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