2. THE GENERAL LEAST SQUARES ADJUSTMENT TECHNIQUE

Transcription

1 . HE GENERAL LEAS SUARES ADJUSMEN ECHNIUE A common tratmnt of th last squars tchniqu of stimation starts with simpl linar mathmatical modls havin osrvations (or masurmnts) as xplicit functions of paramtrs with non-linar modls dvlopd as xtnsions. his adjustmnt tchniqu is nrally dscrid as adjustmnt of indirct osrvations (also calld paramtric last squars). Cass whr th mathmatical modls contain only masurmnts ar usually tratd sparatly and this tchniqu is oftn dscrid as adjustmnt of osrvations only. Both tchniqus ar of cours particular cass of a nral adjustmnt modl, th solution of which is st out low. h nral adjustmnt tchniqu also assums that th paramtrs, if any, can tratd as "osrvals" i, thy hav an a priori covarianc matrix. his concpt allows th nral tchniqu to adaptd to squntial procssin of data whr paramtrs ar updatd y th addition of nw osrvations. In nral, last squars solutions rquir itration, sinc a non-linar modl is assumd. h itrativ procss is xplaind low. In addition, a propr tratmnt of covarianc propaation is prsntd and cofactor matrics ivn for all th computd and drivd quantitis in th adjustmnt procss. inally, th particular cass of th nral last squars tchniqu ar dscrid... h Gnral Last Squars Adjustmnt Modl Considr th followin st of non-linar quations rprsntin th mathmatical modl in an adjustmnt d lx, i (.) whr l is a vctor of n osrvations and x is a vctor of u paramtrs; l and x rfrrin to stimats drivd from th last squars procss such that l l + v (.) x x + δ x (.3) whr v is a vctor of rsiduals or small corrctions and δx is a vctor of small corrctions. 3, R.E. Dakin Coordinat ransformations (3)

2 As is usual, th indpndnt osrvations l hav an a priori diaonal cofactor matrix containin stimats of th variancs of th osrvations, and in this nral adjustmnt, th paramtrs x ar tratd as "osrvals" with a full a priori cofactor matrix. h diaonal lmnts of contain stimats of variancs of th paramtrs and th off-diaonal lmnts contain stimats of th covariancs twn paramtrs. Cofactor matrics and ar rlatd to th covarianc matrics Σ and Σ y th varianc factor ll σ Σ ll ll σ ll (.4) ll Σ σ (.5) Also, wiht matrics W ar usful and ar dfind, in nral, as th invrs of th cofactor matrics W (.6) and covarianc, cofactor and wiht matrics ar all symmtric, hnc and W whr th suprscript dnots th transpos of th matrix. W Not also, that in this dvlopmnt whr and W ar writtn without suscripts thy rfr to th osrvations, i.., and W W ll ll Linarizin (.) usin aylor's thorm and inorin nd and hihr ordr trms, ivs d i a f d i lx, lx, + l l + x x (.7) l l, x l, x and with v l l and δx x x from (.) and (.3), w may writ th linarizd modl in symolic form as Av + B δ x f (.8) Equation (.8) rprsnts a systm of m quations that will usd to stimat th u paramtrs from n osrvations. It is assumd that this is a rdundant systm whr n m u (.9) 3, R.E. Dakin Coordinat ransformations (3)

3 h rdundancy or drs of frdom is r m u (.) In quation (.8) th cofficint matrics A and B ar dsin matrics containin partial drivativs of th function valuatd usin th osrvations l and th "osrvd" paramtrs x. A mn, l lx, (.) B mu, x lx, (.) h vctor f contains m numric trms calculatd from th functional modl usin l and x. m a fr (.3) l x f m,,.. h Last Squars Solution of th Gnral Adjustmnt Modl h last squars solution of (.8), i, th solution which maks th sums of th squars of th wihtd rsiduals a minimum, is otaind y minimisin th scalar function ϕ ϕ vwv+ δxwδx k Av+ Bx δ f a f (.4) whr k is a vctor of m Laran multiplirs. ϕ is a minimum whn its drivativs with rspct to v and δx ar quatd to zro, i. ϕ vw ka v ϕ δxw kb δx hs quations can simplifid y dividin oth sids y two, transposin and chanin sins to iv Wv + A k (.5) W δ x + B k (.6) 3, R.E. Dakin Coordinat ransformations (3) 3

4 Equations (.5) and (.6) can comind with (.8) and arrand in matrix form as L NM W A A B B W L P NM v k δx L P f NM P (.7) Equation (.7) can solvd y th followin rduction procss ivn y Cross (99, pp. -3). Considr th partitiond matrix quation P y which can xpandd to iv or L NM P P u ivn as P P y u P L N M y P L N M u P (.8) P y + P y u y P u P y Eliminatin y y sustitutin (.9) into (.8) ivs Expandin th matrix quation ivs L NM P P P P L P NM (.9) P u P y y u P L N M u P PP u Py + Py u PP u PP Py + Py u and an xprssion for y is ivn y cp PP Phy u PP u Now partitionin (.7) in th sam way as (.8) v can liminatd y applyin (.) LL N MNM B B W L NM W A A B B W L P NM v k δx L P f NM P A W A k f A W P L N M P P L N M x P L N M P L N M P δ (.) (.) 3, R.E. Dakin Coordinat ransformations (3) 4

5 Rmmrin that W th quation can simplifid as L NM AA B B W k f P L N M x P L N M P δ Aain, applyin (.) to th partitiond quation (.) ivs and r-arranin ivs th normal quations W BcAA h Bj δx BcAA h c h j c h B AA B + W δx B AA f f (.) (.3) Mikhail (976, p. 4) simplifis (.3) y introducin quivalnt osrvations l whr l Al (.4) Applyin th matrix rul for cofactor propaation (Mikhail 976, pp ) ivs th cofactor matrix of th quivalnt osrvations as AA (.5) With th usual rlationship twn wiht matrics and cofactor matrics, s (.6), w may writ W AA c h (.6) Usin (.6) in (.3) ivs th normal quations as c BWB+ Whδx BWf (.7) With th auxiliaris N and t N B WB (.8) t B Wf (.9) th vctor of corrctions δx is ivn y (.3) δx N + W t 3, R.E. Dakin Coordinat ransformations (3) 5

6 h vctor of Laran multiplirs k ar otaind from (.) y applyin (.9) to iv c h a f a f and th vctor of rsiduals v is otaind from (.) as ivin k AA f Bδx W f Bδ x (.3) Wv + A k v W A k A k (.3).3. h Itrativ Procss of Solution Rmmrin that x x + δ x, s (.3), whr x is th vctor of a priori stimats of th paramtrs, δx is a vctor of corrctions and x is th last squars stimat of th paramtrs. At th innin of th itrativ solution, it can assumd that x quals th a priori stimats x and a st of corrctions δx computd. hs ar addd to x to iv an updatd st x. A and B ar rcalculatd and a nw wiht matrix W computd y cofactor propaation. h corrctions ar computd aain, and th whol procss cycls throuh until th corrctions rach som prdtrmind valu, which trminats th procss. x x + δ (.33) n+ n x n.4. Drivation of Cofactor Matrics In this sction, th cofactor matrics of th vctors x, δx, vand l will drivd. h law of cofactor propaation will usd and is dfind as follows (Mikhail 976, pp ). Givn a functional rlationship z axf (.34) twn two random vctors z and x and th varianc-covarianc matrix Σ, th varianccovarianc matrix of z is ivn y Σ J Σ J (.35) zz zx zx 3, R.E. Dakin Coordinat ransformations (3) 6

7 whr J is a matrix of partial drivativs zx J zx L NM z z z z z z z z z m m m Usin th rlationship twn varianc-covarianc matrics and cofactor matrics, s (.5), th law of cofactor propaation may otaind from (.35), as n n n P J J (.36) zz zx zx or a function z containin two indpndnt random varials x and y with cofactor matrics and yy th law of cofactor propaation ivs th cofactor matrix as zz z ax, yf (.37) J J + J J (.38) zz zx zx zy yy zy.4.. Cofactor Matrix for x Accordin to quations (.33) and (.3) with (.9) th last squars stimat x is (.39) x x + N + W B Wf and x is a function of th a priori paramtrs x (th "osrvals") and th osrvations l sinc th vctor of numric trms f contains functions of oth. Applyin th law of propaation of cofactors ivs h partial drivativs of (.39) ar I K J I K J I + K J x x x x x + + I N W B W f x x I K J (.4) (.4) 3, R.E. Dakin Coordinat ransformations (3) 7

8 a lf (.4) x + N W B W f l f rom (.3), f x, th partial drivativs and f, ar th dsin matrics A and B ivn y (.) and (.) f x B (.43) f l A (.44) Sustitutin (.43) and (.44) into (.4) and (.4) with th auxiliary N x + I N W B W B x I N + W N B WB ivs (.45) Sustitutin (.45) and (.46) into (.4) ivs (.46) x + N W B WA l o t o o t o I N + W N I N + W N + N + W B WA N + W B WA With th auxiliary N N + W (.48) and notin that th matrics I, NN, and W ar all symmtric, (.47) may simplifid as I N N I NN N B WA A WBN I K J Rmmrin that AA and W I K J + H G I K J H G I K J t t (.47) NN N N + N N NN + N NN (.49) h last two trms of (.49) can simplifid as follows 3, R.E. Dakin Coordinat ransformations (3) 8

9 N N NN + N NN N N NN + W N N N N + W N N N NN I K J N N and sustitutin this rsult into (.49) ivs NN N N + N N NN (.5) urthr simplification ivs I NN N N N N + W N N W H N I K I K J (.5) and sinc W I th cofactor matrix of th last squars stimats x is N N + W (.5).4.. Cofactor Matrix for l Binnin with th final adjustd osrvations ivn y (.) and usin (.3) and (.3) w hav l l A + k l l + v (.53) l + A W f Bδx l + A Wf A WBδx a f 3, R.E. Dakin Coordinat ransformations (3) 9

10 Sustitutin th xprssion for δx ivn y (.3) with th auxiliaris t and N ivn y (.9) and (.48) rspctivly ivs l l A W f A + W B N + W t l + A Wf A WB N + W B Wf l + A Wf A WBN B Wf (.54) and l is function of th osrvals x and th osrvations l sinc f of propaation of cofactors to (.54) ivs ll I KJ I + KJ l x l l and th partial drivativs ar otaind from (.54) as I KJ ax, lf. Applyin th law I l KJ (.55) l A W f A WBN B W f l + I A W f A WBN B W f l With f x B and f l A, and with th auxiliary N B WB th partial drivativs com l A WBN B W A WB A WBN N A WB (.56) l I+ AWBN BWA AWA (.57) Sustitutin (.56) and (.57) into (.55) ivs ll m r m r st nd trm + trm (.58) whr 3, R.E. Dakin Coordinat ransformations (3)

11 st m trmr A W B N N N N B W A A W B N NB WA A W B N N B W A + A W B B W A nd m trmr + A WBN B WA A WA + + A W B N B W A A WBN B WAA WBN B WA A WBN B WAA WA A WA h st trm can simplifid as A W AA W B N B W A + A W AA W A st m trmr A W B N N N N N N N N + B W A I K J + A W B N N N N N N B W A ut w know from (.5) that NN, and from (.5) that N so st m trmr A W B N N B WA A W B N N N N B W A A W B N I N N B W A h trm in rackts has n simplifid in (.5) as W N which ivs th st trm as I K J I K J I K J I K J I K J st m trmr A WBN W N B WA (.59) h nd trm of (.58) can simplifid y rmmrin that AA W so that aftr som cancllation of trms w hav nd m trmr + A WBN NN B WA A WA (.6) Sustitutin (.59) and (.6) into (.58) ivs th cofactor matrix of th adjustd osrvations as + A WB N + W B WA A WA ll (.6) 3, R.E. Dakin Coordinat ransformations (3)

12 .4.3. Cofactor Matrix for δx rom (.3) and (.9) δx N + W B Wf N B Wf and applyin th law of propaation of cofactors ivs δxδ x H G I K J H G I ff K J N BW N BW (.6) (.63) h cofactor matrix is otaind from f x, l ff ff I K J f x f fi K J f I K J a f a f a f a f (.64) B B + A A B B + AA B B + a f as I K J + Sustitutin (.64) into (.63) and simplifyin ivs δδ N + W N N N + W + N + W N N + W (.65) Equation (.65) can simplifid furthr as δδ N N NN + N NN N N NN + W N N N N + W N N N NN I K J or δδ N N N + W N (.66) 3, R.E. Dakin Coordinat ransformations (3)

13 .4.4. Cofactor Matrix for v rom (.3), (.3) and (.3) w may writ th followin v A k A W f Bδx A W f A W Bδx and with (.9) and th auxiliary N N + W x x A W f A W B N + W t a f v A Wf A WBN B Wf (.67) v is a function of th osrvals x and th osrvations l sinc f x, l of propaation of cofactors ivs h partial drivativs of (.67) ar vv I K J v x v I K J + vi K J a f and applyin th law v I K J (.68) v A W f A WBN B W f v A W f A WBN B W f l With f x B and f l A, and with th auxiliary N B WB th partial drivativs com v A WBN N A WB (.69) v A WBN B WA A WA (.7) Sustitutin (.69) and (.7) into (.68) ivs m r m st vv trm + nd trm r (.7) whr 3, R.E. Dakin Coordinat ransformations (3) 3

14 st m trmr A W B N N N N B W A A W B N NB WA A W B N N B W A + A W B B W A nd m trmr A WBN B WAA WBN B WA + A W B N B W AA W A A W AA W B N A W AA W A B W A h st trm aov is idntical to th st trm of (.58) which simplifis to (.59) as st m trmr A WBN W N B WA (.7) h nd trm aov can simplifid y rmmrin that AA W so that aftr som manipulation w hav h trm in rackts can xprssd as and th nd trm coms nd m trmr A W B N N N N B W A A W B N B W A + A W A N NN N N N N N N N N + W N N W N I K J H I K c h nd m trmr A WBN W N B WA A W B N B W A + A W A (.73) Sustitutin (.7) and (.73) into (.7) ivs th cofactor matrix of th rsiduals v as and y inspction of (.64) and (.74) AWBN+ W BWA+ AWA vv (.74) 3, R.E. Dakin Coordinat ransformations (3) 4

15 (.75) vv ll.4.5. Covarianc Matrix Σ Σ σ (.76) h stimatd varianc factor is σ vwv+ δxwδx r (.77) and th drs of frdom r ar r m u + u x (.78) whr m is th numr of quations usd to stimat th u paramtrs from n osrvations. u is th numr of wihtd paramtrs. [Equation (.78) is ivn y Krakiwsky (975, p.7, qn -6) who nots that it is an approximation only and dircts th radr to Bosslr (97) for a complt and riorous tratmnt.] x.5. Gnration of th Standard Last Squars Cass Dpndin on th form of th dsin matrics A and B, and also on whthr th paramtrs ar tratd as osrvals, i, is W, thr ar svral diffrnt possiilitis for th formulation and solution of last squars prolms. h standard cass ar listd low..5.. Comind Cas with Wihtd Paramtrs A, B, W, W Av + B δ x f h nral cas of a non-linar implicit modl with wihtd paramtrs tratd as osrvals is known as th Comind Cas with Wihtd Paramtrs. It has a solution ivn y th followin quations (.3), (.8), (.9), (.6), (.3), (.3), (.3), (.), (.65), (.5), (.74), (.6), (.64), (.77) and (.78). δx N + W t (.79) with N B WB (.8) t B Wf (.8) 3, R.E. Dakin Coordinat ransformations (3) 5

16 W AA c h (.8) x x + δ x (.83) k W f Bδx a f (.84) v W A k A k (.85) l l + v δδ N + W N N N + W + N + W N N + W N + W N N + W (.86) (.87) (.88) A WA A WB N + W B WA vv + A WB N + W B WA A WA ll (.89) (.9) ff BB + (.9) σ vwv+ δxwδx r (.9) r m u + u x (.93) Σ δ x δ x σ δ x δ x (.94) Σ σ (.95) Σ vv σ vv (.96) Σ ll ll σ (.97) Σ ff σ ff (.98) 3, R.E. Dakin Coordinat ransformations (3) 6

17 .5.. Comind Cas A, B, W, W Av + Bδx f h Comind Cas is a non-linar implicit mathmatical modl with no wihts on th paramtrs. h st of quations for th solution is dducd from th Comind Cas with Wihtd Paramtrs y considrin that if thr ar no wihts thn W and. his implis that x is a constant vctor (dnotd y x ) of approximat valus of th paramtrs, and partial drivativs with rspct to x ar undfind. Sustitutin ths two null matrics and th constant vctor x x into quations (.) to (.78) ivs th followin rsults. δx N t (.99) with N B WB (.) t B Wf f x lh c, W AA c h (.) (.) (.3) x x + δ x (.4) k W f Bδx c h (.5) v W A k A k (.6) l l + v (.7) δδ N (.8) A WA A WBN B WA vv (.9) + A WBN B WA A WA ll f f (.) (.) σ vwv r (.) 3, R.E. Dakin Coordinat ransformations (3) 7

18 r m u (.3) Σ Σ σ δδ (.4) Σ vv σ vv (.5) Σ ll ll σ (.6) Σ f f f f σ (.7).5.3. Paramtric Cas A I, B, W, W v + B δ x f h Paramtric Cas is a mathmatical modl with th osrvations l xplicitly xprssd y som non-linar function of th paramtrs x only. his implis that th dsin matrix A is qual to th idntity matrix I. Sttin A I in th Comind Cas (with no wihts) lads to th followin quations. δx N t (.8) with N B WB (.9) t B Wf f x lh c, (.) (.) x x + δ x (.) v f Bδ x (.3) l l + v (.4) δδ N (.5) BN B vv (.6) ll BN B (.7) f f (.8) 3, R.E. Dakin Coordinat ransformations (3) 8

19 σ vwv r (.9) r n u (.3) Σ Σ σ δδ (.3) Σ vv σ vv (.3) Σ ll ll σ (.33) Σ f f f f σ (.34).5.4. Condition Cas A, B, W, W Av f h Condition Cas is charactrisd y a non-linar modl consistin of osrvations only. Sttin B in th Comind Cas (with no wihts) lads to th followin quations. k Wf (.35) with W AA f af l v W A k A k c h (.36) (.37) (.38) l l + v (.39) vv A WA (.4) A WA ll (.4) σ vwv r (.4) r m (.43) 3, R.E. Dakin Coordinat ransformations (3) 9

20 Σ vv σ vv (.44) Σ ll ll σ (.45) 3, R.E. Dakin Coordinat ransformations (3)