MULTIVARIATE BAYESIAN REGRESSION ANALYSIS APPLIED TO PSEUDO-ACCELERATION ATENUATTION RELATIONSHIPS

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1 h 4 th World Confrnc on Earthquak Enginring Octobr -7, 8, Bijing, China MULIVARIAE BAYESIAN REGRESSION ANALYSIS APPLIED O PSEUDO-ACCELERAION AENUAION RELAIONSHIPS D. Arroyo and M. Ordaz Associat Profssor, Dpto. D Matrials, Univrsidad Autónoma Mtropolitana, México DF, México Profssor, Instituto d Ingniría, Univrsidad Nacional Autónoma d México, México DF, México arsda@corro.azc.uam.mx, MOrdaz@iingn.unam.mx ABSRAC : h application of a linar multivariat Baysian rgrssion modl to comput psudoacclration (SA) attnuation rlationships is prsntd. h modl is abl to includ th corrlation btwn obsrvations for a givn arthquak, th corrlation btwn SA ordinats of diffrnt priods and th corrlation btwn cofficints of th rgrssion modl. hrough comparisons btwn rsults obtaind with th last-squars mthod and th on-stag maximum-liklihood mthod, th advantags of th Baysian modl ar discussd. KEYWORDS: Baysian analysis, attnuation rlations, ground motions. INRODUCION In th past, mpirical attnuation rlationships of SA wr fit to availabl data trough th last squars mthod. Svral authors obsrvd that in som cass th attnuation of SA with distanc could not b corrctly dtrmind with this mthod sinc it disrgardd th corrlation btwn obsrvations rcordd at diffrnt sits for a givn arthquak (Campbll 98 and Joynr and Boor). Indd, th two-stag rgrssion mthod and th on-stag maximum liklihood mthods wr dvlopd to solv this problm (Joynr and Boor 993). Also, attnuation rlationships hav bn drivd through univariat Baysian analysis (Vnziano and Hidari 985, Ordaz t al. 994 and Rys 999). Ordaz t al. (994) discussd th advantags of th Baysian analysis with rspct to th last squars mthod. h rsults that w prsnt in this papr can b considrd an xtnsion of th original work of Ordaz t al. (994), sinc w also usd a Baysian approach. Nvrthlss, our modl is mor gnral and is abl to includ th corrlation btwn obsrvations rcordd at diffrnt sits for a givn arthquak, th corrlation btwn SA ordinats at diffrnt priods, and th corrlation btwn rgrssion cofficints. In this papr w brifly compar th rsults obtaind with rsults obtaind with th last-squars and th on-stag maximum liklihood mthods. For th comparisons w us as bnchmark a st of synthtic SA spctra with prdfind statistical paramtrs. In ordr to fulfill th lngth rquirmnts of th confrnc w hav only includd som basic findings of svral rsults that w hav obtaind with th Baysian modl prsntd. Othr issus such as convrgnc of th Gibbs sampling mthod, a sound discussion on how th prior information should b dfind whn working with actual ground motions, diffrncs btwn multivariat and univariat analysis and thir possibl implications on th stimation of sismic paramtrs through attnuation rlationships will b prsntd in two manuscripts which ar in prparation for a spcializd journal.. HE REGRESSION MODEL For a givn, th standard shap shown in quation () was adoptd as attnuation modl for rgrssion analysis. ( ) α ( ) + α ( )( M 6) + α ( )( M 6) + α ( ) ( R) + α ( )R y = w 3 w 4 ln 5 (.) Whr stands for structural priod, y is th natural logarithm of SA, M w is th momnt magnitud, R is

2 h 4 th World Confrnc on Earthquak Enginring Octobr -7, 8, Bijing, China som masur of th distanc to th fault ara and α i ar th cofficints dtrmind by rgrssion analysis. Although in this papr w hav usd th function shown in quation (.) as attnuation modl, th procdur prsntd can b radily applid to othr linar forms of attnuation rlationships. h multivariat rgrssion modl considrd is Y = X α + E, whr symbol stands for transpos, Y is a known n o x n matrix which includs n o obsrvations of y for th n priods considrd, X is a known n o x n p matrix which is composd by th n o obsrvations of th n p paramtrs considrd in th modl (not that according to quation (.) th lmnts of th first column of th matrix X ar qual to unity), α is a unknown n x n p matrix which compriss th cofficints dtrmind by rgrssion analysis (ach row of α contains th α i cofficints for a givn ) and E is a unknown n o x n matrix which is comprisd by th rgrssion rsiduals. It is assumd that th lmnts E ar corrlatd, normally distributd random variabls with zro man. h corrlation btwn lmnts of E is dfind through an unknown n o n x n o n matrix Ω =Φ Σ, whr Φ is an unknown n o x n o matrix which accounts for th corrlation btwn th rows of Y, Σ is an unknown n x n matrix which accounts for th corrlation btwn spctral ordinats, and th symbol stands for Kronckr product. 3. HE ONE-SAGE MAXIMUM LIKELIHOOD MEHOD For th modl dscribd in last sction, th liklihood of Y is dfind in quation (3.), whr r dnots trac and th symbol stands for proportionality, sinc w hav omittd th normalization constant. [ ] n n ( α, Σ, Φ, X ) Σ Φ xp r Φ ( Y Xα ) Σ ( Y Xα ) L Y (3.) Following Joynr and Boor (993) w considrd that th lmnts of E, say ε ij, can b xprssd as th sum of arthquak-to-arthquak variability (ε ) and rcord-to-rcord variability (ε r ). In addition, th following considrations wr mad: ) For a givn arthquak and a givn sit, th cofficint of corrlation btwn rsiduals for two diffrnt priods, say and, is qual to ρ, ) For a givn arthquak th cofficint of corrlation btwn rsiduals for th sam priod at diffrnt sits is qual to γ. 3) For a givn arthquak th cofficint of corrlation btwn rsiduals for two diffrnt priods, say and, at diffrnt sits is qual to γ ρ,. 4) Rsiduals rlatd to diffrnt arthquaks ar indpndnt. According to ths assumptions Φ is a block diagonal matrix dfind in (3.) whr n is th numbr of arthquaks considrd, and th squard submatrix φ i rlatd to arthquak i is dfind in (3.3). h siz of φ i is th numbr of rcords of th arthquak i. Not that γ is qual to paramtr γ in Joynr and Boor (993) whil ρ, is th cofficint of corrlation btwn spctral ordinats SA for a givn pair of priods, namly and. In summary, th sam structur of matrix Φ usd by Joynr and Boor (993) for th univariat cas was usd in th multivariat cas.

3 h 4 th World Confrnc on Earthquak Enginring Octobr -7, 8, Bijing, China φ L φ L Φ = M M O M L φ n (3.) γ L γ γ L γ φ = i M M O M γ γ L (3.3) For a givn γ th valus of α and Σ which maximiz th liklihood ar th wll known wightd last squars stimators dfind by quations (3.4) and (3.5). ( X Φ X ) X Φ Y ˆ α = (3.4) Σ= ˆ ( Y X ˆ α ) Φ ( Y X ˆ α ) n In th maximum liklihood mthod, th valu of γ which maximizs th liklihood is found itrativly and th valus of α and Σ for th rgrssion analysis ar th valus rlatd to th γ of maximum liklihood. 4. HE BAYESIAN MODEL In th Baysian approach α, Σ and Φ ar rgardd as matrix random variabls with known joint prior dnsity p(α, Σ, Φ). his prior dnsity is updatd through Bays thorm and hnc th postrior dnsity is givn by th product btwn th liklihood and th prior dnsity. In standard Baysian analysis, thr typs of p(α,σ, Φ) ar commonly usd: vagu dnsitis, conjugat dnsitis and gnralizd conjugat dnsitis. Vagu dnsitis ar usd whn prior knowldg about paramtrs is diffus, whil conjugat and gnralizd conjugat dnsitis ar usd whn prior information about paramtrs is availabl. A mor dtaild dscription of ach family of probability dnsity functions and thir implications in th rgrssion analysis can b found lswhr (s for xampl, Bromling 985, Row ). In this papr w adoptd a gnralizd conjugat probability dnsity function as basic dnsity. Howvr, in ordr to kp th structur of Φ shown in quation (3.) w usd a scalar bta dnsity for γ, thus th prior dnsity of Φ is not of standard form. hs dnsitis wr chosn sinc w found that thy wr suitabl to proprly includ our prior knowldg of th paramtrs in th rgrssion analysis. (3.5) h prior joint probability dnsity usd is givn by p( α Σ, Φ) = p( α ) p( Σ) p( Φ) V,, whr α V = vc(α). Following Row (), w assum that th prior dnsity of α V is th Normal dnsity dfind in quation (4.) with man α V and covarianc matrix Δ. hus, α V =vc(α ) whr α is th prior xpctd valu of α and th positiv n n P x n n P matrix Δ is th prior covarianc matrix of α V. p( α ) Δ ( ) Δ ( ) V xp (4.) V

4 h 4 th World Confrnc on Earthquak Enginring Octobr -7, 8, Bijing, China For Σ w usd as prior dnsity th invrtd Wishart dnsity (Row ) shown in quation (4.) with paramtrs ν and Q. p ν ( Σ) Σ xp r[ Σ Q] (4.) h positiv n x n matrix Q can b computd from th prior xpctd valu of Σ as it is shown in quation (4.3): ( n ) Σ Q ν (4.3) = whr Σ is th prior xpctd valu of Σ. Not that Q also dpnds on th scalar ν, which is a masur of our dgr of crtainty on Σ. In ordr to giv a finit valu to th varianc of th lmnts of Σ th valu of ν should b gratr than (n +4); th largr th valu of ν, th gratr th dgr of crtainty on Σ. In Baysian analysis usually an invrtd Wishart dnsity is also usd for Φ (Row ). Nvrthlss, if it is dsird that Φ has th structur shown in quation (3.) th invrtd Wishart dnsity cannot b usd. W not that, for a givn st of data, Φ is only a function of γ, hnc w dcidd to us a scalar bta dnsity for γ according to quation (4.4). a ( ) b γ ( γ ) p γ (4.4) whr paramtrs a and b can b computd from th prior xpctd valu of γ and its prior standard dviation. In summary, th prior information about rgrssion paramtrs is includd in th analysis through α V, Δ, Q, v, a and b, which ar known as hyprparamtrs, and quations (4.) to (4.4). Multiplying th prior joint dnsity function by th liklihood function w obtain th postrior joint dnsity of th rgrssion paramtrs: ( no + ν ) n a (,,, ) b Σ Σ Φ ( ) p α γ X Y γ γ xp ( α ) Δ ( ) V xp r[ Σ (( Y Xα ) Φ ( Y Xα ) + Q )] (4.5) his joint dnsity should b marginalizd in ordr to obtain th postrior marginal xpctd valus of α, Σ and γ. Howvr, for this dnsity, it is not possibl to obtain marginal distributions in an analytical closd form. Nvrthlss, marginal xpctd valus can b numrically computd through th stochastic intgration mthod known as Gibbs sampling. 5. SYNHEIC DAA In ordr to assss th prformanc of th last-squars mthod, th on-stag maximum liklihood mthod and th Baysian mthod w gnratd diffrnt sts of synthtic SA spctra with prdfind statistical proprtis. W considrd 5 structural priods ranging btwn and 5. sconds. W gnratd 6 sts of synthtic spctra assuming th numbr of arthquaks shown in tabl. It was considrd that th momnt magnitud (M w ) of th vnts follows a modifid Gutnbrg-Richtr distribution, with minimum valu of M w =6, maximum valu of M w =8. and β=.

5 h 4 th World Confrnc on Earthquak Enginring Octobr -7, 8, Bijing, China Also, w considrd that ach arthquak was rcordd at th numbr of sits shown in tabl. W assum that R followd a uniform distribution btwn 5 and 4 Km. Hnc, th numbr of rcords for a givn st is th product btwn th numbr of arthquaks and th numbr of sits. abl Synthtic sts of SA spctra usd St Earthquaks Sits n o St Earthquaks Sits n o h prdfind statistical proprtis of th st of ground motions wr α p, Σ p and Φ p. W st α p as th valu of th attnuation modl proposd by Rys (999) for th CU station of Mxico City. h diagonal trms of Σ p wr st as th varianc of th rsiduals (σ ) rlatd to th Rys (999) modl, whil th off-diagonal trms wr computd through th quation proposd by Bakr and Cornll (6) to stimat th cofficint of corrlation ρ,. For Φ p w usd th structur shown in quation (3.) with γ =.34 which is a valu that w infr from th rsults prsntd by Joynr and Boor (993). hus, givn th numbr of arthquaks and th numbr of rcords shown in tabl, a n o x n matrix random variat from th matrix normal distribution was gnratd. h man valu of th distribution was computd with α p with a matrix covarianc dfind by Φ p Σ p. Not that rgardlss of th st considrd, α p, Σ p and Φ p rprsnt th statistical proprtis of th ntir population of SA spctra; hnc ths paramtrs wr usd as bnchmarks for th rgrssion analysis prsntd in th following sctions. 6. RESULS FOR ONE-SAGE MAXIMUM LIKELIHOOD MEHOD Figur shows a comparison btwn rgrssion paramtrs obtaind with th on-stag maximum-liklihood mthod and th bnchmarks. Although not shown, th rsults wr vry similar to thos obsrvd for th last squars mthod. In gnral, th on-stag maximum liklihood mthod is not abl to attain th bnchmark valus, xcpt for α and σ. 7. RESULS FOR HE BAYESIAN MEHOD 7. Prior Information h lmnts of matrix α wr st as follows. With th amplitud Fourir spctra dfind by th Brun s modl and using random vibration thory, w obtaind SA spctra rlatd to svral valus of M w and R (McGuir and Hanks 98). hn, w applid th last squars mthod to comput th valu of α. his implis that a priori w bliv that th attnuation of SA spctra could b proprly charactrizd by th Brun s modl. Sinc th trms α and α 5 in quation (.) dpnd on sit ffcts (Ordaz t al. 994) w assignd to thir varianc a larg valu with rspct to thir prior xpctd valu. his implis that α and α 5 could attain any valu which yilds th bst fit in th rgrssion analysis. In th cas of α, α 3 and α 4 w assignd to thir covarianc a valu that implis a cofficint of variation of.7, as it was usd in a prvious study (Ordaz t al. 994). W st th diagonal lmnts of Σ qual to.49, which mans that a priori w bliv that th standard dviation of th rsiduals is qual to.7 indpndntly from. In addition, th off-diagonal trms wr dfind through th cofficint of corrlation shown in quation (7.): ( q ) ρ (7.), xp =

6 h 4 th World Confrnc on Earthquak Enginring Octobr -7, 8, Bijing, China According to quation (7.) th cofficint corrlation varis from unity whn priods ar qual to narly.5 whn th diffrnc btwn priods is about 3 sconds and for q=.. It must b acknowldgd that quation (7.) is quit arbitrary. his corrlation structur was cratd only for th synthtic xampl prsntd in this papr. Whn working with actual ground motions it could b mor rasonabl to us th corrlation cofficints dfind by Bakr and Cornll (6) as prior valus. In th xampl prsntd in this papr w dcidd not to us th quations of Bakr and Cornll (6) as prior information sinc thy wr usd to gnrat th synthtic data. α 5 α 5 α (s) (s) (s) 5 α 4 α 5 σ (s) (s) (s) 5 Figur Rsults for maximum liklihood mthod (Symbols: black circls n o =, whit circls n o =5, black triangls n o =, whit triangls n o =, black squars n o =5, whit squars n o = and thick lin bnchmark) h dgr of crtainty of Σ dpnds on paramtr ν; as it has bn discussd, th largr th valu of ν, th largr th dgr of crtainty on Σ. h postrior valu of Σ can b xprssd as a wightd avrag btwn prior information and th conditional wightd last squars stimat. h wighting factors ar ν/(n+ν) for th prior information and n/(n+ν) for th conditional wightd last squars stimat. In th computations w hav usd a valu qual to 57 (th minimum rquird for n =5 in ordr to giv a finit valu to th covarianc of Σ ). It mans that w bliv that Σ is uncrtain. Finally, our prior information of γ is vagu, so w hav st a=b=.5, which is a vry flat dnsity, with xpctd valu of.5, in ordr to forc γ to tak th valu which yilds th bst fit to data. 7. Rsults h rsults obtaind with th Baysian modl ar prsntd in figur. Convrsly to what happnd with th othr modls, th Baysian was abl to attain bnchmark valus, xcpt for paramtr α 3. Not that with n o btwn and vry accurat stimats of th rgrssion paramtrs ar obsrvd. Although not shown, small valus of standard dviations ar obsrvd in th Baysian modl, which is an advantag ovr th othr

7 h 4 th World Confrnc on Earthquak Enginring Octobr -7, 8, Bijing, China mthods, sinc scattrs of th Baysian rgrssion paramtrs ar smallr than thos obsrvd with othr mthods. α 5 α 5 α (s) (s) (s) 5 α α (s) (s) (s) 5 Figur Rsults for Baysian mthod (Symbols: black circls n o =, whit circls n o =5, black triangls n o =, whit triangls n o =, black squars n o =5, whit squars n o = and thick lin bnchmark) W notd that th information containd in data was not nough to compltly dfin paramtr α 3 ; in othr words, paramtr α 3 has littl ffct on y and almost any valu could hav bn usd. Hnc, th Baysian mthod, instad of lading to any valu of α 3 (such as th othr mthods) lads to valus of α 3 that ar clos to its prior valu. Also, it was obsrvd that accurat stimats of γ wr obtaind for n o gratr than 5. In spit of our us of a prior valu of γ =.5 th data shiftd th prior valu to th corrct valu of γ. On th othr hand, to obtain accurat stimats of γ valus of n o gratr than wr rquird for th on-stag maximum liklihood mthod. 8. DISCUSSION AND CONCLUSIONS W notd that whil valus of M and R li in th rangs obsrvd in th sampl, rsults for th thr mthods yild th sam lvl of accuracy, vn whn som cofficints sm thortically unaccptabl. Howvr, w dcidd to compar th prdictd SA spctra, obtaind with cofficints associatd to n o = and diffrnt rgrssion mthods, with th corrsponding bnchmark spctra. W choos M=7 and four diffrnt valus of R: and 5 Km. hs valus ar out of th rang of data containd in th synthtic st; thus, this comparison can b rgardd as an valuation of th possibility of xtrapolating th rsults from th rgrssion analysis. h rsults ar summarizd in figur 3. As can b obsrvd, only th Baysian rgrssion yilds accptabl rsults. On th othr hand, th last squars and maximum liklihood mthods might lad to vry inaccurat rsults. Not that for R=, larg diffrncs ar obsrvd for th last-squars and maximum-liklihood mthods, in spit of th fact that this distanc is only % lowr than th minimum valu of R includd in th synthtic st. σ

8 h 4 th World Confrnc on Earthquak Enginring Octobr -7, 8, Bijing, China W hav prsntd a linar multivariat Baysian rgrssion mthod which includs th corrlation btwn obsrvations for a givn arthquak, th corrlation btwn SA ordinats at diffrnt priods, and th corrlation btwn cofficints of th rgrssion modl. SA(cm/s ) M= 7, R= SA(cm/s ) 8 M= 7, R= (s) (s) 5 Figur 3 Comparison btwn SA spctra obtaind through diffrnt mthods. (Symbols: black circls last squars mthod, whit circls maximum liklihood mthod, black triangls Baysian mthod and thick lin bnchmark valu) hrough comparisons btwn rsults obtaind with th last squars and th maximum liklihood mthods w hav shown than multipl solutions clos to minimum rror could xist and that th Baysian mthod could b usd to obtain rgrssion paramtrs consistnt with sismological thory. In addition, for th synthtic xampl prsntd, it is shown that attnuation rlationships obtaind through Baysian analysis yild mor accurat rsults than othr mthods whn attnuation rlationships ar xtrapolatd. Howvr, th Baysian mthod rquirs significantly mor analytical and computational work than traditional mthods. REFERENCES Bakr, J. W. and Cornll, A. C. (6). Corrlation of rspons spctral valus for multi-componnt ground motions, Bull. Sism. Soc. Am. 96, 5-7 Bromling, L. D. (984). Baysian analysis of linar modls, Marcl Dkkr Inc., Nw York, 454 pp. Boor, D. M. (983). Stochastic simulation of high-frquncy ground motions basd on sismological modls of th radiatd spctra, Bull. Sism. Soc. Am. 73(6), Campbll, K. W. (98). Nar-sourc attnuation of pak horizontal acclration, Bull. Sism. Soc. Am. 7, 39-7 Joynr, W. B. and Boor, D. M. (993). Mthods for rgrssion analysis of strong-motion data, Bull. Sism. Soc. Am. 83(), Ordaz, M., Singh, S. K. and Arciniga, A. (994). Baysian attnuation rgrssions: an application to Mxico City, Gophys. J. Int. 7, Rys, C. (999). El stado límit d srvicio n l disño sísmico d dificios, Ph. D. hsis, School of Enginring, UNAM. Row, D. B. (). Multivariat Baysian statistics: modls for sourc sparation and signal unmixing, Chapman & Hall/CRC, Nw York, 39 pp.