A Differential Evaluation Markov Chain Monte Carlo algorithm for Bayesian Model Updating M. Sherri a, I. Boulkaibet b, T. Marwala b, M. I.

Transcription

1 A Dfferental Evaluaton Markov Chan Monte Carlo algorth for Bayesan Model Updatng M. Sherr a, I. Boulkabet b, T. Marwala b, M. I. Frswell c, a Departent of Mechancal Engneerng Scence, Unversty of Johannesburg, PO Box 54, Auckland Park 006, South Afrca. b Insttute of Intellgent Systes, Unversty of Johannesburg, PO Box 54, Auckland Park 006, South Afrca. c College of Engneerng, Swansea Unversty, Bay Capus, Swansea SA 8EN, Unted Kngdo. Abstract: The use of the Bayesan tools n syste dentfcaton and odel updatng paradgs has been ncreased n the last ten years. Usually, the Bayesan technques can be pleented to ncorporate the uncertantes assocated wth easureents as well as the predcton ade by the fnte eleent odel (FEM) nto the FEM updatng procedure. In ths case, the posteror dstrbuton functon descrbes the uncertanty n the FE odel predcton and the experental data. Due to the coplexty of the odeled systes, the analytcal soluton for the posteror dstrbuton functon ay not exst. Ths leads to the use of nuercal ethods, such as Markov Chan Monte Carlo technques, to obtan approxate solutons for the posteror dstrbuton functon. In ths paper, a Dfferental Evaluaton Markov Chan Monte Carlo (DE- MC) ethod s used to approxate the posteror functon and update FEMs. The an dea of the DE-MC approach s to cobne the Dfferental Evoluton, whch s an effectve global optzaton algorth over real paraeter space, wth Markov Chan Monte Carlo (MCMC) technques to generate saples fro the posteror dstrbuton functon. In ths paper, the DE-MC ethod s dscussed n detal whle the perforance and the accuracy of ths algorth are nvestgated by updatng two structural exaples. Keywords: Bayesan odel updatng; Markov Chan Monte Carlo; dfferental evoluton; fnte eleent odel; posteror dstrbuton functon.. Introducton Durng the last thrty years, the applcaton of the fnte eleent ethod (FEM) [-3] has exponentally ncreased where ths nuercal technque has becoe one of the ost popular engneerng tools n systes odellng and predcton. In the doan of structural dynacs, the FEM tools are wdely appled to odel coplex systes where ths technque can produce results wth hgh accuracy, especally when the odelled syste s sple. However, the results attaned by the FEM can be relatvely naccurate and the satches between the FEM results and the results attaned fro experental studes are relatvely sgnfcant. Ths s due to the errors assocated wth the odellng process as well as the coplexty of odelled structure, whch ay reduce the accuracy of the odellng process. Consequently, the odel obtaned by an FEM needs to be updated to reduce the errors between the experental and odelled outputs. The procedure of nzng the dfferences between the nuercal results and the easured data s known as the fnte eleent odel updatng (FEMU) [4, 5], where the FEMU ethods can be dvded nto two an classes. In the frst class, whch s the drect ethods, we equate the experental data drectly to the FEM outputs resultng n a procedure that constrans the updatng to the FE syste atrces (ass, stffness) only. Ths knd of approach ay produce non-realstc results where the resultng updatng paraeters ay not have physcal eanng. In the second class, whch s also known as the teratve (or ndrect) approaches, the FEM outputs are not drectly equated to the experental data, but nstead, an objectve functon s ntroduced and teratvely nsed to reduce the errors between the analytcal and experental results. Thus, we vary the syste atrces and the odel output durng the nsaton process, and realstc results are often expected at the end of the updatng process.

2 Generally, several sources of uncertanty are assocated wth the odellng process, such as the atheatcal splfcatons ade durng the odellng, where ths knd of uncertanty ay affect the accuracy of the odellng process. Moreover, the nose that contanates the experental results ay also have a sgnfcant pact on the updatng process. To deal wth such uncertanty probles, another class of ethods called the uncertanty quantfcaton ethods accoplshes the updatng process. The ost coon uncertanty quantfcaton ethod s known as the Bayesan approach n whch the unknown paraeters and ther uncertanty are dentfed by defnng each unknown paraeter wth a probablty densty dstrbuton (PDF). Recently, the use of the Bayesan ethodology has assvely ncreased n the doan of syste dentfcaton and uncertanty quantfcaton. In ths approach, the uncertantes assocated wth the odelled structure are expressed n ters of probablty dstrbutons where the unknown paraeters are defned as a rando vector wth a ult-varable probablty densty functon, and the resultng functon s known as the posteror PDF. Solvng the posteror PDF helps n dentfyng the unknown paraeters and ther uncertantes. Unfortunately, the posteror PDF cannot be solved n an analytcal way for suffcently coplex probles whch s the case for the FEMU probles snce the search space s usually nonlnear and hgh densonal. In ths case, saplng technques are eployed to dentfy these uncertan paraeters. The ost recognsed saplng ethods are these related to Markov chan Monte Carlo (MCMC) ethods. Generally, the MCMC ethods are very useful tools that can effcently cope wth large search spaces and generate saples fro coplex dstrbutons. These ethods draw saples wth an eleent of randoness whle beng guded by the values of the posteror dstrbuton functon. Then, the drawn saples are accepted or rejected accordng to the Metropols crteron. Unfortunately, the updated odels, wth relatvely large coplextes, ay have ultple optal (or near optal) sple MCMC algorths cannot easly dentfy soluton, and ths. In ths paper, another verson of the MCMC algorths, known as the Dfferental Evoluton Markov Chan (DE-MC) [6, 7] algorth, s used to update FEMs of structural systes. The DE-MC algorth cobnes the abltes of the dfferental evaluaton algorth [8, 9], whch s one of the genetc algorths for global optzaton, wth the Metropols-Hastngs algorth. In ths algorth, ultple chans are run n parallel, and the exploraton and explotaton of the search space n the current chan are acheved by the dfference of two randoly selected chans, ultpled by the value of the dfference wth a preselected factor and then the result s added to the value of the current chan. The value of the current chan s then accepted or rejected accordng to the Metropols crteron. In ths paper, the effcency, relablty and the ltatons of the DE-MC algorth are nvestgated when the Bayesan approach s appled for FEMU. Ths paper s organzed as follows: n the next secton, the Bayesan forulatons are ntroduced. Secton 3 descrbes the DE-MC algorth whle secton 4 presents the results when a sple ass-sprng structure s updated. Secton 5 presents the updatng results of an unsyetrcal H-shaped Structure. The paper s concluded n secton 6.. Bayesan forulatons In ths paper, the Bayesan approach s adopted to copute the posteror dstrbuton functon n order to update the FEMs. The posteror functon can be represented by Bayes rule [0-4]: P(θ D, M) P(D θ, M) P(θ M) (.) where M descrbes the odel class for the target syste where each odel class M s defned by certan updatng paraeters θ Θ R d. The experental data D of the structural syste, whch s represented by the natural frequences f and ode shapes φ, are used to prove the FEM results. P(θ M) s the pror probablty dstrbuton functon (PDF) that represents the ntal knowledge of the uncertan paraeters gven a specfc odel M, and n the absence of the easured data D. The functon P(D θ, M) s known as the lkelhood functon and represents the dfference between the experental data and the FEM results. Fnally, the probablty dstrbuton functon P(θ D, M) s the posteror functon of the unknown paraeters gven a odel class M and the easured data D. The odel class M s used only when several classes are nvestgated for both odel updatng and odel selecton. In ths paper, only one odel class s consdered, and therefore, the ter M s otted n order to splfy the Bayesan forulatons. In ths paper, the lkelhood functon s gven by:

3 P(D θ) = ( π N ) β c N = f exp ( β N c (f f f ) ) (.) where N s the nuber of easured odes, β c s a constant, f and f are the th analytcal and easured natural frequences. The ntal knowledge of the updatng paraeters θ, whch s defned by a pror PDF, s gven by the followng Gaussan dstrbuton: P(θ) = (π) Q Q = α Q exp ( α θ θ 0 ) = (π) Q Q = exp ( (θ θ 0 )T Σ (θ θ 0 )) α (.3) where Q s the nuber of the uncertan paraeters, θ 0 represents the ean value of the updatng paraeters, α, =,, Q are the coeffcents of the updatng paraeters and the Eucldean nor s gven by the notaton:. After substtutng Eqs. (.) and (.3) nto the Bayesan nference defned by Eq. (.), the posteror P(θ D) of the unknown paraeters θ gven the experental data D s characterzed by: P(θ D) Z s (α, β c ) exp ( β N c (f f Q f ) α θ θ 0 ) (.4) where Z s (α, β c ) = ( π N ) β c N f = (π) Q/ Q = α (.5) Generally, the coplexty of the posteror PDF, whch depends on the odal paraeters of the analytcal odel, s related to the coplexty of the analytcal odel, and for certan relatvely coplex structural odels the analytcal results for the posteror dstrbuton are dffcult to obtan due to the hgh densonalty of the search space. In ths case, saplng technques [5, 0,, 3, 4] are the only practcal approaches n order to approxate the posteror PDF. The an dea of saplng technques s to generate a N s sequence of vectors {θ, θ,, θ Ns } and use these saples to approxate the future response of the unknown paraeters at dfferent te nstances. The ost recognzed saplng technques are Markov Chan Monte Carlo (MCMC) ethods [5, 3-8]. In ths paper, the cobnaton of one of the basc MCMC algorths, known as the Metropols-Hastng algorth, wth one of the genetc algorths, known as dfferental evoluton (DE), s used to generate saples fro the posteror PDF n order to update structural odels. 3. The Dfferental Evoluton Markov Chan Monte Carlo (DE-MC) ethod In ths paper, the DE and MCMC ethods, whch are extreely popular ethods n several scentfc doans, are cobned to prove the convergence of the saplng procedure. In ths approach, ultple chans are run n parallel n order to prove the accuracy of the updatng paraeters, whle these chans learn fro each other nstead of runnng all the chans ndependently. Ths ay prove the effcency of the searchng procedure and avod saplng n the vcnty of a local nu. The new chans are then accepted or rejected accordng to the Metropols-Hastngs crteron. The Metropols-Hastngs (M-H) [8, 9, 0] algorth s one of the coon MCMC ethods that can be used to draw saples fro ultvarate probablty dstrbutons. To saple fro the posteror PDF P(θ D), where θ = {θ, θ,, θ d } s a d-densonal paraeters vector, a proposal densty dstrbuton q(θ θ t ) s used to generate a

4 proposed rando vector θ gven the value at the prevous accepted vector θ t at the teraton t of the algorth. Next, the Metropols crteron s used to accept or reject the proposed saple θ as follows: α(θ θ t ) = n {, P(θ D) q(θ t θ ) P(θ t D)q(θ θ t ) } (3.) On the other hand, the Dfferental Evoluton (DE) [8] s a very effectve genetc algorth n solvng varous realworld global optzaton probles. As one of the genetc algorths, the DE algorth begns by randoly ntalsng the populaton wthn certan search area, and then these ntal values are evolved over the generatons n order to fnd the global nu. Ths can be acheved usng genetc operators such as: utaton, selecton, and crossover. By ntegratng the Metropols-Hastngs crteron wthn the search abltes of the DE algorth, the resulted MCMC ethod can be ore effcent n deternng where other chans can be eployed to create the new canddates for the current chan. In the DE-MC algorth, the new value of the chan s obtaned by a sple utaton operaton where the dfference between two randoly selected chans (dfferent fro the current chan) are added to the current chan. Thus, the proposal for each chan depends on a weghted cobnaton of other chans whch can be easly defned as [6, 7]: θ = θ + γ(θ a θ b ) + ε (3.) where θ represents the new proposed vector, θ s the current state of the -th chan, θ a and θ b are randoly selected chans, γ s a tunng factor that always take a postve value and can be set to vary between [0.4, ]. Note that the vectors: θ θ a θ b. Fnally, the nose ε, whch s defned as a Gaussan dstrbuton ε~n p (0, σ ) wth a very sall varance vector σ, s added to the proposed vector to avod degeneracy probles. The factor γ can be seen as the agntude that controls the jupng dstrbuton. The an dea of the DE-MC algorth can be llustrated n Fgure 3.b. (a): Metropols-Hastngs (b): DE-MC Fgure 3.: Proposed vector generaton n the M-H and DE-MC ethods Fgure 3. explans the way to generate proposed vectors for the M-H ethod (Fgure 3.a) and for the DE-MC ethod (Fgure 3.b). As llustrated, the dfference vector between the two randoly selected chans θ a and θ b represents the drecton of the new proposed vector, where ths dfference s ultpled by the factor γ to defne the ovng dstance. The ovng dstance s then added to the current chan θ to create the proposed vector. Note that, the DE-MC ethod has only one tunng factor γ n coparng to other versons of evolutonary MCMC ethods. Fnally, the new proposal θ of the -th chan s accepted or rejected accordng to the Metropols crteron whch s gven as:

5 r = n {, P(θ D) P(θ D) } (3.3) The steps to update FEMs usng the DE-MC algorth are suarzed as follows: - Intalze the populaton θ,ο, {,,, N}. - Set the tunng factor γ. In ths paper, γ =.38/ d and d s the denson of the updatng paraeters. 3- Calculate the Posteror PDF for all chans. 4- For all chans {,,, N}: 4. Saple unforly two rando vectors θ a,θ b where θ a θ b θ. 4. Saple the rando value ε wth sall varance ε~n p (0, σ ). 4.3 Calculate the proposed vector θ = θ + γ(θ a θ b ) + ε. 4.4 Calculate the Posteror PDF for the vector θ. 4.5 Calculate the Metropols rato r = n {, P(θ D) P(θ D) } 4.6 Accept the proposed vector θ θ wth probablty n (, r), otherwse θ s unchanged. 5- Repeat the steps 4. to 4.6 untl the nuber of saples requred s acheved. In next two sectons, the DE-MC perforance s hghlghted when two structural exaples are updated. 4. Applcaton : Sple Mass-Sprng syste In ths secton, a fve degrees of freedo ass-sprng lnear syste, as presented n Fgure 4., s updated usng the DE- MC algorth. Fgure 4.: The fve degrees of freedo ass-sprng syste The syste contans 5 asses connected to each other usng 0 sprngs (see Fgure 4.). The deternstc values of the asses are: =.7 kg, =.7 kg, 3 = 6. kg, 4 = 5.3 kg and 5 =.9 kg. The stffness of the sprngs are: k 3 = 300 N/, k 5 = 840 N/, k 7 = 00 N/, k 9 = 800 N/ and k 0 = 000 N/. The sprng stffnesses k, k, k 4, k 6, and k 8 are consdered as the uncertan paraeters where the updatng vector s: θ = {θ, θ, θ 3, θ 4, θ 5 } = {k, k, k 4, k 6, k 8 }.

6 Snce the DE-MC ethod s used for the updatng procedure, the populaton used by the algorth s selected to be N = 0. The updatng vectors are bounded by θ ax and θ n whch are set to {4800, 600,670, 3400, 750} and {300, 800,600, 800, 050}, respectvely. The tunng factor s set to γ =.38/ d whle d = 5, the ntal vector of θ s set to θ 0 = {4600, 580, 680, 300, 350} and the nuber of generatons (nuber of saples) s set to N s = The obtaned saples are llustrated n Fgure 4. whle the updatng paraeters, as well as the ntal and updated natural frequences, are shown n Tables 4. and 4., respectvely Fgure 4.: The scatter plots of the saples usng the DE-MC algorth Fgure 4. shows the scatter plots of the uncertan paraeters usng the DE-MC algorth. The confdence ellpses (error ellpse) of the saples are also shown n the sae fgures (n red) where these ellpses vsualze the regons that contan 95% of the obtaned saples. As expected, the fgure shows that the DE-MC algorth has found the hgh probablty area after only few teratons. Table 4. contans the ntal values, the nonal values and the updated values of the uncertan paraeters. The coeffcent of varaton (c.o.v) values, whch are estated by dvdng the standard devatons σ by the updated vectors θ (or μ ), are also presented n Table 4. and used to easure the errors n the updatng. It s clear that the obtaned values of the c.o.v when the DE-MC algorth s used to update the structure are sall and less than.5% whch eans that the DE-MC algorth perfored well and was able to dentfy the areas wth hgh probablty. Ths also can be verfed fro the sae table where the updatng paraeters are close the nonal values. Table 4.: The updatng paraeters usng DE-MC technque Unknown paraeters (N/) Intal Nonal values Error DE-MC (μ ) Error σ μ c.o.v %) θ θ θ θ θ

7 Table 4. contans the ntal, nonal and updated natural frequences. Furtherore, the absolute errors, whch are estated by f f f, the total average error (TAE), whch s coputed by TAE = N f N = f, N = 5, and the c.o.v values are also dsplayed. Obvously, the updated frequences obtaned by the DE-MC are better than the ntal frequences, and alost equal to the nonal frequences. f Table 4.: The updated natural frequences and the errors obtaned usng the DE-MC Modes Nonal Frequency Intal Frequency Error Frequency DE-MC c.o.v Error TAE The total average error of the FEM output was reduced fro.98% to 0.0%. On the other hand, the values of the c.o.v for all updated frequences are saller than 0.5% whch ndcates that the DE-MC technque effcently updated the structural syste. Fgure 4.3 shows the evaluaton of the total average error at each teraton. The TAE n Fgure 4.3 s obtaned as follows: frst, the ean value of the saples at each teraton s coputed as θ = E(θ) = j= θ where s the current teraton. Next, the ean value s used to copute the analytcal frequences of the FEM, and then the total average error s calculated as: TAE() = effcently after the frst 000 teratons. N f j fj N j= f j N s. As a result, t s clear that the DE-MC algorth converges 0 0 Total average error Iteraton Fgure 4.3: The evaluaton of the TAE usng the DE-MC ethod Fgure 4.4 llustrates the correlaton between the updatng paraeters where all paraeters are correlated (the values are dfferent fro zero). Moreover, the ajorty of these paraeters are weakly correlated (sall values <0.3) except the pars (θ, θ ) and (θ 4, θ 5 ) whch are hghly correlated (values >0.7).

8 .5 Correlaton Fgure 4.4: The correlaton between the updatng paraeters. In the next secton, the DE-MC ethod s used to update an unsyetrcal H-shaped alunu structure wth real experental data. 5. Applcaton : The FEMU of the unsyetrcal H-Shaped structural syste. In ths secton, the perforance of the DE-MC algorth s exaned by updatntg an unsyetrcal H-shaped alunu structure wth real easured data. The FEM odel of the H-shaped structure s presented n fgure (5.) where the structure s dvded nto eleents, and each eleent s odelled as an Euler-Bernoull bea. The locaton dsplayed by a double arrow at the ddle bea ndcates the poston of exctaton whch s produced by an electroagnetc shaker. An acceleroeter was used to easure the set of frequency-response functons. The ntal analytcal natural frequences are 53.9, 7.3, 08.4, 54.0 and Hz. In ths exaple, the oents of nerta I xx and the cross-sectonal areas A xx of the left, ddle and rght subsectons of the H-shaped bea are selected to be updated n order to prove the analytcal natural frequences. Thus, the updatng paraeters are: θ = {I x, I x, I x3, A x, A x, A x3, }. Fgure 5.: The Unsyetrcal H-Shaped Structure The rest of the H-shaped structure paraeters are gven as follows: The Young s odulus s set to N/ and the densty s set to 785 kg/ 3. The updatng paraeters θ are bounded by axu and nu vectors gven by: [ , , , , , ] and [ , , ,.6 0 4,.6 0 4, ], respectvely. These boundares are used to ensure that the updatng paraeters are physcally realstc. The nuber of saples s set to N s = 5000, the factor β c of the lkelhood

9 functon s set equal to 0, the coeffcents α of the pror PDF are set to σ where σ s the varance of the th uncertan paraeters, and σ = [5 0 8, 5 0 8, 5 0 8, 5 0 4, 5 0 4, ]. Fgure 5. llustrates the scatter plots of the updatng paraeters. The confdence ellpse that contans 95% of saples are also ncluded n the fgure. The updatng paraeters were noralzed by dvdng the paraeters by k = [0 8, 0 8, 0 8, 0 4, 0 4, 0 4 ]. As expected, the DE-MC algorth was able to fnd the area wth hgh probably after a few teratons. The rest of the updatng paraeters are shown n Table 5. as well as the ntal values, the c.o.v values and the updatng paraeters obtaned by the M-H algorth / k / k / k / k 3 Fgure 5.: The scatter plots of the saples usng the DE-MC algorth Table 5.: The ntal paraeters, the c.o.v values and the updatng paraeters usng the DE-MC and M-H algorths Intal DE-MC (μ ) σ μ M-H (μ ) σ μ θ θ θ θ θ θ

10 The results n Table 5. ndcate that the updatng paraeters obtaned by the DE-MC and M-H algorths are dfferent fro the ntal values whch eans that the uncertan paraeters have been successfully updated. Furtherore, the c.o.v values of the updatng paraeters obtaned by the DE-MC algorth are relatvely sall (<.5%) wth verfes that the algorth was able to dentfy the areas wth hgh probablty n a reasonable aount of te; however, the c.o.v obtaned by the M-H algorth are relatvely hgh (>3.08%) whch eans that the M-H algorth does not have the effcency of the DE-MC algorth. Table 5. llustrates the updatng frequences usng the DE-MC and M-H algorths, the errors and the c.o.v values. As expected, the analytcal frequences obtaned by the DE-MC algorth are better than the ntal frequences as well as the frequences obtaned by the M-H algorth. The DE-MC ethod has proved all natural frequences and reduced the total average error (TAE) fro 5.37% to.53%. Also, the c.o.v values obtaned by the DE-MC ethod are relatvely sall (<0.65%). Table 5.: Natural frequences, c.o.v values and errors when DE-MC and M-H technques are used for FEMU Modes Measured Frequency Intal Frequency Error Frequency DE-MC c.o.v Error Frequency M-H c.o.v Error TAE Correlaton Fgure 5.3: The correlaton between the updatng paraeters Fgure 5.3 shows the correlaton of the updatng paraeters usng the DE-MC paraeters where the ajorty of these paraeters are weakly correlated except the pars (θ, θ 5 ) and (θ 4, θ 5 ), where the correlaton between these pars are relatvely hgh (<0.7%).

11 The evaluaton of total average error after each accepted (or rejected) saple s llustrated n Fgure 5.4. The result ndcates that the DE-MC has a fast convergence rate and was able converge after 500 teratons Total average error Iteraton Fgure 5.4: The evaluaton of the TAE usng the DE-MC ethod 6. Concluson In ths paper, the Dfferental Evoluton Markov Chan (DE-MC) algorth s used to approxate the Bayesan forulatons n order to perfor a fnte eleent odel updatng procedure. In the DE-MC ethod, ult-chans are run n parallel whch allows the chans to learn fro each other n order to prove the saplng process where the jupng step depends on the dfference between randoly selected chans. Ths ethod s nvestgated by updatng two structural syste: the frst one s a fve DOF ass-sprng lnear syste and the second one s the unsyetrcal H-shaped alunu structure. In the frst case, the total average error was reduced fro.98% to 0.0%, whle n the second case, the FEM updatng of the unsyetrcal H-shaped structure, the total average error was reduced fro 5.37% to.53%. Also the DE-MC algorth appeared to have better results than the M-H algorth when the unsyetrcal H-shaped structure s updated. In further work, the DE-MC algorth wll be odfed and proved to nclude several steps such as the cross over and the exchange between the parallel chans. These changes ay further prove the saplng procedure. References [] Zenkewcz, O.C., & Taylor, R.L. The fnte eleent ethod, Vol. 3, London: McGraw-hll, 977. [] Hutton, D., Fundaentals of fnte eleent analyss, McGraw-Hll, 004. [3] Cook, R.D., Concepts and applcatons of fnte eleent analyss. John Wley & Sons, 007. [4] Frswell, M. I. and J.E. Mottershead, Fnte eleent odel updatng n structural dynacs. Vol. 38, Sprnger Scence & Busness Meda, 03. [5] Marwala, T., Fnte eleent odel updatng usng coputatonal ntellgence technques: applcatons to structural dynacs, Sprnger Scence & Busness Meda, 00. [6] Ter Braak, C.J., A Markov Chan Monte Carlo verson of the genetc algorth Dfferental Evoluton: easy Bayesan coputng for real paraeter spaces. Statstcs and Coputng, 6(3), pp , 006. [7] ter Braak, C.J. and J.A. Vrugt, Dfferental evoluton Markov chan wth snooker updater and fewer chans. Statstcs and Coputng, 8(4), pp , 008. [8] Storn, R. and K. Prce, Dfferental evoluton a sple and effcent heurstc for global optzaton over contnuous spaces. Journal of global optzaton, (4), pp , 997. [9] Prce, K., R.M. Storn, and J.A. Lapnen, Dfferental evoluton: a practcal approach to global optzaton, Sprnger Scence & Busness Meda, 006. [0] Bshop, C.M., Pattern recognton and achne learnng, sprnger, 006.

12 [] Boulkabet, I., Marwala, T., Mthebu, L., Frswell, M. I., & Adhkar, S. Saplng technques n Bayesan fnte eleent odel updatng, n Topcs n Model Valdaton and Uncertanty Quantfcaton, Vol 4. Sprnger. pp , 0. [] Yuen, K.-V., Bayesan ethods for structural dynacs and cvl engneerng. John Wley & Sons, 00. [3] Boulkabet, I., Mthebu, L., Marwala, T., Frswell, M. I., & Adhkar, S. Fnte eleent odel updatng usng the shadow hybrd Monte Carlo technque. Mechancal Systes and Sgnal Processng, Vol. 5, pp. 5-3, 05. [4] Boulkabet, I., Mthebu, L., Marwala, T., Frswell, M. I., & Adhkar, S. Fnte eleent odel updatng usng Haltonan Monte Carlo technques. Inverse Probles n Scence and Engneerng, 07. 5(7): p [5] Marwala, T. and S. Sbs, Fnte Eleent Model Updatng Usng Bayesan Fraework and Modal Propertes. Journal of Arcraft, Vol. 4(), pp , 005. [6] Bshop, C.M., Neural networks for pattern recognton. Oxford unversty press, 995. [7] Chng, J. & S.-S. Leu, Bayesan updatng of relablty of cvl nfrastructure facltes based on condton-state data and fault-tree odel. Relablty Engneerng & Syste Safety, Vol. 94(), pp , 009. [8] Metropols, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., & Teller, E., Equaton of state calculatons by fast coputng achnes. The journal of checal physcs, Vol. (6), pp , 953. [9] Boulkabet, I., Mthebu, L., Marwala, T., Frswell, M. I., & Adhkar, S., Fnte eleent odel updatng usng an evolutonary Markov Chan Monte Carlo algorth, n Dynacs of Cvl Structures, Vol.., Sprnger. pp , 05. [0] Marwala, T., I. Boulkabet, and S. Adhkar, Probablstc Fnte Eleent Model Updatng Usng Bayesan Statstcs: Applcatons to Aeronautcal and Mechancal Engneerng. John Wley & Sons, 06. [] Boulkabet, I., Fnte eleent odel updatng usng Markov Chan Monte Carlo technques, PhD thess, Unversty of Johannesburg, 04.