Bayesian evidence for finite element model updating

Bayesian evidence for Finite element model updating Linda Mthem bu, PhD student, Electrica l and Information E ngineering, University of t he W itwatersrand, Johannesburg Private B ag 3, Johann esburg, 2050, Sou th Africa Tshilidzi Marwala, Prof essor of Electrical and Informa tion Engineering, University of the W itwatersrand, Private Bag 3, Joh annesburg, 2050, South Af rica Michael I. Friswell, Sir George Wh ite Professor of Aerospace Engineering, Dep artment of Aerospace Engineering, Queens B uilding, University of Bristo l, Bristol BS8 1TR, UK Sondipon Adhika ri, Chair of Aerospa ce Engineering, Sch ool of Engineering, Swanse a University, Singleton Park, Swansea SA2 8PP, Unit ed Kingdom Abstract This paper considers the problem of model selec tion within the cont ext of fin ite elem ent m odel updating. G iven that a num ber of FEM upd ating m odels, with dif ferent updating param eters , can be desig ned, th is pa per pr opos es using th e Ba yesian evidenc e statist ic to assess the probabi lity of each updat ing m odel. T his m ak es it po ssible then to evaluate the need for a lternat ive updating par ameter s in the up dating of the initial F E m odel. T he mode l evidences are com pared u sing th e Ba yes f actor, which is the ratio of ev idence s. The Jeff rey’s sc ale is us ed to determ ine the dif ferences in the m odels. T he Ba yesian e vidence is calculated b y integrating the lik elihood o f the data given the m odel and its par ameter s over the a priori m odel parameter space us ing the n ew neste d sam pling algorithm . The nested a lgorithm samples this likelihoo d dis tribution b y us ing a h ard likelihoo d-valu e c onstraint on the sam pling reg ion w hile pr oviding the poster ior sam ples of the updatin g m odel param eters as a by-product. T his method is us ed to ca lculate the evidenc e of a num ber of plausible f inite elem ent m odels. Nomenclatur e θ Mode l Param eters D Real m easured s ystem data H Finite e lement m odel (Mat hematic al model) Z Evide nce N Num ber of s amples Max Maxim um num ber of iterations L Lik elihood prob abili ty π Prior Probabi lity c Dam ping m atrix k Stiff ness m atrix m Mas s m atrix ( ) t x & & Node ac celerati on ( ) t x & Node v elocit y ( ) t x Node d isplacem ent w Natur al frequenc y vector φ Mode shape vector FEM F inite elem ent mode l FEMUP F inite elem ent mode l updating pr oblem PDF Pr obabilit y Distribution F unction MCMC M arkov Cha in Monte Carlo 1. Introductio n System model ing f orm s an im portant stage of m any engineerin g des ign proble ms. The results from the m odel either confirm or highlight l imitations of the design. A n an al yst is us uall y interes ted in the accurac y, c onfidenc e range a nd mor e c ritically the correc tness of the assumed m athem atical m odel. In this paper the m odel dom ain is structural finite elem ent models (FEMs). Thes e models are used to approximate the dynam ics of structural system s, e.g. train chassis, aircraft fuselages, bicycle frames, civil structur es etc. The finite elem ent model updating problem (FEMUP) arises when a real system ’s dynamic behavior is measured (e.g. the natural frequencies at which part icular system deformations occur) and the results of the mathem atical m odel of that system do not corr espond t o the meas ured data [6, 8]. T his problem is c ompounded b y the fact that a multitude of mathem atical m odels of th e s tructure, with var ying le vels of com plexit y, ca n be deve loped, leading to n on-uniq ue solutions for a particu lar sys tem. Finite elem ent m odels are limited b y definiti on; the y are an approxim ation of a real s ystem and will thus never produce d ynam ic results that are equal to t he m easured s ystem’s data. T he cha llenge is then what c an be done to the initial m odel f or it to better reflect the real system ’s dynamic results? This leads to the need for intelligentl y improved or upd ated m odels. Specifical ly an autom atic m ethodolog y of determ ining salient m odel param eters that require updat ing needs to b e deve loped. This h as to be atta ined whi lst using rea listic charac teristic param eter s of the s y s tem in questio n. T wo main direc tions of r esearch have been establ ished in the area of f inite elem ent m odel updating (FE MU); d irect and indirec t (iterat ive) m ethods [8]. In the direct m odel updat ing para digm [3, 5, 8] t he m odel m odal par ameter s are direc tly equated to t he m easured modal dat a. Model updatin g is then character ized b y the direc t upd ating of m ass or stiff ness m atrix elem ents. T his effec tively c onstr ains the modal properties and frees the sy stem matrices for updatin g. T his appr oach often results in u nrealistic elem ents in th e system matr ices e.g. large and physicall y im possible m ass elem ents. In the indirect or iterative m odel updati ng approach the updating problem is f ormulated as a ‘relaxed’ o ptimization problem , often appr oached by the use of m axim um likelihood m ethods [ 8, 9, 12 and 19]. A s elect few param eters are varied in model ele ments and the res ulting m odel is opt im ized t o minim ize its differ ence from the m eas ured data . Given the n on-uniq ueness of updat ing m odels, this p aper proposes going b ack to the initial stage of modeling an d questions the choice of the initial fin ite element m odel(s) . W e propos e the FEMU P should initiall y be approach ed from a mod el selection [ 2, 10, 1 7] perspec tive. Before m odel updating, a num ber of f undam ental ques tions n eed to b e a ddresse d, thes e are; (i) Is the m odel we want to update the correc t one? (i i) W hich aspects of the m odel do we need to update? (ii i) Having d etermined the up dating param eters, with what certa inty c an we guarantee that our upd ated m odel is corr ect? The first point can be recast as; what ev idenc e do we hav e that our m odel is the corr ect one g iven that a num ber of m odels can be gen erated? T his we propos e ou ght to be an essenti al an d nec essar y statis tic to establ ish bef ore any m odel updating schem e c an proc eed. H aving established th at the mos t proba ble model is found from a set of possible models we can then focus on the updat ing process . In this pa per we f ocus on the firs t problem which lends itself well to Ba yesian inference. Bayesian anal ysis al lows one to deal with initial and subsequent finit e el em ent m odel uncertaint ies in an intuitive and s y stem atic way [ 2, 1 0]. This paper is concer ned with c alculati ng t he ev idenc e of finite elem ent m odels. In the nex t sec tion we formall y prese nt the f inite el ement m odel form ulation and v ery com pactl y review th e c urrent approaches to the F EMUP. In section 3 we re view B ayes theorem and introduc e the evidenc e calcu lation in the Bayesian f ram ework. Sec tion 4 pr esents the evide nce calculation algorithm . Sect ion 5 presents pr oof of conc ept sim ulations of evidence c alc ulation usin g a sim ple beam structur e. We then conc lude the paper. 2. Finite elem ent backg roun d 2.1 Form al definition In engineering, d ynam ic structures are often an alyzed f rom bottom to s y stem s (component) level. At the bottom most level the struc ture is discreti zed into const ituent elem ents, for m athematical anal ysis and com putational feasibilit y, to a s ystem of a second-order m atrix of the form [6, 8]: 0 ) ( ) ( ) ( = + + t Kx t x C t x M & & & (1) where M , C and K are of equal s ize and are c alled the m ass, damping and s tiffness m atrices res pectively or system matr ices . Ass uming eac h finite elem ent’s d ynam ic displacem ent r esponse behaves accor ding to wt e t x − = φ ) ( equation 1 above is tra nsf orm ed to: 0 ] [ 2 = + + − φ K wC M w (2) In cases where th e str uctur e is s m all, lightl y dam ped or undam ped the dam ping m atrix 0 = C the a bove e quatio n reduces to: { } j j j K M w ε φ = + − ] [ 2 (3) where j w and j φ are the j th system natural fr equenc y an d m ode shapes, together k nown as m odal propert ies, { } j ε is the j th error of the err or v ector. G iven a set of m easured r eal s ystem m odal data , the F EMU problem is then for the model to rea listicall y appr oxim ate the mas s and s tiffnes s matr ices that will produc e m odal d ata th at i s as close to the rea l s ystems ’ as poss ible. If the m odel does not result in m atc hing m odal properti es the error vector is non-zero and som e m odel param eters will n eed to be upda ted. Bas ic ph ysics relates elastic e lement s tiffnes s with Young’s m odulus; the mass is a function of its geom etr y and dens ity. By cons ider ing thes e var iables as r andom values th at are defined wit hin cer tain interv als f or particular m aterials, it is pos sible to closel y appr oxim ate the meas ured m odal values in a form al wa y. In this pa per we treat the Young’s m odulus as a variable to be up dated. Measurem ent o f structural dynam ics produces on e set of, not nec essaril y repeatab le data, e.g. buildin g vibrations, earthquakes and vehicle d ynamic perform ances. T raditiona l appro aches to prevent data bi asing models are to repeat the m easurem ents a num ber of times to generate a g eneral trend and a veraging the data. This is not a sim ple proc ess and can becom e very expensive [6]. This situati on is well su ited to t he Ba yesian approach as oppos ed to a freque ntist paradi gm. In Bayesi an inf erence we ar e gi ven one obser ved data set which our model is supposed to approxim ate well. I n th e frequentist approac h t he data ac quis ition or ex perim ent is assumed to be re peatab le, such that a p attern co uld be deter m ined f rom a large dataset. The Ba yesian perspective is bold in that it infers a lot from one dataset [2]. In the next sect ion w e pres ent t he t ools of this paradigm , Bayes theorem and th en explain Bayesian i nference in t he conte xt of the f inite e lem ent updating problem . 3. Bayesian Inference In curr ent ti m es of m assive da ta, the abilit y to quantif y the model’s posterior c onfidence a nd t o com pare one model’s abilit y to approx imate d ata with another has bec ome the focus of data analysis. One paradi gm in this direction that has seen a r ecent resur gence is Ba yesian infer ence [2, 10 an d 20]. Ba yesian inferenc e allo ws one to quantif y uncerta inties in quan tities of interes t in a form al wa y. Bayesian inferen ce is often implem ented in two settings; param eter estim ation and m odel select ion. P aram eter estim ation is con cerned with the p lausibil ity of a given m odel’s param eters and t his is of ten carried out us ing s tandar d s ampling m ethods e.g. Mark ov Chain Monte Carlo (MCM C) [2, 10, 11 and see 16 for rece nt advances in t hese techniq ues]. Model s election on the other h and deals with the evi dence for each cand idate mathem atical m odel to ap proxim ate a part icular obse rved dataset. 3.1 Param eter estim ation In param eter es timation th e m athematical m odel is a ssum ed to be tr ue, the model is th en ‘fitted ’ to th e data and the pos terior plaus ibilit y of the m odel par ameter s c an be inf erred. This pr obability is calculated via Ba yes theorem as follows: ) | ( ) | ( ) , | ( ) , | ( H D P H P H D P H D P θ θ θ = ( 4) where the l eft hand si de is the p osterior pro babi lity of the u pdating param eters for the true m odel H, given som e data D, the prior probabil ity of the model param eters is ) | ( H P θ , ) , | ( H D P θ is called the likelih ood of the model. The denom inator ) | ( H D P is called the m arginal likelihood , or the ev idence , of the model where the param eters have be en m arginalized out. Bayes theorem autom aticall y inc orporates the upd ating of the param eters b y definit ion; it updates the a pr iori pro bab ility distr ibutio n of the m odel par am eters with the likelih ood of the m odel explain ing the m easured dat a. In the F EM c ontext the p aram eter variables that t ypically gover n the dynam ics of the finite ele m ents of the mathem atical m odel are the follo wing: Young’s modulus, cross- sectional areas, dam ping co eff icient etc. T hese par am eters in tur n aff ect the s tiffnes s, m ass and dam ping m atrices in equation 1. I n [11] the paramet er est imation context of the Bayesian fr amework was im plemented to ob tain th e posterior probab ilit y distribution of m odel parameters . By obtaining the poste rior probabil ity of the updating param eters the probabil ity distribution of the modal proper ties m a y then b e c alculated. This gives the r esearc her a quantitat ive m easure of the pro bable ra nges of the m odal and upd ating param eters of the model. In this p aper the interest is not in the pos terior probab ility of the updating param eters per se but in the e videnc e of the c hosen updati ng model t hus a model s election problem although the algorithm implem ented pr ovide s the posterior prob abilities of the up dating param eters as a b y-product. 3.2 Model C om parison In model c ompar ison one dataset is obser ved and a number of possible m odels can be f orm ulated. Bayesian inference provides a platf orm for calculatin g which model is t he m ost plausib le f rom this set. The poster ior probabilit y of each m odel w ithin a set of plausible m odels is given by Ba yes theorem : ) ( ) ( ) | ( ) | ( D P H P H D P D H P i i i = (5) where ) ( i H P is th e pri or pr obabilit y of each m odel and ) ( D P is the proba bilit y of the d ata. Since for all m odels the denom inator is inde pendent of the m odel we m a y ignor e it and t he theor em then r educes to: ) ( ) | ( ) | ( i i i H P H D P D H P ∝ (6) The first term on the right of equation 6 is the e vidence from equation 4. Given that each model ma y be equall y likely t o fit the d ata, t he evidence term is the decidin g factor on which m odel is the mos t plausible f or a parti cular observed d ataset. One s ta ndard m ethod of c om paring models in Bayesian a nalysis is Bayes fac tor def ined as ; ∫ ∫ = = ) | ( ) , | ( ) | ( ) , | ( ) | ( ) | ( j j j j j i i i i i j i H P H D P d H P H D P d H D P H D P Facto r Baye s θ θ θ θ θ θ (7) So by calculating each m odel’s evide nce we can quant ify their ratios. In [1, 4 and 17] the conce pt of m odel selection f or engineeri ng struc tures was us ed. In [1] the poster ior distri bution of the m odel was assum ed to be Gaussian which only work s for certain types of m odels [1, 2, 4, 10 and 1 7]. The method pro posed in this paper does no t pl ace an y ass umption of the form of the posterior distribution of the m odel. In [17] a rece ntl y propose d MCMC type post erior probabilit y sam pling a lgorith m (TMCMC f rom [4]) was im plemented. This a lgorithm estim ates the m odel e videnc e b y sam pling the poste rior pr obabilit y distr ibution of the m odel b y a sequen ce of non-norm alized int erm ediate proba bility f unctions. This algorithm suff ers from the use of m an y f ree param eters; e.g. a vari able to b alance the sam pling s teps, the num ber of intermediate probabilit y distri bution func tions ( PDF), the t em pering param eter and the sam ple weights [4, 17]. The alg orithm propos ed here is bel ieved to be sim pler and to hav e fewer f ree-param eters. In the next sec tion we exp lain the Ba yesian e vidence. W e then intr oduce an alg orithm c alled nested sam pling for efficientl y estimating the mode l evidence that is th eoretical ly and pract ically s impler than TMCMC. 3.3 Ba yesian evidenc e From the pre vious eq uation each m odel’s evid ence is gi ven by equat ion 8: ∫ = = ) | ( ) , | ( ) | ( i i i H P H D P d H D P evide nce θ θ θ (8) This equation m a y be in terpreted as follo ws; given a uni t of param eter s pace θ d with a model h aving a pri or probabilit y of ) | ( i H P θ over this spac e, the posterior pro babili ty d istributi on of the model over the sam e spac e will depend on ho w the m odel p aram eters fit the data ) , | ( i H D P θ . In this paper the likelihood f unction is defined as the norm alized exponentia l error betw een m easure d and real natural freque ncies ( the error g iven in equa tion 1) at the meas ured m odes as in [Ref.13]. T he m odel posterior pr oba bility will b e highest at t he m ost probable param eter set which occup ies a sm all region of the original param eter space. In sim ple models an in itiall y smaller, by d efinition, prior par am eter space would almos t be full y uti lized to f it the data as the pos terior probab ility would occupy a large port ion of the prior param eter s pace r esulting in a correspo ndingl y large e vidence value. C omplex models have lar ger param eter spaces because they have m any fr ee par am eters that allow t hem to fit alm ost an y data and this of ten res ults in o ver-fitting [2, 1 0]. This larger param eters space is ‘wasted’ in accou nting f or the resultant, relativel y narrow posterior density. The Bayesian inference paradigm thus automatically penalizes complex m odels without th e need for a model regularization term often incor porated in non-Ba yesian appr oac hes [2, 10, 19]. Continuing with the e vidence f ormulation, e quation 8 can b e re-written in the fol lowing f orm: ∫ = θ θ π θ d L Z ) ( ) ( (9) Analyticall y e valuating this integr al m a y be diff icult or im poss ible if the product o f the prior and likelihood is not sim ple. T his often happe ns when for exam ple the par am eter spac e is high d imensional w hich requires c alculatin g multidim ensional likelihood s. T he m ost popular appr oach to appr oxim ating s uch integrals has been to apply numer ical techniq ues suc h as im portance s am pling an d ther mod ynamic integration but these confr ont the problem that the prior param eters are distr ibuted in regions wher e the likelihood function is not highl y concentrate d [2, 10,16]. Other r ecent s am pling tech niques for ex ample T MCMC in [4] th at ar e bas ed on MCMC paradigm can be used but these tend to only sam ple the peak s of the posterior distribution which can u nder sample m ost of the narrow bes t-fit regions . Recentl y Sk illing in [18] pr oposed an al gorithm , nested sam pling, which is able to est imate integrals of the form shown in equat ion 9 eff icientl y. T he algorithm w ork s by transforming the multidim ensional parameter space integral to a o ne dim ensional one where classical num erical approxim ation techniques of estimating the area under the f unction can be applied. T he algorithm has been success fully app lied and further modified in a num ber of recent as tronom y and cosm ology pa pers [7, 14, 15, 20] 4. Sampling 4.1 Nested sam pling Nested sam pling is a Mon te Carlo but not a Mark ov Chain m ethod of sampling. T he m ain idea behind the m ethod is to divide the pri or param eter space into ‘e qual m ass’ units a nd to or der the se b y m odel lik elihood. T he total prior mass is denoted X and each unit in t his prior mass i s dX d d H P = = θ θ π θ θ ) ( ) | ( . The likelihood function is written ) ( ) ( ) , | ( X L L H D P = = θ θ in this space. T his form ulation transf orm s the likelihood i nto a function of a one-dim ensio n param eter. The ev idence inte gral can n ow be written: ∫ ∫ = = = 1 0 ) ( ) ( dX L d L Z eviden ce θ θ π θ (10) The algorithm then s uppos es likelihoods c an be evaluated at all i X s o that ) ( i i X L L = , wher e i X is a sequ ence of values that decreas e f rom 1 to 0 s uch that 1 ..... 0 0 1 2 = < < < < < X X X X N as illustrated in the bottom image of f igure 1. The to p im age shows a num ber of s am ples and the l ikelihood iso-c ontours in the 2-dim ensional posterior prob abilit y param eter space. Figure 1 : Sam pling f rom a 2D parameter s pace b y nested sam pling Such a one d imens ional integr al function , equatio n 10, c an e asil y be es tim ated b y an y num erical m ethod such as the trapezoi d rule: ∑ = = N i i Z Z 1 ∑ = = N i i i i b L Z 1 (11) where ) ( 2 1 1 1 + − − = i i i X X b and i L is the likelihoo d at th at sam ple. In the c ontext of f inite elem ent upda ting, the algorithm achieves th e above approxim ation in th e follo wing manner: 1. Sample N u pdating p aram eters f rom the prior pro babilit y distributi on. Evaluate thei r likelihoods. 2. From the N s am ples select the sam ple with the lo west lik elihood ( i L ). 3. Increm ent the evidence by ) ( 2 1 1 + − − = i i i i X X L Z . 4. Discard the sam ple with th e lo west likelihoo d an d replace it with a new point fr om within t he r em aining prior volum e ] , 0 [ i X . The new sam ple m ust satisf y the constra int i new L L > 5. Repeat s teps 2-4 until som e stopping criterion is reached. T his could be the des ired pr ecision on the evidence or s om e iteration count ( Max in our alg orithm ). For further details on neste d sampling see [18]. T he next sec tion pres ents the exper iments per form ed on different finite elem ent m odels and the e valuation of their e vidences . 4.2 Mode l defini tion and com paris on In FEM a mathem atical model ( i H in section 3) i s defined by the quantit y and location of free-parameters (updating p aram eters) . Here finite element m odels were designed t o have a dif ferent numb er of f ree-param eters at d iffer ent posit ions along the sam e structur e, ef fectivel y creat ing diff erent m athem atical m odels for the sam e structure. In m athem atics the num ber of f ree-param eters is related to the com plexity of a m athem atical m odel and to ad vocate Occam ’s razor; a m odel with f ewer para m eters is prefer red. T his is us eful in not onl y de term ining which param eters need to be update d but also how m any can b e deem ed suff icient to appr oxim ate the m easured data well. To deter m ine ho w well diff erent models com pare in fitting the data, Jeffre y’s sc ale [10], sho wn in table 1, was used. Table 1: J effre y's factor scale i j Z Z / ) / ( log 2 i j Z Z ) / ( log i j e Z Z ) / ( log 10 i j Z Z Evidenc e against m odel i H 1 to 3.2 0 to 1 .7 0 to 1. 2 0 to 0. 5 W eak 3.2 to 10 1.7 to 3.3 1.2 to 2.3 0.5 to 1 Subs tantial 10 to 100 3.3 t o 6.6 2.3 to 4.6 1 to 2 Stro ng > 100 > 6.6 > 4.6 > 2 Deci sive > 1000 > 10 > 7 >3 B eyond reas onable dou bt 5. Experimen ts A s imple uns ymm etrical H -beam , with s ix degrees of freedom , previousl y used in [13], is m odeled. The m easured natural fr equencies of this structure occ ur at; 53.9H z, 117. 3Hz, 20 8.4Hz, 254H z and 4 45Hz which correspo nd to modes 7, 8, 10, 11 and 13 r espectivel y. T o validate that m odel evidence calcu lation can reve al th e m ost plausible finite elem ent m odel(s), f our random ly desi gned m odel s of one beam struc ture are develope d and the evidenc e of each is c alculated . The exam ple as sum es no a prior k nowledge of which upd ating param eters should be chos en is a vailable. The objecti ve i s then to deter m ine fr om evidence ratios, the m os t probab le m odel from this random set. The s tructure was m odeled using the SDT ® version 6.0 Matla b® too lbox. Each model used sta ndard iso tropic mater ial pr operties and Euler B ernoul li beam elem ents to approx imate the bea m sections of the str ucture . It is assumed that th e You ng’s m odulus of som e elements is not c ertain and was thus considered a variable to b e updated in the part icular updati ng param eters . 5.1 Uns ymmetr ical H beam Figures 2 an d 3 illustrat e a sk etch of a 600mm long un s y mm etrical a luminium structur e with the lef t vertical b eam of 400m m and the right beam of 200mm . It is divided into 12 elem ents. Eac h cros s-sec tional area is 9.8m m b y 32.2 m m ( see [13] for m ore details on the str ucture and experim ental set-up). T he s tructure’s elem ents are number ed to clarif y the reading of table 2, where t wo sets of models ar e sho wn; mode l 1[A-C] and m odel 2 [A- C]. Figure 2 : Mo del 1A of a 12 Elem ent Unsymm etrical H beam . The updating p aram eters were sam pled f rom a Gauss ian pr ior probab ilit y distr ibution with a m ean of 7.2x 10 10 N.m -2 and an exp onential i nverse var iance of 4.0e- 20 for Young’s m odulus values bet ween 6.8x 10 10 N.m -2 and 8.0x10 10 N.m -2 for aluminiu m. T he number of samples, N , was s et to 100 (to sample the distribution well) an d the sampling algorithm s topping criterion is experim entall y set to a m axim um of 250 iterations. Model 1A in table 2 is the model ed structur e suc h th at the elem ents labele d E1 in figure 2 are set t o a c onstant value of You ng’s modulus f or aluminium ( 7.2x10 10 N.m -2 ). The Young’s m odulus values for elem ents labeled E2 ar e s am pled from a prior with a G aussia n pro bability d istributi on as m ention ed abov e. Mod el 1B is the opposite setting where no w the E2 labeled param eters are f ixed and the E1 p aram eters are s ampled f rom a Gauss ian pr ior distr ibutio n. All the elem ents that are free or update d in each m odel are listed under the head ing “ Param eter label” in t able 2. T he set of updated param eters does no t necess aril y have to be fixed; all par ameter s can be v aried s imultane ously as shown b y models 1C an d 2C in table 2. Fi gure 3 i llustrates m odel 2A f rom the second set of m odels. E2 E2 E2 E2 E1 E2 E1 E1 E1 E1 E1 1 2 3 4 5 6 7 8 9 10 12 11 32.2 9.8 E1 Table 2: U pdated param eters and m odel evidenc es Model Log( Evidenc e) Quant ity of param eters Param eter label 1A 04 . 0 188 . 2 ± − 5 2,3,5, 10,11 1B 12 . 0 40 . 11 ± − 7 1,4,6, 7,8,9,1 2 1C 13 . 0 15 . 12 ± − 12 A ll 2A 13 . 0 95 . 1 4 ± − 7 1,5,6, 7,8,9,10 2B 04 . 0 188 . 2 ± − 5 2,3,4, 11,12 2C 14 . 0 74 . 2 1 ± − 12 All Figure 3 : Mo del 2A of a 12 Elem ent Unsymm etrical H beam . Table 3 s hows the r esults using J eff rey’s sc ale. M odels 1A and 2 A are ver y diff erent; m odel 1A is a better m odel for this data than m odel 2A. It is int eresting to note that the m odels with t he least num ber of updating param eters produced the b est evid ences and this is bec ause it i s relativel y sim ple and in accordance with Oc cam ’s razor which sta tes that th e l east complex m odel that best desc ribes the dat a is the correc t one. Model 1A and 2B are shown to be r elativel y similar while th ere is s trong evidenc e agains t m odel 2A f rom model 1B . So evidence calculation can provid e a m echanism f or eliminating poor m odels f rom the ons et. It also pro vides a platform to determ ine salient par am eters to consider in the upd ating proces s. Table 3: Ba yes factor s for Unsymm etrical beam model q p H H / Bayes Factor Jeffre y’s Scal e Evidence against m odel q H ) / ( q p Z Z ) / ( log i j e Z Z 1A/1B 10122 9.2 Be yond reas onable do ubt 1A/2A 349760 12.7 Be yond reaso nable dou bt 1A/2B 1 0 W eak 1B/ 2A 35 3.5 Str ong 6. Conclusio n In this pap er we have introduced t he m odel selection concept to t he probl em of finite elem ent m odel updating. It was ar gued that plausi ble m odel e vidences shoul d be ca lculated bef ore m odels can be updat ed. A re centl y proposed m ethod of comparing m athem atical m odels was intro duced and im plemented in th e finit e el em ent model context. This algorit hm , nested samplin g, eff icientl y c alcu lates t he Ba yesian evidence of a m odel and provides the poster ior prob ability distrib ution of the model param eters as a b y-product. A sim ple beam s tructure with a n um ber of random m athem atical model formulatio ns, defined by the num ber and pos ition of up dating param eters, is used as proof-of -concept f or the algor ithm in this domain. Clear distinct ions be tween the m odels are evident f rom their relative ev idences. E2 E1 E2 E1 E1 E1 E2 1 E2 E2 E2 E2 1 2 3 4 5 6 7 8 9 10 12 11 32.2 9.8 E1 7. Acknow ledgem ents This resear ch was suppor ted b y t he financi al assista nce of the Nat ional Rese arch Foun dation of So uth Africa. I would lik e to thank mem bers of the W its Com putational Inte lligence gro up for th eir helpf ul discus sions. 8. Referenc es [1] Beck JL and Yuen K-V. Model selectio n using respons e measurem ents : Bayesian Probabilis tic Approach. Journal of Engineeri ng Mec hanics, Vo l. 130, No. 2, pp 192-203, Fe bruar y 2004. [2] Bishop CM . Pattern Rec ognition and Machine Learnin g. Springer, 2006. [3] Car valho J, Datta BN, Gupta A, Lagad apati M. A direct method for m odel up dating with incom plete m eas ured data and without spuri ous m odes . Mechanical S ystems and Signa l Process ing. Vol. 21, pp 2715- 2731, 2 007. [4] C hing JY, C hen YC. T rans itional Mark ov Chain Monte Carlo m ethod for Ba yesian m odel updati ng, model class s election, and m odel a veraging. Jo urnal of Engineer ing Mechanics -ASCE, Vo l.133, No. 7, pp 816- 832, 2007. [5] Datta BN Fin ite elem ent Mode l Updating, E igenstru cture Ass ignment and E igenva lue Em bedding T echniques for vibrating System s. [6] Ewin DJ . M odal T esting: T heory and Pr actice. Pu blis her Letchwor th, Resear ch Stud ies Press , 1984 . [7] F eroz F and Hobson M .P Mu ltim odal nested sam pling: an eff icient and ro bus t alternative to MCMC m ethods for astronom ical data ana lysis. Monthl y Not ices of the R oyal As tronom ical Societ y, Vol. 384 , No. 2, pp 4 49 - 463, 2007. [8] Fris well M I an d Mot ters head JE. Fin ite elem ent m odel upda ting i n s tructural dynam ics. Kluwer Ac ad emic Publishers : Boston, 1995. [9] Kim G-H, Park Y-S. An automated param eter selection proc edure for finite element m odel updat ing and its applications. J ourna l of So und and V ibration. V ol. 309, No. 3-5, pp. 7 78-793, 2 008 [10] MacKa y DJC. Inf orm ation theory, inf erence, a nd lear ning algor ithms . Cam bridge Universit y Press, 200 3. [11] Mar wala T, Sibisi S . Finite Elem ent Model Updating usin g Bayesian Framework and Modal Properties. Journal of Aircraft. Vo l. 42, No. 1, pp 275- 278. Janu ary-F ebruary 200 5. [12] Marwala T. Finite elem ent model upda ting using wavele t da ta and genetic algori thm . J ournal of Aircraf t, Vo l. 39, No. 4, pp 709-710, Jul y –August 200 2. [13] M arwala T. Finite element model updating using response surf ace m ethod. Proceed ings of the 45th AIAA/ASME/ ASCE/AH S/ASC Struc tures, Structur al D ynamic s & Materials Co nf erence, Palm Springs, C alif ornia, USA, April 20 04, AI AA Pap er pp. 516 5-5173, 200 4-2005. [14] Muk herjee P, Park inson D and Li ddle AR. A ne sted sam pling algorithm for cosm ological m odel sel ection. Draft version . ArXiv: astr o-ph/0 508461v2. 11 Janu ary 2006. [15] Park inson D, Mukherj ee P and Liddle AR. A Bay esi an m odel s election analysis of W MAP3. ArX iv: a stro- ph/0605003 v2. 19 th June 2 006 [16] M urra y I. Advanc es in Mark ov C hain Monte Car lo m ethods. PhD thes is, Gatsb y com putationa l ne urosc ience unit, Univers it y College London, 2007. [17] Muto M, Beck LJ. Bayesian updating and model class selection for hysteretic structural m odels using stochastic s imulation. J ournal of Vibration a nd Contro l, Vol. 1 4, No. 1-2 , pp 7-34, 2008 . [18] Sk illing J . Nested sampling for Bayesia n com putations. In J. M. Bernar do, M. J. Ba yarri, J. O . Berger, A. P. Dawid, D . H eck erman, A. F. M. Sm ith, and M. W es t, editors, Ba yesian St atistics 8, Proceed ings of th e Valencia / ISBA 8th W orld Meet ing on Ba yesian Stat istics. Ox ford Uni versit y Press, 2007. [19] Titurus B and Fris well MI. Regul arization in m odel updating. Inter national Journa l for numer ical m ethods in engineerin g.Vol. 75, pp 440-478, 2 008. [20] T rotta R. A pplications of Bayesian m odel selecti on to cosm ologica l param eters. ArXiv : as tro-ph/054 022v3. 2007.