Lecure 5 8. he Normalized Leas-Squares Esimaor wih Expoeial Forgeig his secio is devoed o he mehod of Leas-Squares wih expoeial forgeig ad ormalizaio. Expoeial forgeig of daa is a very useful echique i dealig wih ime-varyig ukow parameers. Is iuiive moivaio is ha pas daa are geeraed by pas parameers ad hus should be discoued whe beig used for he esimaio of he curre parameers. Isead of pealizig all of he pas errors from τ = o τ = ha are due o ˆ θ ( θ (as i is doe whe derivig he pure Leas-Squares Esimaor, he miimizaio problem is posed for he cos fucio wih he so-called expoeial forgeig facor. he cos fucio is chose i he form: ( ˆ θ ( ( x( τ y( τ ( ˆ J θ ( = exp λ( r dr d τ τ 44443 + ( x( τ P( x( τ 444444443 β(, τ =Expoeial Forgeig Facor = Normalizig Sigal 44443 β (, ( ( ( + exp λ( r dr ˆ θ ˆ θ ˆ ˆ Q θ θ (8. he esimaed value ˆ θ ( will be foud such ha: ( ( ( τ ( τ ( ( ( ˆ e J θ =, d, ˆ ˆ Q ˆ ˆ β τ τ + β θ θ θ θ mi (8. x θ y ˆ I (8. (8., P P ˆ θ ˆ = θ is he iiial esimae of he rue ukow cosa parameer vecor θ, Q= Q > is he = > is he ormalizig sigal weighig marix, ermial sae weighig marix, λ : is he so-called ime varyig forgeig facor, + + r R R, ad β(, τ = exp λ( r dr (8.3 τ 5
Remark 8. If λ( r = λ is a cosa he β(, τ o: ( θ ( x τ y( τ ˆ ( ( ( ( x( τ ( = e λ τ, ad he cos fucio (8. simplifies ( ( ( J ˆ θ = e dτ + e ˆ θ ˆ θ Q ˆ θ ˆ θ λ τ λ (8.4 Moreover, seig λ = ad P= Q= resuls i he iegral cos fucio ( ˆ( ˆ ( ( J θ = θ x τ y τ dτ (8.5 ha was used o derive he pure Leas-Squares Esimaor. Goig back o solvig he miimizaio problem i (8., suppose ha P > ad Q >. he cos fucio J ˆ θ ( is covex wih respec o ˆ θ (, for all. hus, ay local miimum is also global ad saisfies ( θ ( J ˆ =, (8.6 Calculaig he gradie of he cos fucio J ˆ θ ( wih respec o ˆ θ (, resuls i: ( ˆ θ ( ( ( τ ( τ x( τ x y J( ˆ θ ( = ( x( τ β(, τ dτ + β(, Q( ˆ θ ˆ θ (8.7 Equaig he gradie o zero while rearragig erms, furher yields: (, ˆ ( ( (, τ ( x( τ β β (, Q+ ( x( τ ( x( τ dτ ˆ θ 4444444444444444443 Γ f (, τ ( x( τ β = β Qθ + x τ y τ dτ 44444444444444443 ( (8.8 Immediaely oe ha sice Q > ad ( x ( x, Γ exiss a each ime. Moreover, by defiiio ad because of (8.3, his marix saisfies he followig IVP: 5
( x ( x ( ( d Γ ( =λ ( Γ ( + d x Γ = Q (8.9 Similarly, he fucio f i (8.8 saisfies f& f x f Q ˆ = θ ( = λ ( ( +( ( y ( x( (8. Usig (8.8, resuls i: ˆ θ ( =Γ ( f ( (8. Based o he ideiy d d ( ( ( ( = Γ Γ =Γ& Γ ( +Γ( Γ ( (8. d d oe ca wrie: d ( ( Γ & =Γ Γ ( Γ( d (8.3 Subsiuig (8.9 io (8.3, yields: ( x( x( x( Γ & ( =Γ( λ ( Γ ( Γ Γ = Q ( (8.4 or, equivalely: ( x ( x ( x( Γ & ( = λ ( Γ( Γ( Γ Γ =Γ = Q ( (8.5 Moreover, differeiaig (8. while usig (8. ad (8.4, oe ges: 5
& ˆ θ ( =Γ & ( f ( +Γ( f& ( x( λ ( ( ( ( λ ( f ( x( ( x( ( x( = Γ Γ Γ f +Γ + ( x( y ( ( (8.6 Collecig erms, yields: ey ( ( x( ˆ 64444744448 & θ ( =Γ( ( x( Γ( f ( y( ( x( 443 ˆ θ ( 4444443 y ˆ( ˆ θ f {{ ( =Γ = ˆ θ Q Q ˆ θ (8.7 Fially, combiig (8.7 wih (8.5, resuls i he coiuous ime recursive versio of he Normalized Leas-Squares Esimaor wih Expoeial Forgeig (NLSEEF: ( x( x( ˆ& θ ( =Γ( e y ( ˆ ˆ θ = θ ( x( ( x( Γ & ( = λ ( Γ ( Γ ( ( x( Γ Γ =Γ = Q ( (8.8 where y ( ( ˆ = ˆ = θ ( θ ( ( ˆ θ( θ ( x( θ ( x( e y y x x = =Δ (8.9 is he oupu predicio error. Remark 8. Seig λ ( = ad P = resuls i he pure Leas-Squares Esimaor. 53
ˆ& θ =Γ ey Γ =Γ & Γ ( (8. O he oher had, wih ( Squares Esimaor. λ = ad P= P >, oe ges he ormalized pure Leas- ˆ& θ =Γ e y Γ & =Γ Γ ( (8. Nex, sabiliy ad covergece properies of he NLSEEF (8.8 will be aalyzed. Usig (8.3 ad (8.8, he esimaor gai ca be explicily wrie as: Cosider ( ( exp ( x( τ ( x( τ λ exp λ τ ( x( τ (8. τ r dr r dr d Γ =Γ + d d & & (8.3 ( ( ( Γ Δ θ =Γ ( Δ θ( +Γ ( ˆ θ( Subsiuig (8.9 ad (8. io (8.3, yields: d ( ( ( ( θ λ θ Γ Δ = Γ + θ λ( θ Δ +Γ Γ Δ = Γ Δ d (8.4 Iegraig boh sides of (8.4, resuls i: Γ Δ θ = exp λ( r dr Γ Δθ (8.5 Cosequely: Δ θ( = λ( r dr Γ( Γ Δ θ =Τ Γ Δθ exp 4444443 Τ ( (8.6 Subsiuig for Γ ( from (8.8, oe ges 54
( x( τ x( τ x( τ ( x( τ x( τ x( τ Δ θ = β(, β(, Q+ β, τ dτ Γ Δθ τ = Q+ exp λ τ Γ Δθ ( r dr d (8.7 As see from (8.6 (8.7, he sabiliy properies of he NLSEEF deped o he ime- λ r. varyig forgeig facor From (8.6 i follows ha i order o show boudedess of he esimaed parameer ˆ θ, i is sufficie o esablish boudedess of ( Τ ( = exp λ ( r dr Γ( (8.8 owards ha ed, differeiaig (8.8 ad subsiuig for he dyamics of Γ ( from (8.8, yields: =exp λ ( r dr Γ Γ d Τ ( =λ( exp λ( r dr Γ ( + exp λ( r dr λ( Γ Γ Γ d (8.9 From (8.9 i follows ha ad herefore ( Τ Τ =Γ = Q <, (8.3 Τ Q <, (8.3 Moreover, sice Τ ( is o-icreasig ad bouded from below, i has a limi: Remark 8.3 lim Τ =Τ < (8.3 55
Alhough ( Τ is uiformly bouded ad has a fiie limi, he esimaio gai Γ ( may grow ubouded. I fac, for a scalar liear i parameers saic model wih a sigle ukow parameer, oe ca show ha if he forgeig facor is cosa, λ( r = λ >, ad if he regressor vecor ( x( = e he Γ, as. O he oher had, if he regressor vecor ( x( is PE he o oly he esimaio gai remais bouded bu also he parameer errors coverge o zero expoeially wih he rae λ. Usig (8.3, a upper boud for he parameer esimaio error i (8.6 ca be calculaed: Q = Q 64748 Δθ Τ Γ Δ < } ( ( ( θ ( (8.33 his proves ha he esimaed parameer errors are uiformly ulimaely bouded, ad hece ˆ θ L. Also because of (8.3, akig he limi i (8.6 gives: Q lim Δ θ = lim Τ Γ Δ θ =Τ Δ θ < (8.34 ad hus ˆ Q limθ = θ +Τ Δ θ = θ < (8.35 Nex, we show ha he ormalized predicio error is uiformly ulimaely bouded ad e y square-iegrable, ha is L L. owards ha ed, cosider he fucio (, V Differeiaig (8.36 alog he rajecories of Δ θ =Δθ Γ Δ θ (8.36 ˆ& θ =Γ Δθ d ( Γ Δ θ =λ( Γ Δθ d (8.37 yields: 56
ˆ d V& &, Δ θ = θ Γ Δ θ +Δθ ( Γ ( Δθ d = Γ Δ Γ ( Δ ( Δ Γ Δ 443 444444443 V(, Δθ θ θ λ θ θ ey Δθ Δ θ= (8.38 or, equivalely ( θ λ V ( θ e y V &, Δ =, Δ (8.39 Sice V ad V& he V is a o-icreasig fucio of ime ad cosequely i has a fiie limi: ( θ lim V, Δ = V < (8.4 Usig (8.39 ad iegraig boh sides of e y V& (8.4 proves ha he ormalized predicio error e y is square-iegrable: ( τ x( τ e y ey d V V < L (8.4 he uiform ulimae boudedess of he ormalized predicio error follows direcly from: y θ θ θ e Δ Δ Δ = = + P λ P mi ( P λmi (8.43 ad from he fac ha e y Δθ L. hus, L L. Γ Γ for all, (sufficie codiios for his will be esablished i Secio 9. he usig (8.8 yields Moreover, suppose ha ( 57
ˆ& ey θ ( =Γ e y Γ e y Γ {{ L L L (8.44 herefore ˆ& θ L L (8.45 Summarizig, yields: heorem 8. Recursive versio of he NLSEEF is give by (8.8 ad he esimaor has he followig properies: ˆ θ L e y L L lim ˆ θ( = θ < If, i addiio Γ( ˆ& L L θ Γ he Remark 8.4 Followig he lies of Secio 6 (NGE, a se of sufficie codiios ca be foud ha guaraee asympoic covergece of he predicio error e y. 9. he Normalized Leas-Squares Esimaor wih Bouded Gai Forgeig Daa forgeig feaure of he NLSEEF resuls i he esimaor abiliy o rack slowly- Γ may grow ubouded if varyig parameers. However, he esimaor gai marix he regressor vecor ( x is o PE, (read Remark 8.3. hus, i is desirable o ue λ e τ variaio of he forgeig facor β(, τ =, (i.e., choosig ( r forgeig is acivaed whe ( x is PE ad suspeded whe ( x r dr λ, so ha daa is o PE. owards 58
ha ed, oe ha he magiude of he gai marix Γ is a idicaor of he exciaio level of x(. hus, i is reasoable o correlae he forgeig facor variaio wih Γ (. A specific echique, (bouded gai forgeig, for achievig his purpose is o choose: λ( λ Γ = k (9. I (9., λ is he maximum forgeig rae, ad k is he pre-specified boud for gai marix magiude. Basically, if he orm of Γ is small, (idicaig srog PE, he λ. As Γ becomes larger forgeig facor i (9. discards he daa a maximum rae he forgeig speed is reduced ad whe he orm reaches he pre-specified upper boud k he forgeig is suspeded. Noe ha iiially he gai marix i (8.8 mus be chose such ha Γ k. Nex, covergece properies of he Normalized Leas-Squares Esimaor wih Bouded Gai Forgeig (NLSEBGF will be aalyzed. We show ha he form (9. guaraees x. ha he resulig gai marix Γ is upper bouded regardless of he PE of his is i coras o a cosa forgeig facor ad cosiues he mai beefi of he variable forgeig mehod. Subsiuig he bouded forgeig relaio (9. io he gai updae dyamics (8.9, resuls i: ( d Γ ( Γ =λ Γ + d k Γ = Q IN N k (9. his leads o Sice ( ( τ x x λ λ τ τ Γ ( = Qe + e Γ Γ + d k x λ ( τ τ τ τ (9.3 59
is posiive semi-defiie he Usig (9.5 i (9.3, oe ges: ( ( IN N ( IN N Γ Γ =Γ Γ Γ Γ (9.4 Γ Γ I N N (9.5 λ λ ( x τ λ ( τ ( x( τ ( ( τ ( x τ ( x( τ ( ( τ Γ Qe + e IN N+ dτ k x = + + λ λ Q IN Ne IN N e d k k x ( τ τ (9.6 he esimaio gai iiial codiio from (9. guaraees ha: Q I N N (9.7 k Cosequely, for all he followig lower boud akes place: or, equivalely: ( Γ I N N (9.8 k Γ k I N N (9.9 he upper boud (9.9 implies ha he bouded forgeig relaio (9. always resuls i: λ ( r (9. his proves uiform boudedess of he esimaio gai Γ ad he o-egaive aure of he fucio λ ( r. Furhermore, suppose ha he regressor vecor ( x is PE. he by Defiiio 6., here mus exis cosas α, > such ha for all + I ( x ( x d I α τ τ τ α (9. 6
I addiio, suppose ha x bouded, ha is x R : he for all : ad ( x is coiuous. he ( x L ( x C is uiformly (9. ( x( τ x( τ x( τ P x ( ( τ + + dτ ( x( τ ( x( τ dτ ( + λ max + 444444443 P C α I = λ I + λmax P C 4444443 λ (9.3 his PI codiio ca equivalely be wrie as: d I (9.4 τ λ, Based o (9.4, a ew lower boud for Γ ca be compued. I fac, usig (9.6 i follows ha for all : λ( τ λ( τ Γ ( N N+ τ N N k + k I e d I e dτ + + k k λ λ IN N e dτ e λ I (9.5 herefore, for all : k Γ( I k I λ < + e λ k (9.6 his, i ur, leads o uiform lower boudedess of he ime-varyig forgeig facor λ by a posiive cosa: ( λ Γ λe λ k λ( = λ = λmi > λ k + e λ k (9.7 6
Recall from (8.6 ha ( λ( r dr ( θ Δ θ = e Γ QΔ (9.8 ad ha he esimaio gai Γ ( is uiformly bouded as i (9.9. he he orm of he parameer esimaio error Δ θ ( saisfies: λ( r dr e k Q e k Q ( λ mi τ Δθ Δθ Δ θ (9.9 which immediaely implies he expoeial covergece of he esimaed parameers o heir rue ukow cosa values, ha is: lim Δ θ = (9. he properies of he NLSEBGF are summarized below: heorem 9. I addiio o he properies of he NLSEEF (8.8 saed i heorem 8., usig he bouded forgeig facor (9. resuls i he uiformly bouded esimaio gai marix: Γ L If he regressor vecor ( x is PE ad bouded he he esimaed parameers coverge expoeially o heir rue ukow cosa values: limθ = θ ( 6