APPLICATION OF A SUPPORT VECTOR MACHINE FOR LIQUEFACTION ASSESSMENT
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1 38 Journal of Marne Scence and echnology, Vol., o. 3, pp (03) DOI: 0.69/JMS APPLICAIO OF A SUPPOR VECOR MACHIE FOR LIQUEFACIO ASSESSME Chng-Ynn Lee and Shuh-G Chern Key ords: A, CP, lquefacton, SVM. ABSRAC hs study presents a support vector machne (SVM)-based approach for predctng earthquake lquefacton. he SVM model database ncludes fve ndexes: earthquake magntude, total overburden pressure, effectve overburden pressure, q c values from cone penetraton tests (CP), and peak ground acceleraton. he proposed model as traned and tested on a dataset comprsng 466 feld lquefacton performance records and CP measurements. A grd search method th k-fold cross-valdaton as also used to verfy the feasblty. Compared th an artfcal neural netork (A) based method, the SVM-based method has the advantage of ncreased accuracy and smpler operaton. Expermental results sho that the proposed SVM approach can ncrease the classfcaton accuracy rate to a standard of 98.7%. I. IRODUCIO Lquefacton s one of the most destructve phenomena caused by earthquakes, and often occurs n loose, saturated sol deposts. Examples of lquefacton nclude the earthquakes n gata, 964; Alaska, 964; angshan, 979; Loma Preta, 989; Kobe, 995; urkey, 998; Ch-Ch, aan, 999; and Honshu, Japan, 0. In ve of the serous damage caused by earthquake-nduced lquefacton, geotechncal engneers are actvely engaged n the study of the sol lquefacton caused by earthquakes, and have developed many assessment methods for evaluatng sol lquefacton. Hoever, the hgh uncertanty n earthquake envronments and sol characterstcs makes t dffcult to choose a sutable emprcal equaton for regresson analyss. herefore, many scholars and experts have attempted to develop scentfcally Paper Submtted //; revsed 04/4/; accepted 05/8/. Author for correspondence: Shuh-G Chern (e-mal: sgchern@mal.ntou.edu.t). Department of Harbor and Rver Engneerng, atonal aan Ocean Unversty, Keelung, aan, R.O.C. derved analytcal models that are smpler, easer to mplement, and more accurate than tradtonal emprcal equatons for sol lquefacton analyss. Many of the exstng assessment methods ere developed from observatons of the behavor of stes durng earthquakes. Geotechncal engneers have often used the smple lquefacton analytcal model developed by SP-, because of ts computatonal speed and analytcal ablty. Based on recent mprovements n data processng and analytcal ablty, the cone penetraton test (CP) offers the advantage of fast, contnuous, and accurate sol parameter measurements. Related testng data has also contnued to accumulate. herefore, the potental of applyng CP to lquefacton research has gron sgnfcantly. hs study presents a relatvely ne soft computng method knon as a support vector machne (SVM) [, 5]. SVMs have been dely used n recent years n areas such as mage dentfcaton and facal recognton. An dentfcaton model that adopts SVM analyss s an effectve method for accurately predctng lquefacton, and can be used n practcal applcatons. Prevous studes have shon that the SVM method s a poerful and effectve tool for dealng th lquefacton problems, and s more accurate and relable than conventonal methods [8, 0]. II. OVER VIEW OF SVM hs secton presents the basc SVM concepts for typcal bnary classfcaton problems.. Lnear SVM A support vector machne, as presented by Vapnk [], s a machne learnng algorthm based on the statstcal learnng theory. he dagram n Fg. shos the basc concepts of ths approach. he crcles and the damonds n ths fgure represent to samples, and H s a labelng lne separatng the to samples. he H and H dashed lnes pass through the nearest samples to the labelng lne. he nearest data ponts used to defne the margn are called support vectors (SV), and the dstance beteen H and H s called the margn. he
2 C.-Y. Lee and S.-G. Chern: Applcaton of a Support Vector Machne for Lquefacton Assessment 39 H: x + b = H: x + b = 0 H: x + b = - y = - Support vectors Support vectors y = + Margn = L (, b, ) = y ( x + b) P α α = = yx b y+ α α α = = = (7) here α 0 ( =,,, ) are the Lagrangan multplers. he goal here s to fnd and b hch mnmzes, and the α hch maxmzes Eq. (7). hs can be done by dfferentatng L p th respect to and b and settng the dervatves to zero Fg.. Optmal hyperplane for a lnear SVM. separatng hyperplane H that has the maxmum dstance beteen the nearest data (.e., the maxmum margn) s called the optmal separatng hyperplane. As Fg. shos, the data patterns can be shon as {x, y }, =,,, k, here x R s an -dmensonal data vector th each sample belongng to ether of the to classes labeled as y {-, +}, and the decson functon (hyperplane) can be expressed as x+ b= 0 () here x s an nput vector, s an adaptve eght vector, b s a bas, and x s an nner product of and x. For the lnearly separable class, a separatng hyperplane for the to classes can be defned as x+ b, y=+ () x+ b, y= (3) Eqs. () and (3) can be combned nto y ( x + b) 0 (4) he goal of the SVM s to fnd and b for the optmal separatng hyperplane to maxmze the margn (Fg. ). Hence, the hyperplane that optmally separates the data s the one that mnmzes. he optmal separatng hyperplane can be obtaned by solvng the follong convex quadratc optmzaton problem []: Mnmze = (5) subject to y ( x ) + b, (6) he above equaton can be transformed nto the equvalent Lagrangan dual problem as Lb (,, α) = 0 = α y x (8) = = Lb (,, α) = 0 αy = 0 (9) b Based on Eq. (9), the thrd term on the rght hand sde of Eq. (7) s zero. Multplyng Eq. (8) by leads to = α = αα j j j = = j (0) y x y y x x Eq. (7) can then be reformulated as L subject to () D ( α) = α αα jyyjx xj = = j () = α y = 0 () α 0, () he problem s no re-cast as fndng the optmum Lagrangan multplers that maxmze the objectve functon Eq. () subject to Eq. (). hs s a convex quadratc optmzaton problem, and requres a quadratc program (QP) solver that returns α ι. he soluton α ι for the dual optmzaton problem determnes the parameter * and b * of the optmal hyperplane. hus, the optmal hyperplane decson functon can be rtten as * * * * f( x) = sgn( x + b ) = sgn α y x xj + b (3) = here sgn s the sgnum functon. If the result s postve, then t s classfed x as class, and classfed as class otherse.. Lnearly Inseparable SVM he soft margn method, hch ntroduces an addtonal cost functon assocated th msclassfcaton, s an appro-
3 30 Journal of Marne Scence and echnology, Vol., o. 3 (03) X x φ z 3 F x - Orgn b - ξ z z Fg.. Hyperplane through to lnearly nseparab classes. Fg. 3. Mappng from the data space X to the feature space F. prate ay to extend the SVM methodology to data that s not lnearly separable. Cortes and Vapnk [5] ntroduced postve slack varables ξ and a penalty factor C. As Fg. shos, data ponts on the ncorrect sde of the margn boundary have a penalty that ncreases th dstance. o reduce the number of msclassfcatons, modfy the constrants of Eq. (5) for the non-separable case as follos: mnmze l + C ξ (4) = subject to y ( x ) + b + ξ 0, (5) here ξ s called a slack varable used to account for the effects of msclassfcaton. C s called a penalty factor, a parameter defnes the trade-off beteen the number of msclassfcaton n the tranng data and margn maxmzaton. As before, reformulatng ths as a Lagrangan requres the mnmzaton of, b, and ξ, and the maxmzaton of α (here α 0): Lb (,, ξ, α, β) = + C ξ = y( x b) = = (6) α + + ξ + βξ subject to α, β 0 ( =,,..., ) (7) Dfferentatng L th, b, and ξ, and settng the dervatves to zero leads to Lb (,, ξ, α, β) = 0 = α y x (8) = = Lb (,, ξ, α, β) = 0 αy = 0 (9) b Lb (,, ξ, α, β ) = 0 C α β = 0 ξ (0) After substtutng these values n, L D has the same form as Eq. (0), Eq. (). Agan, maxmze L b yyxx D (,, ξ, α, β) = α αα j j j = = j subject to () () = α y = 0 () 0 α C, () he equatons are almost the same dual problem as before, th a slght dfference beng that the multplers α have an extra constrant. 3. onlnear Separable SVM he concepts can also be extended to the case of a nonlnear separatng hyperplane by mappng the nput space onto a hgh dmensonal space, x φ( x), here the data can be lnearly classfed (Fg. 3). he key property of ths mappng s that the functon φ must be subject to the condton that the dot product of the to functons φ(x ) φ(x j ) can be rtten as a kernel functon K(x, x j ) he decson functon then becomes f( x) = yαk( x, xj) + b (3) = Dfferent kernel functons can construct varous learnng machnes. Some typcal kernel functons are as follos: Lnear kernel: K( x, x ) = x x (4) j j d Polynomal kernel: K( x, x ) = ( γx x + r), γ > 0 (5) Radal bass functon (RBF): j j ( γ ) K( x, x ) = exp x x, γ > 0 (6) j j
4 C.-Y. Lee and S.-G. Chern: Applcaton of a Support Vector Machne for Lquefacton Assessment 3 Randomly Mxed Data Dvson nto k-folds st Learnng est Learnng Learnng Learnng st Learnng Learnng est Learnng Learnng Error Rate Accuracy rate % Accuracy rate % Fg. 4. A k-fold cross-valdaton procedure. Sgmod kernel: K( x, x ) = tan( γ x x + r) (7) j j In the questons above, γ, r and d are kernel parameters. III. CROSS-VALIDAIO Cross-valdaton s a technque for assessng ho the results of statstcal analyss can be generalzed to an ndependent dataset. hs technque s manly used n stuatons here the goal s predcton, and one ants to estmate ho accurately a predctve model ll perform n practce. hs study adopts a k-fold cross-valdaton technque that randomly parttons the orgnal sample nto k subsamples. A sngle subsample s retaned as valdaton data for testng the model, and the remanng k subsamples are used as tranng data. he cross-valdaton process s repeated k tmes (the folds), th each of the k subsamples used exactly once as the valdaton data. he k results from the folds can be averaged (or otherse combned) to produce a sngle estmaton. Fg. 4 provdes an example of a k-fold cross-valdaton procedure. he advantage of ths method over repeated random subsamplng s that all observatons are used for both tranng and valdaton, and each observaton s used for valdaton exactly once. he man draback of ths method s that t requres ntense computaton. Fg. 5 shos the k-fold cross-valdaton error versus k for a bg data set, and ndcates that a k value beteen 4 and 0 s a good trade-off: ncreasng ths value sgnfcantly ncreases computaton tme and does not sgnfcantly mprove results []. hus, ths study adopts 5-fold crossvaldaton. hs approach may not be useful n achevng hgh tranng accuracy, but t can prevent the over-fttng problem. IV. GRID SEARCH he grd search algorthm performs an exhaustve search through the parameter space of a learnng algorthm to solve the problem of model selecton (.e., fndng the optmal parameters for a dataset) k Fg. 5. A plot of k-fold cross-valdaton error vs. k []. Researchers have proposed four basc kernel functons for SVM models. Frst, decde hch one to use, and then choose the penalty C and kernel parameters. For example, there are to parameters for an RBF kernel: C and γ. Varous pars of (C, γ) values are tred th a grd search procedure and the one th the best cross-valdaton accuracy s chosen. estng exponentally grong sequences of C and γ s a practcal method for dentfyng good parameters (e.g., C = -4, -3.5,, 4 ; γ = -4, -3.5,, 4 ). V. APPLICAIOS OF SVM CLASSIFICAIO he case records n ths study ere evaluated usng the MALAB (R00a) program and tool box [, 6]. Fg. 6 shos the flochart of the proposed SVM system. he database ncludes 466 CP-based feld lquefacton records from over major earthquakes beteen 964 and 999. he data conssts of case records from Japan, 85 from Chna, 7 from Canada, 9 from the USA, and 34 from aan. hs represents 50 stes that lquefed and 6 stes that dd not lquefy. Fve parameters that ere recorded n all 466 stes are () earthquake magntude, M; () total overburden pressure, σ 0 ; (3) effectve overburden pressure, σ 0 ; (4) q c values from CPs; and (5) peak acceleraton, a max able summarzes the maxmum and mnmum values of each parameter. he parameter values for all 466 case records are presented n a paper rtten by Chern et al. [3]. he nput representng the lquefacton potental s gven a bnary value of for lquefed stes and a value of for non-lquefed stes. Before the datasets ere used to tran the SVM model, they ere preprocessed usng Eq. (8). Each parameter s normalzed beteen 0 and, th x x y = x max mn xmn (8) n hch y s a normalzed nput parameter, x s the orgnal nput parameter, and x max and x mn are the maxmum and mnmum parameters, respectvely.
5 3 Journal of Marne Scence and echnology, Vol., o. 3 (03) able. he maxmum and mnmum values of the reference datasets. Parameter M σ 0 (kpa) σ 0 (kpa) q c (kpa) a max (kpa) Max Mn ranng Data Decson Kernel k-fold Cross-Valdaton Data Preprocessng ormalzaton estng Data Accuracy (%) Cross-Valdaton Accuracy = 95.79% Best C = Best gamma = 3 Best C, gamma 5 0 logg logc Fg. 7. Parameters C and γ versus the accuracy rate. o mprove SVM classfcaton accuracy, the grd search procedure plays an mportant role n the performance of the SVM. Fg. 7 also shos that the parameters C and γ greatly affect the classfcaton accuracy of the SVM. 5 Learnng set est set VI. RESULS AD DISCUSSIO he procedure for usng the SVM s descrbed belo: Grd Search Best Parameter ranng Data SVM Model Accuracy Rate % Fg. 6. Flochart of the proposed SVM system. he man advantage of scalng s to avod attrbutes th greater numerc ranges domnatng those th smaller numerc ranges. Another advantage of ths method s to avod numercal dffcultes durng calculaton. A normal SVM model randomly selects kernel parameters usng a tral-and-error method [8, 0]. he grd search approach n ths study s an alternatve ay to fnd the best parameters for the SVM classfer. hs approach avods the over-fttng problem of the SVM model occurrng because of the mproper determnaton of these parameters. he RBF kernel s a reasonable frst choce for an SVM model [9]. Hence, the proposed SVM model as frst constructed by a radal bass functon (RBF) kernel. here are to parameters, C and γ, to be determned. After the grd search procedure, the optmal parameters th maxmal classfcaton accuracy ere selected. As shon n Fg. 7, the best (C, γ) s (.5, 5 ) th a cross-valdaton rate of 95.79%. In ths result, the optmal parameters are used to tran the SVM model to generate the fnal classfer.. ransform data to the format of the SVM package.. Conduct smple scalng on the data. 3. Consder the RBF kernel. 4. Use cross-valdaton and grd searchng to fnd the best parameters C and γ. 5. Use the best C and γ to tran the hole tranng set. 6. est. 7. Fnd the best accuracy rate. After the tranng procedure, the best (C, γ) s (.5, 5 ) th a cross-valdaton rate of 95.79%. Out of the 466 datasets used, only 6 cases ere msclassfed, achevng an overall classfcaton accuracy rate of 98.7%. In addton to verfyng the effectveness of the proposed method, ths study compares t th an A method n the reference [3, 4]. he A model proposed n that paper combnes fuzzy theory th a subtractve clusterng algorthm to form a fuzzy-neural netork system. o verfy the feasblty of the A model, ths study compares that A model th the B5 model employed n Goh [7] usng the same 09 data groups, ncludng 74 tranng data groups and 35 test data groups. he results of ths analyss are presented n able. he A-G5 model [3] performs better than Goh s optmal B5 model n both the tranng and testng segments. herefore, the 466 collected CP datasets are used n ths study to compare the SVM model th the A (C4, C4H6, C5, and C5) models [3]. Results are lsted n able 3, t shos that the SVM model acheves better results than the A models because of ts loer total error rate of.9%.
6 C.-Y. Lee and S.-G. Chern: Applcaton of a Support Vector Machne for Lquefacton Assessment 33 able. Result of the G5 model and B5 model [7]. Model Input varables o. of elements n every o. of Error o. of hdden neurons tranng cluster ranng estng otal error rate (%) B5 M, D 50, σ 0, q c, a max 5.75 G5 M, D 50, σ 0, q c, a max 53, 3, able 3. Comparson beteen SVM and A models [3]. Model Input varables o. of elements n every tranng cluster o. of hdden neurons o. of Msclassfed otal error rate (%) C4 M, σ 0, q c, a max 7,6,8, C4H6 M, σ 0, q c, a max 7,6,8, C5 M, σ 0, σ 0, q c, a max 90,93,4, C5 M, σ 0, σ 0, qc, a max 90,9,3, SVM M, σ 0, σ 0, q c, a max 6.9 able 4. he classfcaton accuraces versus C for dfferent kernel functons. C Lnear Poly RBF Sgmod As ndcated prevously, there are four types of basc kernel functons: lnear, RBF, second order polynomal, and sgmod. hs study employs the accuracy rate as a crteron to fnd the optmal kernel functon. able 4 shos the accuracy rate versus the C parameter from 0.0 to 00 for dfferent kernel functons. he RBF kernel functon th parameter C =.5 provdes the best performance for the SVM model. he excellent classfcaton accuracy of an SVM suggests ts practcalty for engneerng applcatons. herefore, ths study develops a lquefacton assessment algorthm based on SVM theory, called LA-SVM. he graphcal user nterface (GUI) of ths algorthm as mplemented n a MALAB/GUI. hs nterface provdes an ntutve and user-frendly means of nteracton. Users do not need any dagrams, formulae, or manuals. By smply usng a mouse cursor to select optons and nput tranng data and parameter ranges, they can receve the classfcaton results and accuraces of the testng data n a short CPU runtme. LA-SVM greatly smplfes the lquefacton assessment process and produces extremely accurate results. he operaton steps are lsted as follos:. Launch LA-SVM program (Fg. 8).. Select the nput button, and nput the tranng data and testng data. 3. Select the data s normalzaton and ts ranges (specfed by users). 4. Select the grd search method and specfy the parameter ranges. Fg. 8. Easy to use LA-SVM/GUI nterface. Fg. 9. Expermental Result of LA-SVM. 5. Specfy the number of folds for cross-valdaton. 6. Select the run button, and start the analyss. 7. When analyss s complete, obtan the optmal kernel functon parameters and the classfcaton accuracy (Fg. 9).
7 34 Journal of Marne Scence and echnology, Vol., o. 3 (03) VII. COCLUSIO SVM has been successfully appled n many applcatons, but t s less dely appled n the geotechncal feld. he results n ths study sho that SVM s a poerful computatonal tool that can be used to analyze the complex relatonshp beteen sol and sesmc parameters n lquefacton assessment. he expermental results n ths study ndcate that an SVM acheves greater classfcaton accuracy than an A. In addton to ts hgher accuracy rate, the SVM model requres only to parameters, as compared to the A, hch requres multple parameters. In concluson, the SVM model s more effectve and feasble than the conventonal A. An SVM not only has a sold foundaton n statstcal learnng theory, but can also effectvely handle nonlnear classfcaton. herefore, t s regarded as one of the most effectve classfcaton methods. he expermental results and dscusson above sho that the proposed LA-SVM can be effectvely appled to lquefacton assessment. he LA-SVM program has an ntutve nterface that s easly understandable. As ne lquefacton assessment cases are collected to expand the database, the classfcaton accuracy of LA-SVM can be further ncreased. hus, LA-SVM s a novel lquefacton assessment tool orthy of promoton and support. REFERECES. Boser, B. E., Guyon, I. M., and Vapnk, V.., A tranng algorthm for optmal margn classers, Proceedngs of the Ffth Annual Workshop on Computatonal Learnng heory, ACM Press, pp (99).. Chang, C. C. and Ln, C. J., LIBSVM: A Lbrary for Support Vector Machnes, (00). 3. Chern, S. G., Lee, C. Y., and Wang, C. C., CP-based lquefacton assessment by usng fuzzy-neural netork, Journal of Marne Scence and echnology, Vol. 6, o., pp (008). 4. Chern, S. G., Lee, C. Y., and Wang, C. C., CP-based smplfed lquefacton assessment by usng fuzzy-neural netork, Journal of Marne Scence and echnology, Vol. 7, o. 4, pp (009). 5. Cortes, C. and Vapnk, V., Support-vector netork, Machne Learnng, Vol. 0, pp (995). 6. Faruto and Lyang, LIBSVM-faruto Ult- mate Verson, a toolbox th mplement for support vector machne based on lbsvm, matlabsky.com (0). 7. Goh, A.. C., eural netork modelng of CP sesmc lquefacton data, Journal of Geotechncal Engneerng, ASCE, Vol. 9, o., pp (996). 8. Goh, A.. C. and Goh, S. H., Support vector machnes: ther use n geotechncal engneerng as llustrated usng sesmc lquefacton data, Computer and Geotechncs, Vol. 34, pp (007). 9. Hsu, C. C., Chang, C. C., and Ln, C. J., A Practcal Gude to Support Vector Classfcaton, ape (00). 0. Samu, P. and Stharam,. G., Machne learnng modelng for predctng sol lquefacton susceptblty, Journal of atural Hazard and Earth System Scence, Vol., pp. -9 (0).. Sterln, P., Overfttng Preventon th Cross-Valdaton, Pars (007).. Vapnk, V.., he ature of Statstcal Learnng heory, Sprnger, e York (995).
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