Estimation Theory. Chapter 12

Transcription

1 timtio heoy hpte

2 Lie Byei timto Optiml MMS Byei etimto i geel e difficult to compute i cloed fom; ecept fo the joitly Gui ce. But i my itutio, we c t mke the Gui umptio. lled iee Filte Ited, we keep the MMS cot fuctio but coti the etimto to be lie. I thi ce, it tu out tht eplicit fom fo the etimto c be detemied which oly deped o t d d momet of pdf. hi i logou to BLU i clicl etimtio.

3 Lie MMS timtio Scl e Poblem: timte cl dom pmete bed o [..-] by coideig the cl of lie ffie etimto ˆ whee the epecttio i d miimizig BMSˆ [ ˆ ] w..t p, ote: llow fo ce of o-zeo-me d LMMS i uboptiml ule optiml MMS etimto / hppe to be lie i the ce of lie model 3LMMS elie o ttiticl depedece coeltio betwee d

4 Detemiig Lie MMS timto [ ] { } - ˆ,...,, lcultig the me e zeo. be zeo if which will ] [ ] [ lculte optimum Bme

5 Bme ˆ Bme ˆ Fo zeo - me ce : to be lie! ce ice the ltte hppe to MMS etimto fo joitly - Gui hich iideticl ˆ ˆ Detemiig Lie MMS timto

6 mple Uig MIL :I : ˆ ; LMMS : pplyiguboptiml itegtio. t be detemied i cloed - fom due to equied etimto c' MMS we foud tht the Fo D leveli G w/ uifom pio PDF, φ I I I

7 we get : Â, whee 3 3 loed fom olutio tht oly deped o kowledge of, d ecod ode mometof mple d me ot the pdf.

8 Geometicl Itepettio Simil to geometicl itepettio of LLS ecept tht vecto i ow dom & ll vecto umed zeo-me o tht ov,yy- YY hece othogolity d ucoeltede become equivlet [ ˆ] MS hee we defie legth of dom vecto om: Vice wo vecto e othogol iff,y Y Geometiclly, the om of the eo vecto i miimized whe ε,, - lo ie poduct : <, Y > Y

9 Geometicl Itepettio [ ] [ ] - m - m m,,... - :,,... - : ˆ,,... - fo - Piciple,,... - fo ˆ Othogolity m m m ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ hi i the fmou oml qutio!

10 oml qutio φ ε m m m m m Bme m m m m ˆ ˆ

11 Vecto LMMS [ ] [ ] ii i ˆ ˆ ˆ ] - [ Uig Othogolity Piciple Fo zeo - me ce: ˆ Bme ˆ ˆ ˆ Mti ovice

12 Popetie of LMMS ˆ ˆ whee ˆ ˆ ˆ if like MMS & MP ove ffie tfomtio LMMS commute ˆ ˆ b if eview poblem lte i give poof b α α α α

13 Byei Gu - Mkov heoem Fo the Byei Lie Model : ˆ defie ε - ˆ εε LMMS - o Gui umptio Reult ideticl to Byei lie model i h. ecept w/o Gui umptio. hi etimto will ot be optimum ule the coditiol epecttio / hppe to be Lie i joitly - Gui ce dvtge of LMMS : loed fom, oly deped o me d covice, vey popul i pctice.

14 BLU v. LLS v. LMMS BLU h.6 : clicl etimto detemiitic pmete, ubied, oly oie umed dom LLS h.8 : o ttiticl umptio, oly lie model umptio LMMS h. : Byei etimto, dom pmete, covege to BLU w/ o pioi ifo. w w BLU ˆ J LLS LLS / ˆ ; to it i equl Fo Lie Model, ˆ

15 fo m R Mot impott pplictio of umptio : Dt i SS w/ zeo me pplictio of Ŝ R SS ` SS R R SS iee Filteig Filteig : S to be etimted bed o m Sm m : R ˆ ` SS LMMS,,..., i.e. bed o peet d pt dt oly. R ' cul filte to dt. R Sˆ R whee [ Symmeticoeplitz S S S ` SS fuctio of ` deote flippig i eveed ode ]

16 Sˆ Ŝ e hve, Poof : whee the k k k R JR h h k, JJ k FIR iee Filte k k k k ime Vyig FIR filte. ' J R ' h whee J whee h It i poible to deive ode - ecuive fomul Levio lgoithm ode filte i computed bed o i evel mti & h h h k k, th ime Reveed Veio ode th J I ode filte

17 Remk : Ŝ ote tht IIR iee Filteig ymptotic e ifiite - ode filte, timte S bed o the peet d ifiite pt h k k Uig othogolity piciple :S -Ŝ k hk - k -l k k [S -Ŝ] -l hk ice it hold oly fo l l k c' t be olved uig Fouie fom. SS ε fo l l fo, -,..,,... S l l,,... he l < llowed we hve the moothig poblem which we'll tudy et e ll come bck to olve it oo!

18 timte Ŝ wo-sided iee Filteig Smoothig :ecove igl fom it etie oiy obevtio pt d futue dt ovice of o RSS RSS R : M Mti Ŝ hece R P f P S S S o ubet of S whee SS < f < - R } h R P [ S R PSS f f P ] [SS ymptote: qutio * hold but fo ll"l". SS f f SS SS SS : f it bed o the etie hi i the iee moothig filte R SS I R SR f : o ul SR f φ : : ] R SS SS SRf φ SRf

19 3 Pedictio : Sice Pedictio iee Filte timte - l fom{,,... -} hi i l - ode pedictio poblem [ l[ ] ] ˆ - l R futue dt fom othe pt dt poit ' R ' - k k k R MMS k h k: k k: hk - k h ' R R ' ' h h h whee h - k [ l l l] k l l l ' e c lo coide itepoltio poblem whee we etimte miig mple withi the dt ecod

20 Filteig Poblem Bck to the filteig poblem :ee SK.6 k hk Defie the oe -ided Z - tfom [z] h [ z P By Spectl Fctoiztio heoem P z l B z z] B z ul mi-phem ti- -Phe ul [ zbzbz ] - P z l : l Z φ k z P SS SS SS φ : cul pt i Z fo whee h i cul :ul

21 Let But Bz hece ow, cul compoet mut be equl to Gz d hece o Gz Bz - - G z i z - tfom of itified iff ice Z Gz zbz - Bz - PSS z - Bz PSS z i two - ided equece, it - Bz { G z} cul fuctio G z Z PSS z B z PSS z - Bz mut be ti - cul ti - cul equece, PSS z Gz - - Bz PSS z - Bz z i lo tictly ti - cul. fo Bz powe of Z oly PSS z B z P P SS z z

22 ul iee Filteig mple S Suppoe whee d.9 P z P z SS.9z. 9 z Fid the cul iee filte to etimte fom S,,... Solutio: P z P z SS.67z.67z.9z. z P z z B z.436.9z

23 G P z B z SS z mple ot d.9 G z.9z.9z.9z.67z z.67z.436 G.436.9z.436.9z z.436 z G z B z z.9z.67z.34.67z z.436 h k k.67 fo.34 k

24 Pedictio Filte mple : w/ uto - coeltio fom d l l SS ' l equece oie i zeo - me white dom poce w/ vice ompute - tp iee pedictio filte to pedict Sol R,,...; l δ l i l. zeo - me dom poce, l l l l l l

25 Pedictio - mple k k ˆ / k

26 Pedictio - mple MMS ' [..] ecie: coide the limitig ce SR become ifiite I Geel:If l l whee & e idepedet l SS dom pocee Pctice Poblem fom SK :.,.,.6,.8

27 Summy of timtio Method Sigl poceig poblem licl v. Byei ppoch Dimeiolity poblem Ye Pio Kowledge Ye Byei ppoch o o Pio Kowledge Ye Byei ppoch ew Dt Model o ke ew dt o ot Poible o licl ppoch

28 Summy of timtio Method Byei ppoch PDF Kow o Fit two Momet kow Ye LMMS timto Ye o ompute Me of Poteio PDF Ye MMS timto ot Poible o Mimize Poteio PDF Ye MP timto o ot Poible

29 Summy of timtio Method licl ppoch PDF Kow o Sigl i oie o ot Poible Ye Ye RLB Stified Ye MVU Stimto Sigl Lie o LS o Ye omplete Sufficiet Sttiitc eit Ye Mke it Ubied Ye MVU Stimto Fit wo oie Momet Kow o o o Ye BLU vlute ML Ye ML o vlute Method of Momet timto Ye Momet timto o ot Poible

30 Review Poblem [ ] - c - b - c b Bme Bme Bme by ; deote - b - c - Bme c b MMS c b c c b b c b

31 Review Poblem ot d ow, if ~ U, φ co π, π, d φ &, co 3 π, Quetio : how doe thi etimto elte to the ylo eie epio of co π / MMS / 9 π, / π b c b 4 π, φ π φ c.4 5 π,

32 Review Poblem ot. MMS c b, but c b c b we hve lie etimto, Uig

33 Review Poblem f f f f f f f f i i i p p p []co ee 4.3 e othogol colum of, i Fo equtio e oml co co co co co co 8.5 Pob 8: hpte π π π π π π π I

34 Review Poblem ot d P i mi, ~ PDF i Fo G, ] [ - - J P P I I P i

35 Review Poblem ot d I I I lie fuctio of i Sice, Sice, w w, ~.

36 Review Poblem [ ] [ ] &.3.9 Uig Bme d whee, Fom.7 Pob.: - µ µ µ µ h h h h h

37 Review Poblem ] [ o, R R whee,. Uig Pob.6 : ww ] [ I w I &

38 Review Poblem ot d I I I I M Fom.

39 Review Poblem Pob.8 α α α i But & α b α α α [ ] [ ] α b b

40 Review Poblem ii α α α if & α α α [ ] - - [ ] - - [ ] α