# Chapter 6 Best Linear Unbiased Estimate (BLUE)

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1 hpter 6 Bet Lner Unbed Etmte BLUE

2 Motvton for BLUE Except for Lner Model ce, the optml MVU etmtor mght:. not even ext. be dffcult or mpoble to fnd Reort to ub-optml etmte BLUE one uch ub-optml etmte Ide for BLUE:. Retrct etmte to be lner n dt x. Retrct etmte to be unbed 3. Fnd the bet one.e. wth mnmum vrnce dvntge of BLUE:eed only t nd nd moment of PDF Ddvntge of BLUE:. Sub-optml n generl. Sometme totlly npproprte ee bottom of p. 34 Men & ovrnce

3 6.3 Defnton of BLUE clr ce Oberved Dt: x [x[0] x[]... x[ ] ] PDF: px;θ depend on unknown θ BLUE contrned to be lner n dt: hooe to gve: θˆ BLU nx[ n] x n0. unbed etmtor. then mnmze vrnce Vrnce Lner Unbed Etmtor BLUE onlner Unbed Etmtor MVUE ote: h not Fg. 6. 3

4 6.4 Fndng he BLUE Sclr e. ontrn to be Lner: ˆ θ n 0 n x[ n]. ontrn to be Unbed: E { θˆ} θ Ung lner contrnt n 0 n E { x[ n ]} θ Q: When cn we meet both of thee contrnt? : Only for certn obervton model e.g., lner obervton 4

5 Fndng BLUE for Sclr Lner Obervton onder clr-prmeter lner obervton: x[n] θ[n] + w[n] E{x[n]} θ[n] hen for the unbed condton we need: ell how to chooe weght to ue n the BLUE etmtor form { ˆ} E θ θ n ow gven tht thee contrnt re met We need to mnmze the vrnce!! 0 eed ˆ θ n n[ n] θ #"! 0 n x[ n] Gven tht the covrnce mtrx of x we hve: vr θˆ { } { } vr x BLU Lke vr{x} vr{x} 5

6 6 Gol: mnmze ubject to ontrned optmzton ppendx 6: Ue Lgrngn Multpler: Mnmze J + λ 0 : Set J λ λ λ \$ #\$"\$! ˆ vr θ x x ˆ θ BLUE ppendx 6 how tht th cheve globl mnmum

7 pplcblty of BLUE We jut derved the BLUE under the followng:. Lner obervton but wth no contrnt on the noe PDF. o knowledge of the noe PDF other thn t men nd cov!! Wht doe th tell u??? BLUE pplcble to lner obervton But noe need not be Gun!!! w umed n h. 4 Lner Model nd ll we need re the t nd nd moment of the PDF!!! But we ll ee n the Exmple tht we cn often lnerze nonlner model!!! 7

8 6.5 Vector Prmeter e: Gu-Mrkov hm Gu-Mrkov heorem: If dt cn be modeled hvng lner obervton n noe: x Hθ + w Known Mtrx Known Men & ov PDF otherwe rbtrry & unknown ˆ hen the BLUE : θ H H H x BLUE nd t covrnce : θ ˆ H H ote: If noe Gun then BLUE MVUE 8

9 Ex. 4.3: DO-Bed Emtter Locton t x,y t t Rx x,y t t Rx t t 3 x,y Rx 3 x 3,y 3 Hyperbol: τ t t contnt Hyperbol: τ 3 t 3 t contnt DO me-dfference-of-rrvl ume tht the th Rx cn meure t O: t hen from the et of O compute DO hen from the et of DO etmte locton x,y We won t worry bout how they do tht. lo there re DO ytem tht never ctully etmte O! 9

10 O Meurement Model ume meurement of O t recever only 3 hown bove: t 0, t,,t - here re meurement error O meurement model: o me the gnl emtted R Rnge from x to Rx c Speed of Propgton for EM: c 3x0 8 m/ t o + R /c + 0,,..., - Meurement oe zero-men, vrnce σ, ndependent but PDF unknown vrnce determned from etmtor ued to etmte t ow ue: R [ x x + y - y ] / t f x, y o + c x x + y y + onlner Model 0

11 Lnerzton of O Model So we lnerze the model o we cn pply BLUE: ume ome rough etmte vlble x n, y n x x n + δx y y n + δy know etmte know etmte θ [δx δy] ow ue truncted ylor ere to lnerze R x n, y n : R Known R n + xn x δx Rn # \$"\$! + yn y δy Rn # \$"\$! B ~ Rn B pply to O: t t o + δ x + δy + c c c known known known hree unknown prmeter to etmte: o, δy, δy

12 O Model v. DO Model wo opton now:. Ue O to etmte 3 prmeter: o, δy, δy. Ue DO to etmte prmeter: δy, δy Generlly the fewer prmeter the better Everythng ele beng the me. But here everythng ele not the me: Opton & hve dfferent noe model Opton h ndependent noe Opton h correlted noe In prctce we d explore both opton nd ee whch bet.

13 3 onveron to DO Model DO rther thn O DO:,,,, ~ ~ t t τ \$"\$! # #\$"\$! \$"\$! # noe correlted known known + + y c B B x c δ δ In mtrx form: x Hθ + w w H B B B B B B c & & & & & & & [ ] τ τ τ ' x [ ] y δx δ θ See book for tructure of mtrx w w } cov{ σ

14 pply BLUE to DO Lnerzed Model θˆ BLUE H H w H w x θˆ H H w H Dependence on σ cncel out!!! H H x σ H Decrbe how lrge the locton error H hng we cn now do:. Explore etmton error cov for dfferent x/rx geometre Plot error ellpe. nlytclly explore mple geometre to fnd trend See next chrt more detl n book 4

15 pply DO Reult to Smple Geometry x R R x α R x α R x3 d d hen cn how: θˆ σ c co 0 α 0 3/ nα Dgonl Error ov lgned Error Ellpe e y nd y-error lwy bgger thn x-error e x 5

16 σ x /cσ or σ y /cσ σ x σ y α degre e Ued Std. Dev. to how unt of X & Y ormlzed by cσ get ctul vlue by multplyng by your pecfc cσ vlue x R R x α R x α R x3 d d For Fxed Rnge R: Increng Rx Spcng d Improve ccurcy For Fxed Spcng d: Decreng Rnge R Improve ccurcy 6

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