Robust estimate of excitations in mechanical systems using M-estimators Theoretical background and numerical applications

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1 Robust estmate of exctatos mechacal systems usg M-estmatos Theoetcal backgoud ad umecal applcatos Paolo Peacch Dpatmeto d Meccaca, Poltecco d Mlao, Va La Masa 34, I-56 Mlao, Italy paolo.peacch@polm.t Abstact Model based methods ae ofte used alog wth least squaes to estmate (o to detfy equvalet but moe egeeg tems) dyamc foces, paametes ad malfuctos mechacal systems, statg fom expemetal vbatos. The effectveess of these methods, boadly pove ad documeted by seveal cases of study, ca be educed f the model of the system s ot accuate o f the expemetal data ae coupted by ose, especally f the mea value of the ose s ot ull o f bases ae peset. A possble soluto s the use of obust estmato techques stead of tadtoal least squaes the ambt of model based detfcato. The autho poposes the applcato of the M-estmatos ad dscusses the poblems elated to the applcato to exctato detfcato mechacal systems. I ths pape the ecessay theoy s peseted detal, toducg seveal cocepts of Statstcs, ode to popely toduce the cocept of obust estmato ad the equed algothms (based o teatve e-weghted least squaes) ae descbed. The the dffeet types of M-estmatos poposed lteatue ae toduced. The pefomaces wth egad to mechacal applcatos ae evaluated by meas of a theoetcal aalyss ad a couple of smple umecal examples: a sgle put sgle output ad a multple puts multple outputs systems. Moeove the poblem of the scale paamete, whch s ot dscussed lteatue fo complex umbes, as the vbatos ae, s aalyzed ad a soluto s poposed usg a cocept elated to the data depth. Keywods: Robust estmato; detfcato; paamete estmato; vese poblems; M-estmatos; least squaes; teated e-weghted least squaes; umecal esults.. Itoducto The least squaes estmate s wdely used mechacal systems to solve detfcato poblems. These ca be fo stace the estmate of the exctatos statg fom the dyamcal espose ad fom the kowledge of the system paametes. Covesely also the system paametes ca be evaluated by detfyg the expemetal fequecy espose fucto (FRF). A compehesve ovevew s gve fo stace [], by lmtg to the mechacal feld oly, whee also some emaks ae gve about the sestvty of least squaes to bas. The sestvty of least squaes to data coupto s the ma dawback of ths method ad seveal mpovemets have bee poposed lteatue. Oe of the smplest s the toducto of weghted least squaes ad

2 some successful applcatos model based detfcato of faults oto-dyamcs ae peseted fo stace [][3]. I these applcatos the faults ae epeseted by meas of equvalet exctatos. Aothe ecet poposal [4] to mpove the method of least squaes method s focused o the applcato of sutable egulazato fltes, amely the tucated sgula value decomposto flte ad the Tkhoov s flte, alog wth total least squaes. Aothe appoach poposed [5] uses o-lea least squaes to estmate paametes of a mult-fequecy sgal fom dscete-tme obsevatos coupted by addtve ose. Some of the poposed techques lteatue have the geeal task to obta the maxmum effcecy the estmate. I [6] a maxmum lkelhood estmato s used to geealze total least squaes ad t s show that ths estmate has the maxmum effcecy. Ayhow, ude a statstcal pot of vew, the maxmum effcecy s atthetc to the obustess of the estmate, as t wll be show late o the pape. I ay case pactcally eveyoe agees about the fact that a estmato should be obust, but also the cocept of obustess s somewhat vague. I some cases the obustess s teded as somethg that makes the estmato less sestve to ose, both o system put ad output. Some algothms that have the task of educg ths sestvty, both toducg the ose model the paametc detfcato (the geealzed total least squaes [7]) ad matag the ealy maxmum lkelhood popety (the bootstapped total least squaes [8]), have bee poposed ad compaed [9] whe they ae appled to modal aalyss. I othe cases the toducto of a goous cocept of obustess s avoded ad the tem mmuty, smla to that of lvg bodes, s used, alog wth the developg of a modfed least squaes algothm []. Ayhow the poblem of the obust estmato s ot ew Statstcs ad useful efeeces ca be foud [][][3][4]. The am of the autho s to gve a pecse defto of the obustess ad to apply obust estmate to mechacal poblems, tyg to peseve a hgh level of effcecy. A sutable tade-off betwee obustess ad effcecy s epeseted by the class of M-estmatos. The applcato to exctato detfcato of the mechacal system s dscussed the pape. M-estmate has eve bee appled to mechacal systems, to the autho s kowledge. I Statstcs the data sets ae geeally composed of eal umbes. I Mechacs vbatos ae coveetly epeseted by complex umbes ad the M-estmate applcato to complex data sets has eve bee peseted befoe. Theefoe ts mplemetato s fully dscussed, by fst evewg the elated theoy ad the toducg the dffeet types of M-estmatos poposed statstcal lteatue. Sce t s ot possble to defe a po whch oe of the M-estmatos s moe sutable to mechacal applcatos, the pefomaces ae evaluated by meas of a theoetcal aalyss ad two smple umecal examples of a sgle put sgle output (SISO) ad a multple puts multple outputs (MIMO) systems, whch the kowledge of the model s pefect ad ose affects oly the output(s). The use of M-estmate mples the evaluato of a scale paamete fo the data sample. Ths poblem s ot dscussed statstcal lteatue because omally data ae eal umbes. Sce

3 vbatos ae coveetly epeseted by complex umbes, data samples ae complex ad the evaluato of the scale paamete s ot tval. Also ths poblem s aalyzed ad a soluto s poposed by usg a cocept elated to the data depth.. Estmato, least squaes, obustess ad fluece fucto Not oly the mechacal feld, thee ae may cases whch two o moe vaables ae elated by themselves ad the elatoshp s made explct by meas of a model. The estmate of model paametes s ofte made usg the least squaes method (descbed appedx A). Ayhow, the estmatos the least squaes sese ely o some fudametal hypotheses o the dstbuto of eos betwee the vaable of teest ad the data. I fact the ose that coupts the data s assumed to have ull mea value ad ths mples the estmate of a ot dstoted paamete. If the vaace of the ose s kow, a mmal vaace estmate of the paametes, ca be obtaed by usg sutable weghts fo the data. Seveal studes have show that least squaes estmatos ae vuleable to the volato of these hypotheses. Fo example the dstbuto of the eos could be asymmetc o poe to exteme outles. Sometmes eve a bad obsevato oly ca completely petub the least squaes estmate. Theefoe may obust techques have bee poposed, but the obustess s a vague popety of statstcal pocedues ad s well outled by Bckel []: A obust pocedue, lke a obust dvdual, pefoms well ot oly ude deal codtos, the model assumptos that have bee postulated, but also ude depatues fom the deal. The oto s vague sofa as the type of depatue ad the meag of good pefomace eed to be specfed. I the followg, the obustess cocept that the autho teds s focused... Evaluato of the fluece fucto, of the obustess ad of the effcecy Oe of the causes of the accuacy of the pobablstc model assumed fo the data s the pesece of eos due to outles,.e. aomalous values, defed as obsevatos fa away fom the majoty of the data. The aalytcal tool that allows the evaluato of the obustess of a estmato pesece of outles ad goss eos s the fluece fucto, toduced by Hampel [5][6]. I ode to allow a moe tutve tepetato of ts defto (as suggested [7] ad [8]), t s toduced as the lmt of the sestvty cuve, poposed by Tukey [9] to evaluate the stablty of a estmato. Let ( X, X,, X ) be a adom sample extacted fom X F( x; θ ) ad the paamete θ s gog to be estmated by meas of the estmato T = T( X, X,, X). Estmato T ca be defed usg the empcal dstbuto fucto: F( u) = I X u ( ), () whch attbutes to evey obsevato equal pobablty /. The opeato I() dcates the umbe of that satsfy the codto sde the paetheses. Obvously, the kowledge of the adom 3

4 sample o of the empcal dstbuto fucto s equvalet, because the adom sample s mmedately feed fom F ( x ) ad, vce vesa, the sample s mmedately feed fom the empcal dstbuto fucto because F ( x ) toduces some jumps coespodece of the sample values. Ths justfes deotg the estmato as a fuctoal of the empcal dstbuto fucto: T = TF [ ]. Now, f a abtay obsevato x s added to the adom sample ( X, X,, X ), a ew sample ( X, X,, X, x) s obtaed, the empcal dstbuto fucto of whch s F ( ; ) + ux. It s easy to show that: F ( u; x) = I X u + I x u = F ( u) + I x u ( ) ( ) ( ). () + = If δ x ( u ) dcates the dstbuto fucto of a adom aomalous vaable x ad ε =, + the: F + ( ux ; ) = ( ε) F( u) + εδ ( u) = ( ε) F+ εδ. (3) x x Smlaly, whe the estmato T = TF [ ], defed o ( X, X,, X ), shfts o the exteded sample ( X, X,, X, x), the fuctoal becomes: [ ] [ ] T ( x) = T F ( ux ; ) = T( ε) F+ εδ. (4) + + x The codto that makes T a acceptable estmate fo θ s that the addto of a obsevato x to the adom sample does ot stogly modfy ts value. The most staghtfowad way to measue the effect of ths addto s to cosde the dffeece betwee T ( ) + x ad T ad to compae t to the weght of the added obsevato (measued by ε ), that obvously s vesely popotoal to the sample umbe. Ths defes the sestvty cuve SC of the estmato T [9]: [ ε εδ ] T ( ) [ ] ( ) T F x TF + x T + SC( x, T ) = =. (5) ε ε The sestvty cuve ca be studed as a fucto of x fo a estmato T. If the obsevato x s fa away fom the majoty of the data, the cuve of sestvty shows what happes to the estmato whe ths outle s peset the sample. Theefoe t s ecessay that SC( x, T ) s lmted, so that the effect of a outle o the estmato s always estcted wth defed lmts. If, the Glveko-Catell s theoem states that F ( x ) coveges ufomly ad dstbuto to F( x ). Moeove fo Fshe s cosstet estmatos TF [ ] TF [ ] =θ ad TF [ ] ca be eplaced asymptotcally by TF. [ ] The fluece fucto IF of the estmato T wth espect to F( x; θ ) s: [ ] T( ε) F+ εδx TF [ ] IF( x, T, F) = lm. (6) ε ε The fluece fucto shows the asymptotc vaato of TF [ ] due to a ftesmal cotamato of the dstbuto F( x ), elated to the cotamato etty. The fluece fucto depeds o x, as the sestvty cuve, but also o F( x ), that s o the paametc model supposed 4

5 fo the data. Theefoe t epesets a paametc tool that ca be used to vefy the behavou of a estmato the cases whch the actual dstbuto s smla to the hypotheszed dstbuto F( x ). The fluece fucto wll be used paagaph 3.. to dscuss the obustess of the least squaes. It allows also a obustess measue to be toduced ad the obustess cocept to be fally poted out. The goss eo sestvty γ [][6] s defed as: γ ( T, F) = sup IF( x, T, F). (7) x If γ ( T, F ) < +, the estmato s obust wth espect to outles,.e. to aomalous values. The obustess eques stead that the fluece fucto s supeoly lmted, but ths equemet s cotast to the estmato effcecy, whch s ofte the taget of the detfcato methods mechacal systems [6]. I fact a theoem peseted [6] states: let X F( x, θ ) ad T = T( X, X,, X) be a estmato fo θ ad let V ( θ ) be the elated scoe fucto; f T s Fshe s cosstet, ude egulaty codtos vald fo Camé-Rao s equalty [], the T s effcet fo θ f IF( x, T, F) V ( θ ). The scoe fucto V ( θ ) s defed as: f ( x; θ ) V ( x; θ) = log f( x; θ) = θ f( x; θ ), (8) whee f s the pobablty desty fucto. The scoe fucto s omally ubouded, so that the fluece fucto has to be both lmted ad ulmted (popotoal to the scoe fucto) to acheve espectvely estmato obustess ad effcecy. To solve ths paadox, the M-estmato class has bee toduced [] [] [3]... Defto of M-estmate Istead of estmate paamete ˆ θ by meas of the mmzato of the quadatc eo: ( ) x = = = θ, (9) as t s doe the stadad least squaes, the followg quatty s mmzed: ( x ) ρ θ = ρ ( ), () = = whee the type of fucto ρ wll be dscussed secto 3. To obta the mmum, eq. () s deved wth espect to ad put equal to zeo. Let θ = [ θ,, θ m] be the vecto of the paametes to be estmated. The M-estmato of θ based o fucto ρ ( ) s the vecto θ soluto of the m equatos: whee the devate: = ψ ( ) = fo j =,..., m, () θ j 5

6 d ρ( ) ψ ( ) =, () d s popotoal to the fluece fucto of ρ. Ths s show appedx B. If the estmato s obust, the fluece of a sgle obsevato s suffcet to cause a sgfcat eo. A weght fucto s the defed as: so that eq. () s ewtte as: = ( ) w ( ) = ψ, (3) w ( ) = fo j=,..., m. (4) θ j The equato system (4) s coespodg to that obtaed whe the teated e-weghted least squaes (IRLS) poblem [4][5] s solved: m t ( ) ( ) whee t s the umbe of teato dex ad the weghts w ( t ) ( ) w, (5) = have to be calculated pe each teato. The detaled descpto of the IRLS algothm s peseted paagaph Codtos o the ρ fuctos I ode to be obust ad to have good computatoal chaactestcs, a M-estmato should comply wth some codtos that ae eflected o ts ρ fucto: ) The fluece fucto eq. () has to be bouded, as show paagaph. [8]. ) ρ fucto must have these popetes [4]: ρ () ; ρ () = ; ρ() = ρ ( ) ; ρ( ) ρ ( ), > ; ρ s less ceasg tha a quadatc fucto. I pactce t s equed that the ρ fucto has the same good popetes of the least squaes wth the addtoally obvous codto that ρ s less ceasg tha a quadatc fucto ode to lmt the fluece of the outles. 3) The obust estmato should be uque,.e. the objectve fucto eq. () should have a uque mmum. Ths eques that the dvdual ρ fucto s covex the vaable θ, whch s equvalet to mpose that ρ θ s o-egatve defte [6]. 4) A pactcal equemet s that wheeve ρ θ s sgula, the objectve fucto should have a gadet, that s ρ θ. Ths avods havg to seach fo a mmum though the complete paamete space. 6

7 Not all the M-estmatos poposed lteatue actually comply wth all of these codtos. I the followg secto 3, the types poposed ae aalyzed detal ad some pefomace chaactestcs ca be foecasted, whle the applcatos sectos 4 ad 5 make explct the advatages ad the dawbacks mechacal applcatos. 3. Scale estmate ad types of M-estmatos Befoe cosdeg the possble types of M-estmatos, t s ecessay to toduce a dscusso about the scale of the sample. I paagaph. t has bee mplctly assumed a utay scale, whle geeal the scale of the sample s ot utay ad t should also be estmated. The poblem of eq. () should be stated as: x θ j ψ = ˆ = σ θ. (6) j Obvously the same scale estmate ˆ σ has to be obust. Sce the cases commoly aalyzed Statstcs, the obseved data ae eal values, x, the estmates of scale ae eal values too ad omally the MAD (meda absolute devato) s used [][4][6]. Ayhow, paamete estmate mechacal systems uses vbato data that ae complex umbes, x. No studes exst egadg the applcato of M-estmate to complex quattes to the autho s kowledge. Nevetheless a ch lteatue exsts about the o-paametc estmate of obust locato paametes multdmesoal dstbutos [7][8][9][3][3] ad may of them ae based o the cocept of data depth [3]. These studes ca be exteded to the peset case, whch the esdues ae complex umbes,, ad a scale estmate fo bvaate data should be employed. A poposal, whch wll be adopted followg by the autho, s based o the exteso of the MAD to the complex feld usg the Tukey s meda stead of the covetoal meda. Let X = { x, x,, x}, x, be the bvaate data vecto composed by the complex vbato measues ad T * () the Tukey s meda opeato. The exteso of the MAD s hee defed as TMAD (Tukey s meda absolute devato): * ( T ) TMAD( X) = Med X ( X ). (7) Detals o Tukey s meda calculato ae peseted [33][34]. 3.. Least Powes Ths s a wde class of ρ fuctos that does ot deped o the pe-emptve kowledge of a measue of scale. The geeal defto s: ν ( ) ( ) ν ρ x θ = x θ ρ =. (8) Depedg o the value of the expoet ν, seveal estmatos ae defed. Note that the codtos expessed pot of paagaph.3 ae satsfed also f. 7

8 3.. L - Least Absolute o Absolute Value I ths case ν = ad: ρ() = ; ψ () = sg(); w () =. (9) Ths estmato has a bouded fluece fucto ψ (), thus educes the fluece of lage eos, but has the dsadvatage of possble umecal stablty, because the ρ fucto s ot stctly covex, sce secod devatve s ubouded ad a detemate soluto may esult. 3.. L - Least Squaes The classcal least squaes ae obtaed fo ν = ad esults: ρ() = ; ψ () = ; w () =. () Although ths estmato s covex, the fluece fucto ψ ( x) s a staght le, sofa ψ ( x) s ot bouded ad ths estmato s ot obust, as well-kow L L Ths estmato does ot deve dectly fom eq. (8), but s teded to mata the advatages of both L to educe the fluece of lage eos ad L to be covex. It behaves lke L fo small esdues ad lke L othewse. ρ () = + ; () = ; w () = ψ. + + () 3..4 L p - Least powe Fom the cosdeato of L ad L, t appeas that as smalle the expoet ν of eq. (8) s, as smalle the cdece of geat esdues s o the estmate of θ ; that s ν has to be eough small to gve obust estmatos o, othe wods, to gve a estmato pooly petubed by the outles. The fuctos ae: ν ρ() = ; ψ() = sg() ; w () = ν ν ν. () The vestgatos about the selecto of a optmal ν have dcated that. s a sutable value [3][8][35]. 8

9 3.. Hube s fucto The am of ths estmato s to fd the smplest magable fucto that s cosstet wth the codtos of obustess. Hube toduced t [3][][] to gve the m-max soluto to eq. (6) fo omal dstbutos of data affected by ose, ude the hypothess of kow scale paamete ˆ σ ad the exteded ts use to geeal dstbutos. He stated fom the maxmum lkelhood estmato (MLE) ad educed ts sestvty to the outles. Fo a data sample wth dstbuto desty f, the MLE maxmzes: log f( x θ ). (3) It ca be defed by cosdeg that, f the sample dstbuto s ukow, the most easoable assumpto s to suppose the symmety: f( x θ) f( θ x). (4) The Taylo s expaso poxmty of the cete of the symmetc dstbuto s: ( ) I the poxmty of the cete ca be wtte that: plus a costat. Theefoe the paabola: f( x) f( θ) x θ. (5) ( x θ ) log f( x θ ), (6) ( x θ ) ρ( x θ ) =, (7) s the optmal choce the poxmtes of the cete. Ths otwthstadg eq. (7) cocdes to the least squaes, whch ae ot obust. A possblty s to lmt the fluece fucto f the esdue exceeds a ceta value c, called tug paamete [5]. The esultg fuctos ae: f c f c f c ρ() = ; ψ () = ; w () = c. (8) c csg( ) f > c f > c c f > c Ths ρ fucto s so satsfactoy that has bee ecommeded fo ay stuato. Howeve depedg o the value of the tug paamete, dffeet estmatos ca be obtaed. Sce the estmato has bee developed ogally fo omal dstbutos, the optmal tug paamete c s calculated ode to have the 95% of asymptotc effcecy wth espect to a omal dstbuto ad esults c =.345 ˆ σ, whee ˆ σ s the scale paamete. Ths value ad the calculato of the tug paametes of the othe fuctos ca be foud by statg fom the scoe fucto of eq. (8) as show [3][6]. 9

10 3.3. Modfed Hube s fucto Eve f Hube s estmato has outstadg pefomaces, t could cause calculato poblems elated to lack of stablty of the gadet values (as oted [8]) due to the dscotuty of the secod devatve of the ρ fucto. I fact: d ρ() f = d f The modfcato poposed [8] s: c. (9) > c π c c cos f π π cs f s f c c c c c c ρ() = ; () ψ = ; w () =. (3) c π π c c f π π + > csg( ) f > f > c c c esults The 95% of asymptotc effcecy wth espect to a omal dstbuto s obtaed ths case c =.7 ˆ σ, whee ˆ σ s the scale paamete. Ayhow the modfed Hube s fucto ths fom s ot sutable case of complex data ad esduals, sce the tgoometc fuctos eq. (3) have complex agumets ad ths detemes the weghts w () to be complex ad ρ () s ot complat wth the codtos of pot of paagaph.3. Fo the mechacal systems peseted the pape we use a coected modfed Hube s fucto that avods the poblem of complex weghts: π π c cos f csg( ) s f c c c c ρ() = ; ψ() = ; π π π c+ c f > csg( ) f > c c c π s f c c w () =. c π f > c (3) 3.4. Fa fucto It has bee aleady poted out that also the scale paamete ˆ σ should be estmated ad that may cases, lke Hube s estmato defto, a aveage scale facto s used. The Fa fucto has bee defed [8] wth the am to have low sestvty to the scale facto, so that the estmato has low sestvty to the tug paamete. It s defed as: ρ() = c log + ; ψ () = ; w () =. c c + + c c (3)

11 Ths ρ fucto s obust as well as yelds to a uque soluto (sce t has eveywhee defed cotuous devatves up to the thd ode). The 95% asymptotc effcecy wth espect to a omal dstbuto s obtaed wth a tug paamete c =.3998 ˆ σ Cauchy s fucto The ame of ths fucto deves fom the fact that t s optmal fo data havg a Cauchy s dstbuto. Ayhow t does ot guaatee a uque mmum. The deceasg fst ode devatve ca yeld to eoeous soluto that caot be obseved ad the fluece of lage eos oly deceases lealy wth the sze. The defto s: c ρ() = log + ; ψ () = ; w () =. c + + c c The 95% of asymptotc effcecy o a omal dstbuto s obtaed fo c =.3849 ˆ σ, whee ˆ σ s the scale paamete. By studyg the elatoshps betwee tug costat ad effcecy, ths fucto appeas to be the best amog the fuctos poposed lteatue that does ot comply wth the satsfacto of the popety 3 of paagaph.3 (the best amog the wose oes). (33) 3.6. Welsch s fucto Ths fucto has bee toduced ode to futhe educe the effect of lage eos, but has ot a uque mmum. It s defed as: c ρ( ) = exp ; ψ ( ) = exp ; w ( ) = exp. (34) c c c The tug paamete fo 95% asymptotc effcecy wth espect to a omal dstbuto s c =.9846 ˆ σ Tukey s fucto Ths fucto has bee poposed by Tukey ad s also called bweght fucto [36]. Its am s to suppess the outles as show by the defto of w ():

12 3 c f c 6 c f c ρ() = ; ψ() = c ; c f > c f > c 6 f c w () = c. f > c (35) It s msleadg due to the lack of a uque mmum ad the 95% asymptotc effcecy wth espect to a omal dstbuto s obtaed wth a tug paamete c = ˆ σ Gema-McClue s fucto The am of ths fucto s educe the effect of lage eos wthout toducg a tug paamete, but t has ot a uque mmum. It s defed as: ρ() = ; ψ () = ; w () =. (36) ( ) ( ) Due to the ot complace wth codto 3) of paagaph.3, the pefomaces of ths fucto wll ot be good a po. It s cluded oly fo completeess, sce t has bee used some applcatos elated to mage ecogto [37][38]. 4. Numecal applcato to a SISO system The smplest mechacal dyamcal system s epeseted by a sgle degee of feedom (d.o.f.) system show fgue. The paametes of the system ae the mass m, the dampg c ad the stffess k. The system d.o.f. s descbed by meas of the system dsplacemet xt (). A exteal hamoc exctato F causes the foced vbato of the system. The equato of moto of the cosdeed system s smply: By cosdeg that the focg system s hamoc: also the soluto of eq. (38) has to be hamoc: mx + cx + kx= F() t. (37) Ft () = Fe = Fe ϕ e, (38) Ω t Ω t xt () Xe Xe φ e Ω t Ω t = =. (39) Replacg eq. (39) eq. (37), the well-kow steady-state soluto ca be obtaed as: ad fally: ( ) Ω m+ Ω c+ k X = F e ϕ (4)

13 F e Ω m+ Ω c+ k ϕ X = = H Ω F ( ) e ϕ. (4) Fgue. Sgle degee of feedom system. Note that the solutos gve by eq. (4) as fucto of the fequecy Ω of the focg system ae complex. Sce eq. (4) gves the espose of the SISO system to the exteal foce, a detemstc cotext, whch the mass, the dampg ad the stffess of the system ae exactly kow, the kowledge of the system dsplacemet ampltude ad phase at a sgle fequecy oly allows to deteme the ampltude ad the phase of the focg system. Now, let us cosde a stochastc evomet, whch the dsplacemets of the mass ae measued fo a gve set of fequeces, oce the system has eached the steady-state. Measues could be coupted by ose, bases, systematc eos ad so o. Fo each measue x of the dsplacemet ( ampltude ad phase) at the fequecy Ω, t s possble to wte the equato: x = h Ω F =. (4) ϕ ( ) e,..., The system of all the equatos (4) has the oly ukow epeseted by the foce ( ampltude ad phase) ad theefoe t s ove-detemed. Nomally ths system s solved by meas of least squaes. If the measues ae petubed by whte-ose, usually qute accuate esults ae obtaed ay case. Covesely f a systematc eo affect the measues, L estmate does ot poduce ay moe accuate esults. Ths fact ca be show smply by meas of a smulated case. Let us cosde a system whch m = kg, k = N/m ad c = 6 Ns/m (the dampg s equal to 3% of the ctcal dampg). If the focg system has the ampltude F equal to N ad the phase ϕ of 45, the omal,.e. o-coupted, system espose calculated the fequecy age fom to 3 ad/s wth a step of. ad/s s show as Bode plot fgue. Now a systematc eo s appled o the system espose. Systematc eos ca be chose fte ways. To test the obustess, the cteo selected s to use a fxed step the fequecy. Ths epoduces the pesece of electomagetc dstubace o the expemetal sgals, at a ceta fequecy ad ts multples. A costat magfcato of ampltude ad phase shft s used to cease the systematc chaactestc of the eo: 3

14 Each value, the odeed vecto of the measues, statg fom ad/s wth a step of ad/s, has the ampltude ceased of 5% ad the phase otated of +45. Ths coespods to the x k whch k = + j, j. Each value statg fom.4 ad/s wth a step of ad/s has the ampltude educed of 5% ad the phase otated of. Ths coespods to the x k whch k = 5 + j, j. The esultg coupted system espose s show fgue 3. If L estmate s used, the ampltude ad the phase of the exteal foce esults: F = 5.64[N], ϕ = (43) The eo s cosdeable o both ampltude ad phase.. Ampltude [m] Ω [ad/s] 5 3 (a) 8 Phase [ ] Ω [ad/s] 5 3 (b) Fgue. Nomal espose of the system: (a) ampltude, (b) phase. 4

15 Ampltude [m].5 st sample th sample Ω [ad/s] 5 th sample 5th sample (a) 8 Phase [ ] Ω [ad/s] (b) Fgue 3. Coupted espose of the system: (a) ampltude, (b) phase. 4.. Calculato of the M-estmate by the mplemetato of the ILRS algothm The exctato, the exteal foce, s ow estmated usg M-estmate. The objectve fucto to be mmzed s: x h( Ω ) F e ϕ ρ( ) = ρ = = ˆ σ whee ˆ σ s the scale estmate calculated usg eq. (7). 5, (44) The soluto of the mmzato of eq. (44) follows the theoetcal agumets peseted paagaph.. Let ψ be the fst devatve of ρ wth espect to the ukow foce: ρ() ψ () =. (45) F e ϕ The mmzato of eq. (44) s obtaed by dffeetatg the objectve fucto wth espect to the ukow foce ad settg the patal devatves to : x h( Ω ) F e If the weght fucto w () s defed as: the the equatos (46) ca be wtte as: ϕ ψ h( Ω ) = = ˆ σ. (46) ψ () w () =, (47)

16 whch s equvalet to mmze the least squaes poblem: ϕ x h( Ω) F e w = ˆ = σ, (48) w. (49) = Ayhow, the weghts w () deped upo the esduals, the esduals deped upo the estmated exctato F ad the estmated exctato depeds upo the weghts w (). To solve ths loop, a teatve soluto, called IRLS, s used. The algothm s the followg: whee:. The tal estmate of the foce ampltude ad phase s selected usg the esults of least squaes calculato, thus s that of eq. (43). (). At each teato t, the esduals t () ad the assocated weghts w t ae calculated fom the pevous teato. 3. The ew weghted least squaes estmate s: ( ) ( t ) t F e ϕ + = HW H HW X, (5) T () T () t () t () t W = dag w. (5) Steps ad 3 ae epeated utl the estmated coeffcets covege. The ma advatage of the ILRS algothm s ts smplcty, pactcally stadad umecal methods to calculate weghted least squaes ae teatvely used. The fst dawback s that, oce a ρ fucto s chose, the weght fucto w desceds automatcally. Theefoe ρ fucto should be good (see paagaph.3). The secod s the teato stop codto. I ths case, beg the estmated coeffcet complex, the algothm s epeated utl the maxmum omalzed dffeece betwee the eal ad the magay pats of the coeffcet value the peset teato ad those of the pevous teato s less tha e 4,.e: ( ) ( ) ( t ) ( t) ( t ) Re( F ) Re( F ) Re( F ) max < e 4. (5) ( t ) ( t) ( t ) Im( F ) Im( F ) Im( F ) If ths covegece s ot eached, the algothm stops afte teatos. Smla stoppg ule s also used [39]. The esults wth dffeet types of M-estmatos ae epoted table. As pedcted by the theoetcal aalyss (secto 3), some of the M-estmatos (Cauchy, Gema-McClue, Welsch ad Tukey) gve bad esults. I all of these fou cases, the algothm has stopped havg eached the maxmum umbe of teato wthout satsfyg the covegece codto. Actually the teds of the estmated ampltude ad phase of the foce peseted a oscllatg behavou as fucto of the 6

17 teato umbe, wth aveage values close to the coect oes. Ths fact wll be dscussed detal the ext secto. The algothm, usg the othe M-estmatos, does ot peseted umecal oscllatos, stops few teatos ad gves emakable good esults. I two cases the esults ae deftely exact eve wth the coupted data. Table. Results of the exctato estmate. Estmato F [N] ϕ L (absolute value) 45 L-L (absolute value ad least squaes) 45. Lp, ν=. 45. Fa Hube 45 Cauchy Gema-McClue Welsch Tukey Weght k=5+j sample k=+j sample Ω [as/s] th sample 5 th sample 5 st sample 4 3 Iteato umbe Fgue 4. Weghts attbuted pe teato fo the Hube s estmato. The explaato of these supsgly esults ca be gve by cosdeg the weghts w that the algothm attbutes to each measue x. Fo example, fgue 4 shows the weghts as a fucto of the measue ode umbe ad of the teato fo the Hube s estmato: afte few teatos the weghts gve to the coupted measues become ea ad the algothm pactcally dscad them. I ths case, the algothm stops 5 steps. 7

18 5. Numecal applcato to a MIMO system I ths example, ot oly the pefomaces of the dffeet types of M-estmatos ae ow evaluated fo a MIMO system, but also the covegece to a stable soluto fo the IRLS s aalyzed depedg o the type of the M-estmato. The cosdeed MIMO system s a smple lea mechacal system wth d.o.f.s, show fgue 5. The physcal paametes of the system ae kow,.e. the model of the system s kow ad elable, ad ae gouped the mass M, the dampg C ad the stffess K matces. The system d.o.f.s ae descbed by meas of the vecto of the mass dsplacemets x. The ukow exteal foce system F, actg o the masses, causes the foced vbato of the system. The equatos of moto of the cosdeed lea system ae smply: Mx Cx Kx + + = m + x c c c + 3 x = k k k x F() t F () t = = F( t). m x c + c c x k + k k x By cosdeg a hamoc focg system: also the steady-soluto of (53) has to be hamoc: (53) ϕ F Ωt F e Ωt F () t = e = e ϕ, (54) F F e φ Ωt X Ωt X e Ωt x() t = X e = e = e φ. (55) X X e Replacg eq. (55) eq. (53), the steady-state soluto ca be obtaed as: ad fally: ( ) Ω M+ Ω C+ K X= F, (56) ( ) X= Ω M+ Ω C+ K F = H( Ω) F. (57) k c m F ( t) F ( t) k k m 3 c c x () t x () t 3 Fgue 5. System wth degees of feedom. Note that the soluto gve by eq. (57) as fucto of the fequecy Ω of the focg system s complex,.e. X,. Smlaly to pevous secto 4, a stochastc evomet s cosdeed. The dsplacemets of the masses ae measued fo a gve set of exctatos/fequeces, oce the system has eached the steady-state ad we wat to estmate (detfy) the focg system. The measues, epeated fo dffeet exctatos/fequeces, could be coupted by ose, bases, systematc eos ad so o. 8

19 Fo each measue { } T x = x x of the dsplacemets ( ampltude ad phase) at the fequecy Ω, t s possble to wte the equato: x = h ( Ω ) F, (58) If a systematc eo affects the measues, L estmate does ot poduce ay moe accuate esults, as aalytcally pove the pevous sectos ad as exemplfed by meas of ths smulated case. Let cosde a system lke that of fgue 5 ad eq. (53) whch, fo smplcty, m = m = kg, k = k = k3 = N/m ad c = c = c3 = 6Ns/m. If the focg system has ampltudes equal to F = N ad F = 5N ad phases ϕ = 45 ad ϕ = 6, the omal,.e. o-coupted, system espose calculated the fequecy age fom to 3 ad/s wth a step of. ad/s s show as Bode plot fgue 6. Now a systematc eo s appled to the system espose: evey value of the system espose the age statg fom ad/s wth a step of.5 ad/s has the ampltude ceased of 5% ad the phase otated of +45,.e. fo x,k : k = + j, j...8. x x.8 Ampltude [m].6.4 Ampltude [m] Ω [ad/s] (a) Ω [ad/s] (b) Phase [ ] Phase [ ] Ω [ad/s] (c) Ω [ad/s] (d) Fgue 6. Nomal espose of the system: (a) ampltude of x, (b) ampltude of x, (c) phase of x, (d) phase of x. 9

20 x x.4 Ampltude [m].3. Ampltude [m] Ω [ad/s] (a) Ω [ad/s] (b) Phase [ ] Phase [ ] Ω [ad/s] (c) Ω [ad/s] (d) Fgue 7. Coupted espose of the system: (a) ampltude of x, (b) ampltude of x, (c) phase of x, (d) phase of x. The esultg coupted system espose s show Fgue 7. If L estmate s used to detfy the ampltude ad the phase of the exteal foces usg the coupted measues, the objectve fucto s: ( ) = ( x h ( Ω) F ) (59) = =, ad t esults: F F = 67.89N, ϕ = (6) = 7.677N, ϕ = The eos ae cosdeable o both the ampltude ad the phase. If M-estmate s used, s the esdue of the -th obsevato coespodg to the fequecy Ω ad the objectve ρ fucto to be mmzed s: ρ ( ) = ρ x h F. (6) = =, ( Ω ) ˆ σ The IRLS algothm s pactcally the same, wth the ecessay adjusts to take to cosdeato the d.o.f.s of the system. Theefoe eq. (5) becomes: whee: ( ) T ˆ t + T ( t ) F = AA AW Y. (6) () A= W t Hv, (63)

21 () t () t W = dag w, (64) ( ) H = dag H Ω, (65) T v =, (66) { x x } T Y =. (67) Matx W s a dagoal matx, the elemets of whch ae the weghts, calculated pe each teato, of the measues of the d.o.f.s, H s a bad matx wth the elemets alog the ma ad secoday dagoals epeseted by the tasfe fucto H coespodg to the fequecy at whch the measues ae acqued ad vecto v s a localzato vecto that dcate the d.o.f.s o whch the exctatos ae actg. The stop codto s smla to that of pevous paagaph 4.: ( ) ( ) max Re( F ) Re( F ) Re( F ) max e 4. (68) ( t ) ( t) ( t ), ( t ) ( t) ( t ) < max Im( F ) Im( F ) Im( F ), The esults obtaed by the dffeet types of M-estmatos, descbed secto 3, ae epoted table, whle the calculated values dug the teatos ae show fgue 8 to fgue 7.

22 Table. Idetfcato esults M-estmato Type F [N] ϕ [] F [N] ϕ [] Iteatos L (least absolute) L L L p, ν =. (least powes) Fa Hube Modfed Hube Cauchy Gema-McClue Welsch Tukey The esults of table cofm what stated about the M-estmato fuctos that do ot have uque soluto. Cauchy s, Gema-McClue s, Welsch s ad Tukey s estmatos ca be badly deemed, ad all the fou cases the algothm has stopped havg eached the maxmum umbe of teato wthout satsfyg the covegece codto. As atcpated Cauchy s estmato behaves bette tha Gema-McClue s, Welsch s ad Tukey s oe. Actually the teds of the estmated ampltude ad phase of the foce peseted a oscllatg behavou as fucto of the teato umbe, wth aveage values close to the coect oes (dash-dot les fgue 4 to fgue 7) tha those of L. Theefoe they could be dealt less seveely wth a dffeet stop codto of the IRLS algothm, based fo stace o the covegece of the aveage value. The othe M-estmatos do ot peset umecal oscllatos (fgue 8 to fgue 3), stop few teatos ad gve good esults, beg vey obust wth espect to data coupto. As expected Hube s ad modfed Hube s have excellet esults whle the good pefomace of L has to be caefully cosdeed due to the dawback udeled paagaph 3... Smlaly to the pevous SISO example, the explaato of these supsgly esults, whch ae ay case based o the pefect kowledge of the system model, ca be gve by cosdeg the weghts w that the algothm attbutes to each measue x each teato. Afte few teatos, the weghts gve to the coupted measues become ad the algothm pactcally ad

23 automatcally dscads them. Ths s evdet f the weghts ae plotted as a fucto of the measue ode umbe ad of the teato (see fgue 8 fo the Hube s estmato). L (Least Absolute) Module [N] (a) Phase [ ] Iteato umbe (b) Fgue 8. Estmated values vs. teato fo the L estmate: (a) foce ampltudes, (b) foce phases. L -L Module [N] (a) Phase [ ] Iteato umbe (b) Fgue 9. Estmated values vs. teato fo the L-L estmate: (a) foce ampltudes, (b) foce phases. 3

24 L p (Least Powes, ν =.) Module [N] (a) Phase [ ] Iteato umbe 8 9 (b) Fgue. Estmated values vs. teato fo the Lp estmate: (a) foce ampltudes, (b) foce phases. Fa Module [N] (a) Phase [ ] Iteato umbe 8 9 (b) Fgue. Estmated values vs. teato fo the Fa estmate: (a) foce ampltudes, (b) foce phases. 4

25 Hube Module [N] (a) Phase [ ] Iteato umbe (b) Fgue. Estmated values vs. teato fo the Hube s estmate: (a) foce ampltudes, (b) foce phases. Modfed Hube Module [N] (a) Phase [ ] Iteato umbe (b) Fgue 3. Estmated values vs. teato fo the Modfed Hube s M-estmate: (a) foce ampltudes, (b) foce phases. 5

26 Cauchy Module [N] 5 Aveage Aveage (a) 5 Aveage Phase [ ] -5 Aveage Iteato umbe (b) Fgue 4. Estmated values vs. teato fo the Cauchy s estmate: (a) foce ampltudes, (b) foce phases. 4 Gema-McClue Module [N] 3 Aveage Aveage (a) Aveage Phase [ ] -5 Aveage Iteato umbe 8 9 (b) Fgue 5. Estmated values vs. teato fo the Gema-McClue s estmate: (a) foce ampltudes, (b) foce phases. 6

27 Welsch Aveage Module [N] 5 Aveage (a) Aveage Phase [ ] -5 Aveage Iteato umbe 8 9 (b) Fgue 6. Estmated values vs. teato fo the Welsch s estmate: (a) foce ampltudes, (b) foce phases. Tukey Aveage Module [N] 5 Aveage (a) Aveage Phase [ ] -5 Aveage Iteato umbe 8 9 (b) Fgue 7. Estmated values vs. teato fo the Tukey s estmate: (a) foce ampltudes, (b) foce phases. 7

28 Iteatos: 8.8 Weght Sample umbe of ode Iteato 3 Fgue 8. Weghs attbuted pe teato fo the Hube s estmate. 6. Coclusos The pape pesets detal the theoy of the M-estmato ad dscusses ts applcato to the detfcato of exctatos mechacal systems, stead of the moe tadto least squaes. The goous defto of estmato obustess s toduced ad t s aalytcally show that least squaes ae deftely ot obust. The dffeet types of M-estmatos poposed lteatue ae dscussed ad the modfcatos equed to be appled to mechacal systems ae peseted. The pefomaces ae theoetcally foecast ad vefed by meas of two applcatos of exctato detfcato a SISO ad a MIMO mechacal systems. I patcula the IRLS algothm ecessay fo the calculatos s peseted. It s vey smple ts mplemetato, sce t epeset a teated applcato of the stadad algothm used fo weghted least squaes. The esults obtaed the umecal applcatos show that few of the poposed M-estmatos ae sutable fo detfcato poblems mechacal system ad that pactcally oly those ae chaactezed by uque soluto obta emakable esults, beg vey obust ot oly ude a aalytcal pot of vew. Refeeces [] Fswell M.I. ad Motteshead J.E. 995, Fte Elemet Model Updatg Stuctual Dyamcs, Kluwe Academc Publshes, Dodecht, The Nethelads. [] Bachschmd N., Peacch P. ad Vaa A.,, Idetfcato of Multple Faults Roto Systems, Joual of Soud ad Vbato, 54(),

29 [3] Peacch P., Bachschmd N., Vaa A., Zaetta G.A. ad Gego L., 6, Use of Modal Repesetato fo the Suppotg Stuctue Model Based Fault Idetfcato of Lage Rotatg Machey: Pat Theoetcal Remaks, Mechacal Systems ad Sgal Pocessg, (3), [4] Lu Y. ad Shepad S. J., 5, Dyamc Foce Idetfcato Based o Ehaced Least Squaes ad Total Leastsquaes Schemes the Fequecy Doma, Joual of Soud ad Vbato, 8(-), [5] Zhu L.-M., L H.-X. ad Dg H., 5, Estmato of mult-fequecy sgal paametes by fequecy doma o-lea least squaes, Mechacal Systems ad Sgal Pocessg, 9(5), [6] Whte P.R., Ta M.H. ad Hammod J.K., 6, Aalyss of the maxmum lkelhood, total least squaes ad pcpal compoet appoaches fo fequecy espose fucto estmato, Joual of Soud ad Vbato, 9(3-5), [7] Vebove P., Gullaume P., Caubeghe B., Paloo E. ad Valadut S., 4, Fequecy-doma geealzed total least-squaes detfcato fo modal aalyss, Joual of Soud ad Vbato, 78(-), -38. [8] Vebove P., Caubeghe B., Gullaume P., Valadut S. ad Paloo E., 4, Modal paamete estmato ad motog fo o-le flght flutte aalyss, Mechacal Systems ad Sgal Pocessg, 8(3), [9] Vebove P., Caubeghe B. ad Gullaume P., 5, Impoved total least squaes estmatos fo modal aalyss, Computes ad Stuctues, 83(5-6), [] Luh G.-C. ad Cheg, W.-C., 4, Idetfcato of mmue models fo fault detecto, Poceedgs of the Isttuto of Mechacal Egees. Pat I: Joual of Systems ad Cotol Egeeg, 8(5), [] Kaske W.S., 983, Robust Regesso, Ecyclopeda of Statstcal Sceces, Eds. S. Kotz, N.L. Johso, C.B. Read, N. Balaksha ad B. Vdakovc, Vol. 8, Joh Wley & Sos, Ltd, [] Bckel P., 983, Robust Estmato, Ecyclopeda of Statstcal Sceces, Eds. S. Kotz, N.L. Johso, C.B. Read, N. Balaksha ad B. Vdakovc, Vol. 8, Joh Wley & Sos, Ltd, [3] Hube P.J., 98, Robust Statstcs, Joh Wley, New Yok. [4] Ba Z.D. ad Wu Y., 997, Geeal M-estmato, Joual of Multvaate Aalyss, 63(), [5] Hampel F.R., 968, Cotbutos o the Theoy of Robust Estmato, Ph.D. dssetato, Uvesty of Calfoa, Bekeley. [6] Hampel F.R., Rochett E.M., Rousseeuw P.J.,ad Stahel W.A., 986, Robust Statstcs: The Appoach Based o Ifluece Fuctos, Joh Wley, New Yok. [7] Pccolo D.,, Statstca, Il Mulo, Mlao, [8] Rey W.J.J., 983, Itoducto to Robust ad Quas-Robust Statstcal Methods, Spge, Bel. [9] Mostelle F. ad Tukey J.W., 977, Data Aalyss ad Regesso, Addso-Wesley, Readg, MA. [] Petttt A.N., 98, Camé-Rao lowe boud, Ecyclopeda of Statstcal Sceces, Eds. S. Kotz, N.L. Johso, C.B. Read, N. Balaksha ad B. Vdakovc, Vol., Joh Wley & Sos, Ltd, 9. [] Ruppet D., 983, M-Estmatos, Ecyclopeda of Statstcal Sceces, Eds. S. Kotz, N.L. Johso, C.B. Read, N. Balaksha ad B. Vdakovc, Vol. 5, Joh Wley & Sos, Ltd, [] Che H. ad Mee P., 3, Robust Regesso wth Pojecto Based M-estmatos, Poceedgs of the Nth IEEE Iteatoal Cofeece o Compute Vso (ICCV 3), , -8. [3] Rousseeuw P.J. ad Leoy A.M., 987, Robust Regesso ad Outle Detecto, Joh Wley, New Yok,. [4] Fox J., Robust Regesso, A R ad S-PLUS Compao to Appled Regesso, [5] Colema D., Hollad P., Kade N., Klema V. ad Petes S.C., 98, A System of Suboutes fo Iteatvely Reweghted Least Squaes Computatos, ACM Tas. Math. Softw., 6, [6] Zhag Z., 997, Paamete estmato techques: a tutoal wth applcato to coc fttg, Image ad Vso Computg, 5, [7] Colls J.R.,, Robustess Compasos of Some Classes of Locato Paamete Estmatos, A. Ist. Statst. Math., 5(), [8] Lu R.Y., Paelus J.M. ad Sgh K., 999, Multvaate Aalyss by Data Depth: Descptve Statstcs, gaphcs ad Ifeece, The Aals of Statstcs, 7(3), [9] Hwag J., Jo H. ad Km J., 4, O the Pefomace of Bvaate Robust Locato Estmatos ude Cotamato, Computatoal Statstcs & Data Aalyss, 44, [3] Ma Y. ad Geto M.G.,, Hghly Robust Estmato of Dspeso Matces, Joual of Multvaate Aalyss, 78(), -36. [3] Dooho D.L. ad Gasko M., 99, Beakdow Popetes of Locato Estmates Based o Halfspace Depth ad Pojected Outlygess, A. Statst., (4), [3] Baett V., 976, The Odeg of Multvaate Data, Joual of Royal Statstcal Socety - Sees A, 39, [33] Tukey J.W., 975, Mathematcs ad Pctug of Data, Poc. of Iteatoal Cogess of Mathematcas, Vacouve, [34] Peacch P., Vaa A. ad Bachschmd N., 5, Bvaate aalyss of complex vbato data: A applcato to codto motog of otatg machey, Mechacal Systems ad Sgal Pocessg, do:.6/j.ymssp [35] Adews D.F., Bckel P., Hampel F.R., Hube P.J., Roges W.H. ad Tukey J.W., 97, Robust Estmates of Locato: Suvey ad Advaces, Pceto Uv. Pess, Pceto. 9

30 [36] Goss A.M., 98, Cofdece Itevals fo Bsquae Regesso Estmates, J. Ame. Statst. Ass., 7, [37] Sgh R. ad Saha M., 3, Idetfyg Stuctual Motfs Potes, Poc. of 8 th Pacfc Symposum o Bocomputg, Lhue, Hawa, USA, Jauay 3-7, 3, -3. [38] De la Toe F., Vta J., Radeva, P. ad Melecho J.,, Egeflteg fo flexble egetackg (EFE), Poc. of IEEE 5 th Iteatoal Cofeece o Patte Recogto, Baceloa, Spa, Volume 3, Septembe 3-7,, 6-9. [39] Lee M.-S., Katkovk V., Km K. ad Km Y.H., 4, Robust M-Estmato fo Multpath CDMA Commucato Systems wth Phased Aay Receve, Poc. of IEEE 5 th Wokshop o Sgal Pocessg Advaces Weless Commucatos, Lsbo, Potugal, July -4, 4, Appedx A I mathematcal tems, cosdeg oly the smplest case wth two vaables (a SISO system), the model that elates a vaable of teest x to the obseved data z s called empcal model. It s easoable to assume that the mea value of the adom vaable x s elated to z by: Ex [ z] = θ + θ z. (69) Although the mea value of x s a lea fucto of z, the actual values obseved of x usually do ot follow exactly a staght le. The appopate way to geealze ths stuato s to assume that the expected value of x s a lea fucto of z, but also that the expected value of x s gve by the lea model wth the addcto of a eo e, fo a fxed value of z. Ths coespods to a smple lea egesso wth a sgle pedcto z ad a espose vaable x : x= θ + θ z+ e. (7) The eo e s assumed to have omal dstbuto N(, σ ), ull mea value ad vaace equal to σ. Oce z s kow, the mea value ad the vaace of x ae sofa: Ex [ z] = θ + θz. (7) Vx [ z] = σ The estmate of θ ad θ s made by meas of the best appoxmato of the data. The most kow ad moe used method was toduced by Gauss, whch poposed the estmate of the paametes θ ad θ by meas of the mmzato of the sum of the vetcal dstaces betwee the obsevatos (the data) ad the le of lea tepolato. Ths s the method of the least squaes. The obsevatos ca be expessed as: x = θ + θ z + e, =,,. (7) The sum of the squaes of the dstaces of the obsevatos fom the actual egesso le s: ( ) θ θ. (73) L= e = x z = = The estmatos of θ ad θ the least squaes sese ae amed ˆ θ ad ˆ θ ad have to satsfy the so called omal equatos of the least squaes: 3

31 L θ L θ ˆ θ ˆ, θ ˆ θ ˆ, θ = = ( x θ θ z ) = = ( θ θ ) = x z z =. (74) The soluto of the omal equatos (74) gves the least squaes estmate of ˆ θ ad ˆ θ. The estmated egesso le s theefoe: ˆx= ˆ θ + ˆ θ z. (75) Note that whateve couple of obsevatos satsfes the equato: whee the quatty the -th obsevato x. x = θ + θ z +, (76) = x xˆ s called esdue: t evaluates the eo the model appoxmato of Appedx B I ode to show that ψ ( ) s popotoal to the fluece fucto of ρ, let s stat fom eq. () that ca be wtte geeal tems as: o: The effect of the added abtay obsevato (see eq. (3)) s: ψ ( X; T) = (77) = ψ ( xt, [ F]d ) F=. (78) ( xt[ F x] ) [ F x] ψ, ( ε) + εδ d ( ε) + εδ =, (79) ε ε = whch becomes, by chagg the ode of the tegato ad dffeetato: ad smplfyg: ( xt, [( ) F x] ) d [ x F] ψ ε + εδ δ + ε =, (8) + ψ( xt, [( ε) F+ εδx] ) T[ F] d F = ε= ε= TF [ ] ε ψ ( xtf, [ ]) d [ δx F] + TF [ ] ψ ( xtf, [ ]) df=. ε (8) ε = Theefoe, IF( x, T, F ) s also equal to (see eq. (6)): TF [ ] 3

32 IF( x, T, F) = T[ F] = ε ε = ψ ψ TF [ ] ( xt, [ F] ) ( xt F ), [ ]df, (8) povdg that the deomato s o-zeo. Theefoe the fluece fucto s popotoal to ( xt, [ F] ) ψ : ( xt F ) IF( xt,, F) ψ, [ ]. (83) 3