STATISTICAL PARAMETER ESTIMATION FROM MODAL DATA USING A VARIABLE TRANSFORMATION AND TWO WEIGHTING MATRICES

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1 SAISICAL PARAMEER ESIMAION FROM MODAL DAA USING A VARIABLE RANSFORMAION AND WO WEIGHING MARICES Haddad Khodapaast, H., Mottshad, J. E., Badcock, K. J. ad Mas, C. Dpatt of Egiig, Uivsity of Livpool, Livpool L69 3GH, UK. School of Egiig ad Dsig, Bul Uivsity, Uxbidg UB8 3PH, UK...ottshad@liv.ac.uk ABSRAC. Stochastic fiit lt odl updatig i stuctual dyaics ds statistical ifoatio o asuts ad stuctual paats. A stochastic odl updatig thod basd o a last squas stiato ad th ptubatio thod is foulatd. h thod is capabl of dtiig th uctaity i stuctual paats usig stablishd popagatio thods such as Mot Calo siulatio ad ptubatio. h poposd thod has b applid to th cas of a siulatd th dg-of-fdo ass-spig syst. h sults a validatd by a scod-od ssitivity thod. h us of wightig atics to balac os btw th two statistical idics of th stiatd paats is itoducd. KEYWORDS: Modl updatig, vaiability, uctaity popagatio, wightig atix INRODUCION Covtioal odl updatig thods us asut ifoatio fo a sigl stuctu [,, 3,, 3] whas stochastic odl updatig thods gally d ultipl sts of tst data fo ay stuctus built i th sa way fo th sa atials, but with aufactuig ad atial vaiability [4, 5, 6]. his lads to ipovd cofidc i th paats of th updatd odl. Stochastic odl updatig ay also iclud th cas of a sigl tst stuctu with vayig vibatio chaactistics du to viotal osio, opatig loads, fatigu, wa tc. [7, 8, 9]. Obsvd vaiability i asud odal data is causd aily by aufactuig tolacs (pstd by uctai paats ad asut ois. It is cla that th us of adoisd stuctual paats lads to icasd coputatio i odl updatig, ad thfo it is ipotat to us statistical stiatio thods so that th coputatioal ffot dos ot bco uasoabl. I this pap w cosid th statistical ivs pobl i which statistical ifoatio fo asuts is usd to idtify adoizd paats. Siila to th covtioal odl updatig pobl, a iitial stiat of systpaat statistical idics ust b chos ad th updatd itativly. h choic of paats i stochastic odl updatig is as ipotat as i covtioal odl updatig ad quis cosidabl physical isight [,, ]. I th statistical odl updatig of Collis t al. [] ad Fiswll [3] th adoss aiss oly fo th asut ois ad th updatig paats hav uiqu valus, to b foud by itativ coctio to th stiatd as, whilst th vaiacs a iiisd. I th thod dscibd i this pap, th adoss aiss fo two soucs, poduct vaiability (picipally du to aufactuig tolacs ad asut ois. I this cas, ultipl tsts a caid out o oially idtical tst stuctus ach havig a st of uiqu valus fo th updatig paats difft fo th oths. hus two spacs a dfid pstig th spac of th asuts ad th spac of pdictios, ad ou pupos is to covg th pdictio spac upo th spac of xpital asuts. his is achivd by a last squas thod which th dfis a coplt spac of updatd paats ad ot ust thi a valus. h assuptio i [, 3] that th xpctd valus of th paats do ot chag fo itatio to itatio is ot

2 appopiat i this cas ad th tasfoatio atix bcos a fuctio of pdictio vaiability as will b xplaid i th squl. h xpctd valu of th tasfoatio atix is xpssd i ts of two wightig atics which allow a balacd stiat of both th a ad stadad dviatio of th paats to b achivd. Stochastic odl updatig by th ptubatio thod [7, 8, 9, 4] ds th scod-od odal ssitivitis, which is ti-cosuig. hs thods hav b validatd by Mot Calo siulatio. W f to ths thods as scod od ssitivity thods ad us th fo validatio i this pap. h thod poposd h ds oly th fist-od ssitivity atix valuatd at th xpctd valus of stiatd paats wh popagatio by th ptubatio thod is usd o by ultivaiat ultipl gssios [5] wh popagatio by th Mot Calo thod is applid. h poposd thod is applid to a siulatd th dg-of-fdo ass-spig syst. h sults obtaid a show to b i good agt with thos obtaid by th scod-od ssitivity thod. h wightig atics itoducd a capabl of balacig th os btw statistical idics of th paats. It is show that us of th wightig atics ca daatically duc th stiatio os fo th stadad dviatio. HEORY Accodig to th covtioal, dtiistic, odl updatig thod [], th stiat b updatd usig pio stiat θ as, θ + ca θ + = θ + ( z z ( wh z is th vcto of stiatd output paats (igvalus ad igvctos, z is th vcto of asud data, θ is th vcto of syst paats ad is a tasfoatio atix. Pvious authos, usig th iiu vaiac stiato [,, 3] with th costat tasfoatio atix, hav supposd that th stiatd paats a ubiasd at ach itatio. Fo th stochastic odl updatig pstd h th assuptio of ubiasdss of th paats,θ, ust b abadod owig to th fo of th, which bcos a fuctio of th odl vaiability. o bgi, w dfi th paats, outputs ad tasfoatio i ts of th xpctd valu E ad th vaiability Δ, [] θ + Δθ [] θ = E ( z z [ z ] + Δz = E (3 [ z ] + Δz = E (4 kk z k k = [ ] + Δ = E (5 Δ z dots vaiability i vibatio spos of th athatical odl at th th itatio. his vaiability aiss fo uctai paats, θ, dd sposibl fo th obsvd vaiability i th asud vibatio data Δz. W sk th statistics (a ad stadad dviatio of ths paats that caus th covgc ot oly of E but also of Δ o Δz. ( z z E o ( z

3 W obsv fo quatio ( that th pdictio spac, dfid i quatio (3, should b ad to covg upo th spac of asud outputs, dfid i quatio (4. h a two asus of this covgc, aly covgc of th as, ad covgc of th stadad dviatios. h covtioal iiu vaiac thod allows oly fo a sigl athatical pdictio, ad ot fo th spac of pdictios dfid i quatio (3 i this ss th iiu vaiac stiato povids oly a icoplt statistical dsciptio. his xplais why th iiu vaiac stiato gally poducs a vy good stiat of th a, but a poo stiat of th stadad dviatio. A cosquc of this udstadig is that th tasfoatio atix should b difft fo difft athatical odl withi th vaiability Δ z. Equatio (5 is th obtaid by a tucatd aylo sis xpasio. A difft atix i th vcto of odl vaiability Δz. kk gally xists fo vy t Followig quatios ay b dvlopd fo quatio (, th xpctatio of quatio ( ad quatios (-(5, [ ] = E[ θ ] + E[ ] ( E[ z ] E[ z ] + [ ( Δ ] E θ (6 + E O [ ]( Δ z Δ z + z Δ ( E[ z ] [ ] + O( Δ Δ + = Δ θ + E kk k E z k= θ (7 wh E [ Δθ ] = E[ Δz ] = E[ Δz ] = Equatio (6 lads to th stiat of th a of th paats ad quatio (7 is usd i th dtiatio of th covaiac atix. A fo siila to that achivd by fist od ptubatio [7, 8, 9, 4] ay b achivd by igoig th scod-od vaiability ts i quatio (6. W itoduc a xpssio fo th xpctd valu of th tasfoatio that aks us of two wightig atics, W ad W, = ( + W [ ] W E[ S ] E[ S ] W E[ S ] E (8 h choic of W = I, W = sults i th psudo ivs ad W = I, W = V givs th tasfoatio atix dfid by Collis t al. []. It will b s that this tasfoatio allows th stiats of th paat as ad stadad dviatios to b balacd, i.., th stadad dviatio ay b ipovd at th xps of th stiat of th a ad vic-vsa. h oth ukow tasfoatio atics, kk, k =,,..., ca b foud by usig a last squas stiato. his is achivd by takig th divativ of th covaiac atix of th paats at th + th itatio with spct to ach tasfoatio atics as follows, ( V + kk =, k =,,... Solvig abov quatios siultaously lads to two cusiv systs of quatio havig th followig fo fo th stiatio of th xpctd valu ad co-vaiac atix of th paats, Δ z k (9

4 [ θ ] = E[ θ ] + W E[ S ]( E[ S ] W E[ S ] + W ( E[ z ] E[ z ] E V + + ( = + V E[ Δθ Δz ]( E[ ] + E[ ] V ( E[ ] E[ ] E[ Δz Δθ ] E[ ] E[ Δz Δz ]( E[ ] ( E[ ] E[ Δz Δz ] E[ Δθ Δz ] ( E[ Δz Δz ] ( E[ ] E[ Δz Δz ] E[ Δθ Δz ] V V + ( wh ad a co-vaiac atix of th paats at th ad + th itatios ad E [ Δ z Δz ], E[ θ Δz ] Δ ca b foud by wll stablishd popagatio tchiqus such as Mot Calo siulatio ad th ptubatio thod [5]. h xpctd valu of ssitivity atix i quatio (8 ca b valuatd by ultivaiat ultipl gssio, as i [5], wh usig th Mot Calo thods. But Mot Calo popagatio usually ds lag us of th fiit lt odls, which ca b xtly ti-cosuig. hfo w popos usig popagatio by a ptubatio appoach, which has siila sults to Mot Calo i lia cass [4]. Ptubatio popagatio thods d th ssitivity atix valuatd at th xpctd valus of th paats E[] θ. W us th followig wightig atics fo quatio (8, basd o th tasfoatio atix itoducd i [], W = V ( W ( E( z = V = α diag (3 wh α is a paat to b slctd by aalyst to appoxiatly pst th lvl of asut ois ad ay b dtid, fo xapl, fo asuts o a sigl tst stuctu. h wightig atix W is ipotat bcaus it accouts fo th diffc, du to asut ois, btw th spac of asuts ad th pdictio spac as will b dostatd lat i a siulatd xapl. h vaiability i asud data aiss fo two soucs, aly asut ois ad odl vaiability du to uctai paats. hfo th co-vaiac of asud data ay thfo b witt i th followig fo: V = V + V (4 u wh V is th asut co-vaiac atix, Vu is co-vaiac atix aisig fo odl uctaity ad V is th co-vaiac atix du to asut ois. h asut ois dpds o th xpital quipt, th tst viot ad data pocssig. hfo odal vaiability du to uctai paats ad asut ois a statistically idpdt. 3 SIMULAED EXAMPLE h poposd thod was applid to th sipl th dg-of-fdo ass-spig syst show i Figu. h oial valus fo th siulatd xpital syst a chos to b th sa as i [5], i =. kg ( i =,, 3, k i =. N / ( i =,..., 5, k 3. N / 6 =

5 k =. N /, k =. N /, k. N 5 = wh, k k ad k5 a oial stadad dviatio of th uctai paats k,k ad k5. Siulatd co-vaiacs of th xpital odal data du to uctai paats ( Vu i quatio (4 ca b foud by [6], u V = S V S (5 wh S is ssitivity atix valuatd at oial xpctd valus of th paats ad V is oial co-vaiac atix of th paats. h oous ado paats a assud as follow, / E( k = E( k = E( k5 =. N / k =.3, k =.3, =. 3 Idd, th asud covaiac atix ( V i quatio (4 is uch o sigificat tha asut ois ( V i quatio (4 i th psc of vaiability i asud data du to aufactuig tolacs, daag ad wa tc. I od to hav a good stiatio of asut ois, th followig idicato is dfid, ( E( z diag E( z V diag V α = = = (6 V Vu + V S V S + α ( wh dot th o of th atix. his idicato shows that how uch of th spac catd by vaiability i asud data is occupid by asut ois. At fist w hav to validat poposd thod with a idal situatio. I this cas w assu that th is a xactly-siulatd full-populatd covaiac atix of asud odl vaiability accodig to quatio (5. h cass of =, = ad = 3 w cosidd. W xpct that th thod should b capabl of gatig th xact valus of siulatd paats wh =. abl shows this to b th cas ad th scod-od ssitivity basd thod vifis th sults. Covgc of th Expctd valu ad stadad dviatio of th uctai paats by usig poposd thod is show i Figu. I th cass of = ad = 3, sults hav w obtaid wh (a th wightig atics w glctd ( W = I, W = ad (b wh wightig atix w dtid accodig to quatios ( ad (3. As ca b s i abl, th wightig atics ca b usd to balac th os i th stiatio of both statistical idics of updatd paats. I oth wods usig th wightig atics ca duc th os i stiatd stadad dviatio whil icasig th os i th stiatio of th xpctd valu. Figus 3-6 show of th xpctd valu ad stadad dviatio of th uctai paats covg fo th cass of = ad = 3.

6 Figu. h dg-of-fdo ass-spig syst [5] Paat ( k ( Iitial Eo ( Eo aft updatig by PM = Eo aft updatig by SSM = E.. E.. k E (.. k 5.. k abl. Rsults by th poposd thod (PM ad th scod-od ssitivity thod (SSM i a idal cas..5 = k = std (k Chag i Expctd valu of paats.5.5 k Chag i SD of paats std (k std ( Itatios Itatios Figu. Covgc of paat stiats i poposd thod- =

7 Paat Eo aft updatig Eo aft updatig Eo aft updatig Eo aft updatig W I, W = = = W = V, W = V = W I, W = = = 3 W = V, W = V = 3 k k k k k abl. Eos aft updatig by th poposd thod.5 = k = std (k Chag i Expctd valu of paats.5.5 k Chag i SD of paats std (k std ( Itatios Itatios Figu 3. Covgc of paat stiats i poposd thod- = - W I, W = Chag i Expctd valu of paats = k k Chag i SD of paats = = std (k std (k std ( Itatios Itatios Figu 4. Covgc of paat stiats i poposd thod - = - W = V, W = V

8 .5 =3 k =3 std (k Chag i Expctd valu of paats.5.5 k Chag i SD of paats std (k std ( Itatios Itatios Figu 5. Covgc of paat stiats i poposd thod- = 3 - W I, W = Chag i Expctd valu of paats =3.5 k k.5.5 Chag i SD of paats = =3 std (k.6 std (k std ( Itatios Itatios Figu 6. Covgc of paat stiats i poposd thod - = 3 - W = V, W = V h ffct of th wightig atics ca b ivstigatd. Figu 7 shows o o i th xpctd valus ad stadad dviatios of th paats vsus asut ois idicato. wo cass a cosidd, W = I, W = ad th wightig atics dtid by quatios (-(3. Figu 7 shows that th latt cas lads to th btt stiatio of th stadad dviatio wh th asut ois is sigificat. Aoth ipotat liitatio i pactical wok is th ffct of os i asud covaiac atix du to uctai paats ( V u i quatio (4. I pactic this atix is usually tak to b diagoal istad of full-populatd. Also th a likly to b os du to th scacity of data fo oially idtical tst pics aufactud withi tolacs. hfo as a xapl of a pactical liitatio, a atix cosistig of th diagoal ts fo quatio (5 cotaiig o is cosidd fo th siulatd co-vaiac atix V u. Figu 8 shows o o of th stiatd paats vsus asut ois idicato. h ffctiv wightig atix ca duc th o causd by iaccuat asut of V u. Although th stiatio of th xpctd valu of th paats a lss accuat wh usig th wightig atics (lft diaga i Figu 8, th ight diaga shows that th stiatio of th stadad dviatio is ipovd. I paticula th cosidabl os i th stiatd stadad dviatio causd by odl vaiability ad asut ois ca b ducd daatically by applyig th wightig atics. Fo istac if w hav.4 asut ois ( α =.4 which lads to =, th o o icass i th a valu by 3.7 (blu cuv istad of zo (d li but dcass i th stadad dviatio (6 (blu li istad of 8 (d li. h stadad dviatio is always calculatd

9 with a vy sigificat o wh th wightig atics a ot usd, as ca b s fo th d li i th ight-had diaga of Figu W=V, W =V W=I, W =. 8 6 W=V, W =V W=I, W =. Eo o (Expctd valu 5 5 Eo o (SD Figu 7. Eo o i xpctd valu ad SD vs. asut ois idicato W=V, W =V W=I, W =. W=V, W =V W=I, W =. Eo o (Expctd valu 5 5 Eo o (SD Figu 8. Eo o i xpctd valu ad SD vs. asut ois idicato 4 CONCLUSION A thod is dvlopd fo stochastic odl updatig usig statistical idics ad a ptubatio appoach with fist-od ssitivitis. Mot-Calo ad ptubatio thods fo uctaity popagatio ay b applid. h thod aks us of a vaiabl tasfoatio atix to covg th spac of odl pdictios upo th spac of asud odal data. Wightig atics a usd to balac th os btw th as ad stadad dviatios of th stiatd stuctual paats. h thod is validatd by usig a scod-od ssitivity appoach ad siulatd xapls with a th dg-of-fdo ass-spig syst povid a dostatio of how th tchiqu could b applid i pactic. ACKNOWLEDGEMENS h sach dscibd i this pap was suppotd by EU Mai Cui Excllc poct ECERA.

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