website chapter 3

8/18/2019 Website Chapter 3

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CHAPTER 3CHAPTER 3

RECURSIVE ESTIMATION FORLINEAR MODELS

•Organization of chapter in ISSO –Linear models

•Relationship between least-squares and mean-square

–LMS and RLS estimation

• Applications in adaptie control –LMS! RLS! and "alman filter for time-ar#ing solution

–$ase stud#% Oboe reed data

Slides for Introduction to Stochastic Search

and Optimization ( ISSO) by J. C. Spall


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Basic Linear ModelBasic Linear Model

•$onsider estimation of ector in model that is linear in

•Model has classical linear form

where z k is k th measurement! hk is corresponding &design ector!'

and v k is un(nown noise alue

•Model used e)tensiel# in control! statistics! signal processing! etc*

•Man# estimation+optimization criteria based on &squared-error'-

t#pe loss functions

– Leads to criteria that are quadratic in – ,nique global. estimate

= + !T k k k z v h


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LeastS!"ares Esti#ationLeastS!"ares Esti#ation

•Most common method for estimating in linear model is b# method

of least squares

•$riterion loss function. has form

where Z n / 0z 1! z 2 !3! z n4T and H n is n × p concatenated matri) of hk T row ectors

•$lassical batchbatch least-squares estimate is

•5opular recursiverecursive estimates LMS! RLS! "alman filter. ma# be

deried from batch estimate

=

− = − −∑2

1

1 1

2 2 . . .

nT T

k k n n n n

k

z n n

h Z H Z H

−≡

. 16 .n T T n n n nH H H Z


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$eo#etric Inter%retation o& LeastS!"ares$eo#etric Inter%retation o& LeastS!"aresEsti#ate '(enEsti#ate '(en p p ) * and) * and nn ) 3) 3


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Rec"rsi+e Esti#ationRec"rsi+e Esti#ation

•7atch form not conenient in man# applications

– 8*g*! data arrie oer time and want &eas#' wa# to updateestimate at time k to estimate at time k 91

•Least-mean-squares LMS. method is er# popular recursie

method

– Stochastic analogue of steepest descent algorithm

•LMS rec"rsion,

•$onergence theor# based on stochastic appro)imation e*g*!

L:ung! et al*! 1;;2< =erencs>r! 1;;?.

– Less rigorous theor# based on connections to steepest descent

ignores noise. @idrow and Stearns! 1;?< Ba#(in! 1;;C.

+ + + += − − >1 1 1 16 6 6 ! D .T k k k k k k a z ah h


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LMS in ClosedLoo% ControlLMS in ClosedLoo% Control•Suppose process is modeled according to autoregressie AR.form%

where x k represents state! γ and βi are un(nown parameters! uk iscontrol! and w k is noise

•Let target &desired'. alue for x k be d k

•Optimal control law (nown minimizes mean-square trac(ing error.%

•Certainty equivalence principle :ustifies substitution of parameter

estimatesestimates for un(nown true parameters – LMS used to estimate γ and βi in closed-loop mode

+ − −= β + β + + β + γ +K 1 D 1 1 !k k k m k m k k x x x x u w

+ − −−β −β − −β=γ

K 1 D 1 1k k k m k m

k

d x x x u


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LMS in ClosedLoo% Control &orLMS in ClosedLoo% Control &or

FirstOrder AR ModelFirstOrder AR Model


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Rec"rsi+e Least S!"ares -RLS.Rec"rsi+e Least S!"ares -RLS.

• Alternatie to LMS is RLS

– Recall LMS is stochastic analogue of steepest descent &firstorder' method.

– RLS is stochastic analogue of Eewton-Raphson &second order'

method. ⇒ faster conergence than LMS in practice

•RLS al/orit(# -* rec"rsions.,

•Eeed P D and to initialize RLS recursions

1 11

1 1

1 1 1 1 1

1

.

T k k k k

k k T k k k

T k k k k k k k ˆ ˆ ˆ z

+ ++

+ +

+ + + + +

= −+

= − −

P h h P P P

h P h

P h h

Dˆ


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Rec"rsi+e Met(ods &or Esti#ation o& Ti#eRec"rsi+e Met(ods &or Esti#ation o& Ti#eVar0in/ Para#etersVar0in/ Para#eters

•Ft is common to hae the underl#ing true eole in timee*g*! target trac(ing! adaptie control! sequential e)perimental

design! etc*.

– Gime-ar#ing parameters implies replaced with k

•$onsider modified linear model

•5rotot#pe recursie form for estimating k is

where choice of Ak and k depends on specific algorithm

1 1 1 1 .!T

k k k k k k k k ˆ ˆ ˆ z + + + += − − A h A

T

k k k k z v = +h


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T(ree I#%ortant Al/orit(#s &or Esti#ationT(ree I#%ortant Al/orit(#s &or Esti#ationo& Ti#eVar0in/ Para#eterso& Ti#eVar0in/ Para#eters

• LMS LMS – =oal is to minimize instantaneous squared-error criteria across

iterations

– =eneral form for eolution of true parameters k

• RLS RLS

– =oal is to minimize weighted sum of squared errors – Sum criterion creates &inertia' not present in LMS

– =eneral form for eolution of k

• Kalman filter Kalman filter – Minimizes instantaneous squared-error criteria

– Requires precise statistical description of eolution of k iastate-space model

• Hetails for aboe algorithms in terms of protot#pe algorithmpreious slide. are in Section I*I of ISSO


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Case St"d0, LMS and RLS 'it( O1oe Reed DataCase St"d0, LMS and RLS 'it( O1oe Reed Data

……an ill win that n!b!y bl!ws "!!#an ill win that n!b!y bl!ws "!!#

J$omedian Hann# "a#e in spea(ing of the oboe in the &Ghe SecretLife of @alter Mitt#' 1;K.

•Section I*K of ISSO reports on linear and curilinear models for

predicting qualit# of oboe reeds

–Linear model has parameters< curilinear has K parameters

•Ghis stud# compares LMS and RLS with batch least-squares

estimates

– 1CD data points for fitting models reeddata-fitreeddata-fit .< D

independent. data points for testing models reeddata-testreeddata-test.

– reeddata-fitreeddata-fit and reeddata-testreeddata-test data sets aailable from

ISSO @eb site


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Oboe withOboe with

Attached ReedAttached Reed


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Co#%arison o& Fittin/ Res"lts &orCo#%arison o& Fittin/ Res"lts &orreeddata-fitreeddata-fit andand reeddata-testreeddata-test

• Go test similarit# of

fit

and

test

data sets! performedmodel fittin" using test data set

• Ghis comparison is for chec(ing consistenc# of the two

data sets< n!t for chec(ing accurac# of LMS or RLSestimates

• $ompared model fits for parameters in

– 7asic linear model eqn* I*2?. in ISSO. p / .

– $urilinear model eqn* I*2C. in ISSO. p / K.

• Results on ne)t slide for basic linear model


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Co#%arison o& Batc( Para#eter Esti#ates &orCo#%arison o& Batc( Para#eter Esti#ates &orBasic Linear Model2 A%%roi#ate 456Basic Linear Model2 A%%roi#ate 456

Con&idence Inter+als S(o'n in 789 8:Con&idence Inter+als S(o'n in 789 8:


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Co#%arison o& Batc( and RLS 'it(Co#%arison o& Batc( and RLS 'it(

O1oe Reed DataO1oe Reed Data

• $ompared batch and RLS using 1CD data points inreeddata-fit and D data points for testing models

in reeddata-test

•Gwo slides to follow present results – irst slide compares parameter estimates in pure linear

model

– Second slide compares prediction errors for linear and

curilinear models


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Batc( and RLS Para#eter Esti#ates &or BasicBatc( and RLS Para#eter Esti#ates &or Basic

Linear Model -Data &ro#Linear Model -Data &ro# reeddata-fitreeddata-fit ..

7atch8stimates

RLS8stimates

$onstant!

θconst −D*1?C −D*D;

Gop close! θT D*1D2 D*1D1

Appearance!θ A

D*D?? D*DKC

8ase of

=ouge! θE D*1? D*11

Nascular! θV D*DKK D*DKIShininess!

θS D*D?C D*D?C

irst blow! θF D*?; D*?KD


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Mean and Median A1sol"te PredictionMean and Median A1sol"te PredictionErrors &or t(e Linear and C"r+ilinear ModelsErrors &or t(e Linear and C"r+ilinear Models-Model &its &ro#-Model &its &ro# reeddata-fit;reeddata-fit; PredictionPrediction

Errors &ro#Errors &ro# reeddata-testreeddata-test..7atch linear

modelRLSlinearmodel

7atchcurilinear

model

RLScurilinear

model

Mean D*2K2 D*2K2 D*2I? D*2I?Median D*2KI D*2?D D*22 D*22K

• Ran matched-pairs t -test on linear ersus curilinearmodels* ,sed one-sided test*

• P -alue for 7atch+linear ersus 7atch+curilinear isD*D

• P -alue for RLS+linear s* RLS+curilinear is D*1D

• Modest eidence for superiorit# of curilinear model

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