, + y, + + (.) The erms c be regroued regrds o he ukow rmeers:

7 ( )( ) ( ) b x, x, + b x, + ( ) (, + ) + (, )(, + ) ( ) + b x b x x x, y, y, x, y, + + y, + ; + (.3)..m rmeer dex (corresods o equo er ech vlue); m umber of ukow rmeers; m- umber of segmes; umber of os -h segme; m umber of os udefed segmes (corresodg sums dser); o dex segme; Jus lke (.), he sums over segmes d m should be omed. Soluo of he orml ler sysem of equo (.3) roduces he swer o he roblem of fg ecewse ler fuco.

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