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Geomatics, Natural Hazards and Risk, 2014 Vol. 5, No. 1, 7–25, http://dx.doi.org/10.1080/19475705.2012.757252 Application of statistical learning algorithms for prediction of liquefaction susceptibility of soil based on shear wave velocity J. KARTHIKEYAN and PIJUSH SAMUI* Center for Disaster Mitigation and Management, VIT University, Vellore - 632014, Tamil Nadu, India (Received 10 August 2012; ﬁnal version received 1 December 2012) The determination of liquefaction susceptibility of soil is an imperative task in earthquake engineering. This study employs two statistical learning algorithms {least square support vector machine (LSSVM) and relevance vector machine (RVM)} for prediction of liquefaction susceptibility of soil based on shear wave velocity (V ) data. This study uses both the algorithms as a classiﬁcation tool. Equations have been also developed for the prediction of liquefaction susceptibili- ty of soil. This study shows that two parameters {peak ground acceleration (PGA) and V } are sufﬁcient for the prediction of liquefaction susceptibility of soil. This paper also highlights the capability of RVM over LSSVM. Introduction Liquefaction in soil is one of the major problems in earthquake geotechnical engi- neering. It is deﬁned as the transformation of a granular material from a solid to a liqueﬁed state as a consequence of increased pore-water pressure and reduced effec- tive stress. The generation of excess pore pressure under undrained loading condi- tions is a hallmark of all liquefaction phenomena. This phenomenon was brought to the attention of engineers more so after Niigata (1964) and Alaska (1964) earth- quakes. Liquefaction is divided into the following two categories (Ishihara 1993, Baki et al. 2012): Flow liquefaction Cyclic mobility. Liquefaction will cause building settlement or tipping, sand boils, ground cracks, landslides, dam instability, highway embankment failures, or other hazards. Dam- ages attributed to the earthquake induced liquefaction phenomenon have cost society hundreds of millions of US dollars (Seed and Idriss 1982). Such damages are general- ly of great concern to public safety and are of economic signiﬁcance. Site-speciﬁc evaluation of liquefaction susceptibility of sandy and silty soils is a ﬁrst step in lique- faction hazard assessment. The geotechnical engineering practitioners commonly use the two penetrations-based methods (standard penetration tests and cone penetra- tion tests) for assessment of liquefaction potential (Seed and Idriss 1967, 1971, Seed et al. 1983, 1984, Robertson and Campanella 1985, Seed and De Alba 1986, Stark *Corresponding author. Email: pijushsamui@vit.ac.in 2013 Taylor & Francis 8 J. Karthikeyan and P. Samui and Olson 1995, Olsen 1997, Robertson and Wride 1998). On the other hand, shear wave velocity (V ) may offer geotechnical engineers a third tool that is lower in cost and provides more physically meaningful measurements. The advantages of using V for evaluating liquefaction potential have been described by many researchers (Dobry et al. 1981, Seed et al. 1983, Stokoe et al. 1988a, Tokimatsu and Uchida 1990). Based on V , Andrus et al. (1999) and Andrus and Stokoe (2000) have evaluat- ed liquefaction potential for different sites. Numbers of approaches based on V , both probabilistic (Cetin et al. 2004) and Artiﬁcial Neural Network (ANN) methods have been proposed. Juang et al. (2001) has proposed a probabilistic framework for liquefaction potential using V data. Goh (2002) successfully used Probabilistic Neural Network (PNN) for assessing liquefaction potential from V data. But, ANN has some limitations such as arriving at local minima, black box approach, slow convergence speed, less generalizing per- formance, and overﬁtting problem (Park and Rilett 1999, Kecman 2001). This study uses two statistical learning algorithms for prediction of liquefaction susceptibility of soil based V data. The ﬁrst algorithm uses Least Square Support Vector Machine (LSSVM). The LSSVM is a statistical learning theory which adopts a least squares linear system as loss functions (Suykens et al. 1999). It is closely relat- ed to regularization networks (Smola and Scholkopf 1998). The second algorithm adopts Relevance Vector Machine (RVM). RVM is based on the Bayesian formula- tion of a linear model with an appropriate prior that results in a sparse representa- tion (Tipping 2000). Two models (MODEL I and MODEL II) have been developed for the prediction of liquefaction susceptibility of soil. For MODEL I, the input parameters are Cyclic Stress Ratio (CSR) and V . MODEL II uses Peck Ground Ac- celeration (PGA) and V as input parameters. This study uses the database collected by Andrus and Stokoe (1997). The paper has the following aims: To determine the capability of LSSVM and RVM for prediction of liquefaction susceptibility of soil based on V data To develop equations based on the developed LSSVM and RVM models To make a comparative study between LSSVM and RVM models. Details of LSSVM model The LSSVM has been used for the prediction of liquefaction susceptibility of soil based on V data. The LSSVM is a statistical learning method which has a self- contained basis of statistical-learning theory and excellent learning performance (Suykens et al. 2002). A binary classiﬁcation problem is considered having a set of training vectors (D) belonging to two separate classes. 1 1 n n n D ¼fðx ; y Þ; :::; ðx ; y Þgx2R ; y2f1; þ1g; ð1Þ where x2R is an n-dimensional data vector with each sample belonging to either of the two classes labelled as y2f1; þ1g, and n is the number of training data. For MODEL I, the input parameters are CSR and V . So, for MODEL I, x ¼½CSR; V . s s MODEL II uses PGA and V as input parameters. Therefore, For MODEL II, x ¼½PGA; V . In the current context of classifying soil condition during an earth- quake, the two classes labelled as (þ1, 1) may mean non-liquefaction and Statistical learning algorithms for prediction of liquefaction susceptibility 9 liquefaction. The Support Vector Machine (SVM) approach aims at constructing a classiﬁer of the form: yðxÞ¼ sign a y cðx; x Þþ b ; ð2Þ k k k k¼1 where, a are positive real constants, b is a real constant and cðx; x Þ is kernel func- k k tion and sign is the signum function. It gives þ1 if the element is greater than or equal to zero and –1 if it is less than zero. For the case of the two classes, one assumes w fðx Þþ b 1; if y ¼þ1ðNo LiquefactionÞ k k ; ð3Þ w fðx Þþ b 1 if y ¼1ðLiquefactionÞ k k which is equivalent to y w fðx Þþ b 1; k ¼ 1; ...; N : ð4Þ k k Where fð:Þ is a non-linear function which maps the input space into a higher dimen- sional space. According to the structural risk minimization principle, the risk bound is minimized by formulating the following optimization problem: 1 g T 2 Minimize : w w þ e : ð5Þ 2 2 k¼1 Subjectedto : y w fðx Þþ b ¼ 1 e ; k ¼ 1; ...; N k k k Where, g is the regularization parameter, determining the trade-off between the ﬁt- ting error minimization and smoothness and e is error variable. In order to solve the above optimization problem (Equation 5), the Lagrangian is constructed as follows: l N X X 1 g T 2 T Lðw; b; e; aÞ¼ w w þ e a y w fðx Þþ b 1 þ e : ð6Þ k k k k 2 2 k¼1 k¼1 Where a are Lagrange multipliers, which can be either positive or negative due to the equality constraints as follows from the Kuhn-Tucker conditions (Fletcher 1987). The solution to the constrained optimization problem is determined by the saddle point of the Lagrangian function L (w, b, e, a), which has to be minimized with respect to w, b, e , and a . Thus, differentiating L (w, b, e, a) with respect to w, k k b, e and a and setting the results equal to zero, the following three conditions have k k 10 J. Karthikeyan and P. Samui been obtained: @L ¼ 0)w ¼ a y fðx Þ k k k @w k¼1 @L ¼ 0) a y ¼ 0 k k @b ð7Þ k¼1 @L ¼ 0)a ¼ ge k k @e @L ¼ 0)y w fðx Þþ b 1 þ e ¼ 0; k ¼ 1; ...; N : k k k @a The above equitation (7) can be written immediately as the solution to the follow- ing set of linear equations (Fletcher 1987) 2 32 3 2 3 I 00 Z w 0 6 76 7 6 7 00 0 Y b 0 6 76 7 6 7 ¼ : ð8Þ 4 54 5 4 5 00 gI I e 0 ZY I 0 a 1 T T Where Z ¼ fðx Þ y ;...; fðx Þ y , Y ¼½y ;...; y , I ¼½1; ...; 1, 1 1 N N 1 N e ¼½e ;...; e , a ¼½a ;...; a . 1 N 1 N The solution is given by 0 Y b 0 ¼ : ð9Þ Y Ω þ g I a 1 Where Ω ¼ Z Z and the kernel trick can be applied within the Ω matrix. Ω ¼ y y fðx Þ fðx Þ kl k l k l ð10Þ ¼ y y Kðx ; x Þ; k; l ¼ 1; ...; N : k l k l Where Kðx ; x Þ is kernel function. k l The classiﬁer in the dual space takes the form yðxÞ¼ sign a y Kðx; x Þþ b : ð11Þ k k k k¼1 The above methodology has been used for the prediction of liquefaction suscepti- bility of soil based on V data by developed two models (MODEL I and MODEL II). The database collected by Andrus and Stokoe (1997) is used in this study. In the MODEL I, CSR versus V data is trained for prediction of liquefaction susceptibility. In the MODEL-II, this is further simpliﬁed by relating PGA versus V for the predic- tion of liquefaction susceptibility. For the purpose of classiﬁcation, a value of 1 is assigned to those sites that did not liquefy while a value of -1 is assigned to those sites that liqueﬁed. The data has been further divided into two subsets; a training dataset, to construct the model, and a testing dataset to estimate the model performance. So, Statistical learning algorithms for prediction of liquefaction susceptibility 11 for our study a set of 134 data is considered as the training dataset and the remaining set of 57 data is considered as the testing dataset. The data is normalized between 0 to 1. In this study, radial basis function is used as the kernel function of the LSSVM. When applying LSVM, the design value of width (s) of radial basis function and g will be determined during the modelling experiment. In MODEL I, CSR is used as one of the input parameters and it has been deter- mined by using the following equation. It is a function of s , s’ depth, magnitude of v v, earthquake, and a and it is deﬁned as (Seed and Idriss 1971): max s a v max CSR ¼ 0:65 r : ð12Þ s g Where, s is total vertical stress, s’ is effective vertical stress a is maximum v v , max horizontal acceleration, g is acceleration due to gravity and r is stress reduction. So, to calculate the value of CSR one has to determine the value of s and s’ Multi- v v. channel analysis of surface wave (MASW) method, spectral analysis of surface wave (SASW) and seismic cone penetration test (SCPT) are being used for the determina- tion of V . But, it is well known that the above mentioned methods do not have the provision to obtain soil sample and calculate the needed soil properties. For this rea- son, determination of CSR from the above mentioned method alone is impossible. The purpose of the development of MODEL II is to simplify and predict the lique- faction based on V and PGA. So, in MODEL II, the input variables are V and s s PGA. In MODEL II, the same training dataset, testing dataset, normalization tech- nique, and same kernel function have been used as used in MODEL I. Details of RVM model The concept of RVM was introduced by Tipping (2000). It allows the computation of the prediction intervals taking into account uncertainties of both the parameters and the data (Tipping 2001). Details about the RVM methodology could be obtained from Tipping (2000); the following paragraph describes the method brieﬂy. A set of targets, Y , for the classiﬁcation problem in the manuscript is occurrence of either Event 0 deﬁned as “liquefaction” or Event 1 deﬁned as “no liquefaction”. The input parameters for MODEL I, x , are CSR and V . For MODEL II, x ¼½PGA; V . i s s These input parameters for each training dataset are represented by a basis function Fj(x). Thereby, the total number of basis functions is the number of samples in the training dataset. Each of the basis function is deﬁned as function of the kernel, K, as Fðx Þ¼ ½1; Kðx ; x Þ; Kðx ; x Þ; :::Kðx ; x Þ: ð13Þ n n 1 n 2 n N The kernel, K, is taken as a radial basis function with mean 0 and variance 1. In RVM methodology, usually the set of targets are related to a set of input vectors using the following equation: ::: Y ¼ w F ðxÞþ w F ðxÞþ w F ðxÞþ þ w F ðxÞð14Þ 1 1 2 2 3 3 k k Where F (x) are the basis functions deﬁned above and w represents the weight i i parameters. Gamma priors are introduced to the model weight, w , which are governed i 12 J. Karthikeyan and P. Samui by a set of hyperparameters (maximum likelihood estimations), one associated with each weight, whose most probable value are iteratively estimated from the data. To obtain the probabilistic estimates for the binomial classiﬁcation problem as given in the manuscript, we follow statistical convention and generalize the linear model in equation 14 by applying a logistic sigmoid function sfYðxÞg ¼ to Y(x). ð1þe Þ This study uses the above methodology for the prediction seismic liquefaction po- tential of soil based on V data. In RVM model, the same training dataset, testing dataset, normalization technique, and same kernel function have been used as used in LSSVM. Results and discussion The design value of g and s has been determined by trial and error approach. Train- ing and testing performance has been determined by using the following equation: Training or Testing performanceð%Þ¼ ð15Þ No of data predicted accurately by RVM Total data For MODEL I, the design value of g and s is 100 and 1, respectively. The perform- ances of training and testing dataset have been determined by equation (15). For MODEL I, the performance of training and testing dataset are 97.76% and 94.73%, respectively. Three data have been misclassiﬁed for both training and testing dataset. Tables 1 and 2 show the performance of training and testing dataset for MODEL I, Table 1. Performance of training dataset for LSSVM model. Predicted class CSR V (m/sec) PGA Actual class Model I Model II .3 136 .36 1 1 1 .29 154 .36 1 1 1 .29 173 .36 1 1 1 .24 177 .32 1 1 1 .24 200 .32 1 11 .14 118 .16 1 1 1 .22 149 .32 1 1 1 .22 158 .32 1 1 1 .13 147 .12 1 1 1 .16 115 .16 1 1 1 .12 122 .12 1 1 1 .14 98 .12 1 11 .14 101 .12 1 1 1 .13 143 .12 1 1 1 .13 127 .13 1 1 1 .26 131 .36 1 1 1 .18 90 .21 1 1 1 .45 126 .51 1 1 1 .09 105 .12 1 1 1 .41 131 .5 1 1 1 .4 164 .5 1 1 1 Statistical learning algorithms for prediction of liquefaction susceptibility 13 Table 1. (Continued ) Predicted class CSR V (m/sec) PGA Actual class Model I Model II .4 173 .5 1 1 1 .08 195 .08 1 1 1 .26 127 .27 1 1 1 .27 115 .27 1 1 1 .18 90 .2 1 1 1 .23 101 .3 1 1 1 .29 105 .36 1 1 1 .02 133 .02 1 1 1 .02 164 .02 1 1 1 .28 107 .36 1 1 1 .29 94 .36 1 1 1 .28 109 .36 1 1 1 .29 122 .36 1 1 1 .26 128 .36 1 1 1 .27 107 .36 1 1 1 .26 122 .36 1 1 1 .29 154 .36 1 1 1 .23 105 .3 11 1 .26 106 .29 1 1 1 .13 143 .16 1 1 1 .42 274 .46 1 1 1 .07 155 .06 1 1 1 .12 152 .16 1 1 1 .33 133 .22 1 1 1 .33 127 .22 1 1 1 .27 146 .18 1 1 1 .27 133 .18 1 1 1 .27 130 .18 1 1 1 .06 146 .04 1 1 1 .06 127 .04 1 1 1 .06 130 .04 1 1 1 .27 133 .18 1 1 1 .27 127 .18 1 1 1 .08 146 .05 1 1 1 .08 133 .05 1 1 1 .08 130 .05 1 1 1 .24 146 .16 1 1 1 .24 127 .16 1 1 1 .33 146 .22 1 1 1 .12 127 .12 1 1 1 .12 124 .12 1 1 1 .1 90 .11 1 1 1 .05 126 .06 1 1 1 .19 105 .24 1 1 1 .02 131 .03 1 11 .02 164 .03 1 1 1 .02 173 .03 1 1 1 .19 124 .2 1 1 1 .2 115 .2 1 1 1 .17 126 .19 1 1 1 .15 101 .2 1 1 1 (Continued on next page) 14 J. Karthikeyan and P. Samui Table 1. (Continued ) Predicted class CSR V (m/sec) PGA Actual class Model I Model II .15 131 .18 1 1 1 .15 133 .18 1 1 1 .15 173 .18 1 1 1 .37 126 .42 1 1 1 .15 157 .14 1 1 1 .15 131 .14 1 1 1 .15 148 .14 1 1 1 .15 137 .14 1 1 1 .15 146 .14 1 1 1 .15 178 .14 1 1 1 .15 154 .14 1 1 1 .12 143 .15 1 1 1 .13 135 .16 1 1 1 .12 117 .16 1 1 1 .12 121 .16 1 1 1 .12 138 .16 1 1 1 .12 145 .16 1 1 1 .12 133 .16 1 1 1 .21 146 .24 1 1 1 .22 148 .24 1 1 1 .21 179 .24 1 1 1 .57 157 .24 1 1 1 .21 145 .24 1 11 .21 176 .24 1 1 1 .41 206 .46 1 1 1 .2 204 .27 1 1 1 .2 116 .27 1 1 1 .19 125 .27 1 1 1 .12 120 .15 1 1 1 .12 105 .15 1 1 1 .12 220 .15 1 1 1 .16 136 .19 1 1 1 .16 161 .19 1 1 1 .16 173 .19 1 1 1 .11 195 .15 1 1 1 .11 200 .15 1 1 1 .11 131 .15 1 1 1 .11 149 .15 1 1 1 .11 168 .15 1 1 1 .24 143 .25 1 1 1 .33 126 .42 1 1 1 .2 97 .27 1 1 1 .11 158 .15 1 1 1 .21 116 .25 1 1 1 .2 130 .25 1 1 1 .26 209 .25 1 1 1 .22 150 .25 1 1 1 .12 120 .15 1 1 1 .19 127 .2 1 1 1 .43 197 .5 1 1 1 .32 149 .48 1 1 1 .15 135 .2 1 1 1 Statistical learning algorithms for prediction of liquefaction susceptibility 15 Table 1. (Continued ) Predicted class CSR V (m/sec) PGA Actual class Model I Model II .33 145 .42 1 1 1 .28 134 .36 1 1 1 .31 135 .42 1 1 1 .44 174 .5 1 1 1 .14 163 .15 1 1 1 .16 154 .19 1 1 1 .49 176 .5 1 1 1 .46 153 .5 1 1 1 .47 183 .5 1 1 1 .49 181 .5 1 1 1 Table 2. Performance of testing dataset for LSSVM model. Predicted class CSR V (m/sec) PGA Actual class Model I Model II .29 161 .36 1 1 1 .24 195 .32 1 1 1 .22 131 .32 1 1 1 .22 168 .32 1 1 1 .24 199 .32 1 1 1 .13 103 .12 1 11 .11 163 .16 1 1 1 .1 101 .12 1 1 1 .41 133 .5 1 1 1 .09 155 .08 1 1 1 .26 124 .27 11 1 .05 126 .06 1 1 1 .02 131 .02 1 1 1 .02 173 .02 1 1 1 .28 102 .36 1 1 1 .13 124 .13 1 1 1 .26 122 .3 1 1 1 .23 105 .29 1 1 1 .17 271 .23 1 1 1 .24 130 .16 1 1 1 .33 130 .22 1 11 .06 127 .18 1 1 1 .06 133 .04 1 1 1 .27 146 .18 1 1 1 .27 130 .18 1 1 1 .08 127 .05 1 1 1 .24 133 .16 1 1 1 .07 150 .1 1 1 1 .1 101 .13 1 1 1 .02 133 .03 1 1 1 .1 90 .2 1 1 1 (Continued on next page) 16 J. Karthikeyan and P. Samui Table 2. (Continued ) Predicted class CSR V (m/sec) PGA Actual class Model I Model II .19 105 .21 1 1 1 .02 164 .18 1 1 1 .15 157 .14 11 1 .15 136 .14 1 1 1 .15 152 .14 1 1 1 .2 212 .27 1 1 1 .06 195 .06 1 1 1 .12 148 .16 1 1 1 .21 134 .24 1 1 1 .21 145 .24 1 1 1 .21 142 .24 1 1 1 .2 193 .27 1 1 1 .12 115 .12 1 1 1 .11 153 .15 1 1 1 .15 130 .14 1 1 1 .11 177 .15 1 1 1 .11 199 .15 1 1 1 .44 116 .42 1 1 1 .13 115 .13 1 1 1 .36 158 .42 1 1 1 .22 162 .25 1 1 1 .22 171 .25 1 11 .16 79 .19 1 1 1 .21 144 .19 1 1 1 .1 179 .12 1 1 1 .46 210 .5 1 1 1 respectively. So, the developed LSSVM model has the ability to predict liquefaction susceptibility of soil. The following equation has been developed for the prediction of status(s) of soil during an earthquake for MODEL I (by putting ðxx Þðxx Þ k k Kðx; x Þ¼ expf g, N ¼ 131, s ¼ 1 and b ¼ 0.2263 in equation 11). k 2 2s "# () 134 T ðx x Þðx x Þ k k s ¼ sign a y exp þ 0:2263 ð16Þ k k k¼1 Figure 1 shows the value of a for MODEL I. For MODEL II, the design value of g and s is 80 and 1.8. The performance of training and testing data has been determined by the design value of g and s. The performances of training and testing data are 97.76% and 92.98%, respectively. Three and four data have been misclassiﬁed for training and testing data, respectively. For MODEL II, the performance of training and testing dataset has been shown in MODELS I and MODEL II, respectively. So, the performance of MODEL I and MODEL II are almost same. The following equation has been developed: Statistical learning algorithms for prediction of liquefaction susceptibility 17 Figure 1. The values of a for MODEL I (LSSVM). For the prediction s of soil during an earthquake for MODEL II (by putting ðxx Þðxx Þ k k Kðx; x Þ¼ exp g, N ¼ 131, s ¼ 1.8 and b ¼ 0.1985 in equation 11). k 2 2s "# () 134 T ðx x Þðx x Þ k k s ¼ sign a y exp þ 0:1985 : ð17Þ k k 3:24 k¼1 The value of a for MODEL II has been shown in ﬁgure 2. For the RVM model, the design value of s has been determined by trail and error approach. For MODEL I, the design value of s and the number of relevance vector is 0.1 and 13, respectively. Equation (15) has been used for the determination of the performance of training and testing dataset. Training and testing performances are 98.50% and 98.24%, respectively. So, two and one data have been misclassiﬁed by training and testing, respectively. Tables 3 and 4 shows the performance of training Figure 2. The values of a for MODEL II (LSSVM). 18 J. Karthikeyan and P. Samui Table 3. Performance of training dataset for RVM model. Predicted class CSR V (m/sec) PGA Actual class Model I Model II .3 136 .36 0 0 0 .29 154 .36 0 0 0 .29 173 .36 0 0 0 .24 177 .32 1 1 1 .24 200 .32 1 1 1 .14 118 .16 0 0 0 .22 149 .32 0 0 0 .22 158 .32 0 0 0 .13 147 .12 0 1 0 .16 115 .16 0 0 0 .12 122 .12 0 0 0 .14 98 .12 0 0 1 .14 101 .12 0 0 0 .13 143 .12 1 1 1 .13 127 .13 1 1 1 .26 131 .36 0 0 0 .18 90 .21 0 0 0 .45 126 .51 0 0 0 .09 105 .12 1 1 1 .41 131 .5 0 0 0 .4 164 .5 1 1 1 .4 173 .5 1 1 1 .08 195 .08 1 1 1 .26 127 .27 0 0 0 .27 115 .27 0 0 0 .18 90 .2 0 0 0 .23 101 .3 0 0 0 .29 105 .36 0 0 0 .02 133 .02 1 1 1 .02 164 .02 1 1 1 .28 107 .36 0 0 0 .29 94 .36 0 0 0 .28 109 .36 0 0 0 .29 122 .36 0 0 0 .26 128 .36 0 0 0 .27 107 .36 0 0 0 .26 122 .36 0 0 0 .29 154 .36 0 0 0 .23 105 .3 0 0 0 .26 106 .29 0 0 0 .13 143 .16 0 0 0 .42 274 .46 1 1 1 .07 155 .06 1 1 1 .12 152 .16 0 0 0 .33 133 .22 1 1 1 .33 127 .22 1 1 1 .27 146 .18 1 1 1 .27 133 .18 1 1 1 .27 130 .18 1 1 1 .06 146 .04 1 1 1 .06 127 .04 1 1 1 .06 130 .04 1 1 1 .27 133 .18 1 1 1 Statistical learning algorithms for prediction of liquefaction susceptibility 19 Table 3. (Continued ) Predicted class CSR V (m/sec) PGA Actual class Model I Model II .27 127 .18 1 1 1 .08 146 .05 1 1 1 .08 133 .05 1 1 1 .08 130 .05 1 1 1 .24 146 .16 1 1 1 .24 127 .16 1 1 1 .33 146 .22 1 1 1 .12 127 .12 1 1 1 .12 124 .12 1 1 1 .1 90 .11 1 1 1 .05 126 .06 1 1 1 .19 105 .24 1 1 1 .02 131 .03 1 1 1 .02 164 .03 1 1 1 .02 173 .03 1 1 1 .19 124 .2 1 1 1 .2 115 .2 1 1 1 .17 126 .19 1 1 1 .15 101 .2 1 1 1 .15 131 .18 1 1 1 .15 133 .18 1 1 1 .15 173 .18 1 0 1 .37 126 .42 0 0 0 .15 157 .14 0 0 1 .15 131 .14 0 0 0 .15 148 .14 0 0 0 .15 137 .14 0 0 0 .15 146 .14 0 0 0 .15 178 .14 1 1 1 .15 154 .14 0 0 0 .12 143 .15 0 0 0 .13 135 .16 0 0 0 .12 117 .16 0 0 0 .12 121 .16 0 0 0 .12 138 .16 0 0 0 .12 145 .16 1 1 1 .12 133 .16 1 1 1 .21 146 .24 0 0 0 .22 148 .24 0 0 0 .21 179 .24 0 0 0 .57 157 .24 0 0 0 .21 145 .24 0 0 0 .21 176 .24 0 0 0 .41 206 .46 1 1 1 .2 204 .27 1 1 1 .2 116 .27 0 0 0 .19 125 .27 0 0 0 .12 120 .15 0 0 0 .12 105 .15 0 0 0 .12 220 .15 1 1 1 .16 136 .19 1 1 1 .16 161 .19 1 1 1 (Continued on next page) 20 J. Karthikeyan and P. Samui Table 3. (Continued ) Predicted class CSR V (m/sec) PGA Actual class Model I Model II .16 173 .19 1 1 1 .11 195 .15 1 1 1 .11 200 .15 1 1 1 .11 131 .15 1 1 1 .11 149 .15 1 1 1 .11 168 .15 1 1 1 .24 143 .25 0 0 0 .33 126 .42 1 1 1 .2 97 .27 0 0 0 .11 158 .15 1 1 1 .21 116 .25 0 0 0 .2 130 .25 0 0 0 .26 209 .25 0 0 0 .22 150 .25 0 0 0 .12 120 .15 1 1 1 .19 127 .2 1 1 1 .43 197 .5 0 0 0 .32 149 .48 1 1 1 .15 135 .2 0 0 0 .33 145 .42 0 0 0 .28 134 .36 0 0 0 .31 135 .42 1 1 1 .44 174 .5 0 0 0 .14 163 .15 0 0 0 .16 154 .19 1 1 1 .49 176 .5 0 0 0 .46 153 .5 0 0 0 .47 183 .5 0 0 0 .49 181 .5 0 0 0 Table 4. Performance of testing dataset for RVM model. Predicted class CSR V (m/sec) PGA Actual class Model I Model II .29 161 .36 0 0 0 .24 195 .32 1 1 1 .22 131 .32 0 0 0 .22 168 .32 0 0 0 .24 199 .32 1 1 1 .13 103 .12 0 0 0 .11 163 .16 1 1 1 .1 101 .12 1 1 1 .41 133 .5 0 0 0 .09 155 .08 1 1 1 .26 124 .27 0 0 0 .05 126 .06 1 1 1 .02 131 .02 1 1 1 .02 173 .02 1 1 1 .28 102 .36 0 0 0 Statistical learning algorithms for prediction of liquefaction susceptibility 21 Table 4. (Continued ) Predicted class CSR V (m/sec) PGA Actual class Model I Model II .13 124 .13 1 1 1 .26 122 .3 0 0 0 .23 105 .29 0 0 1 .17 271 .23 1 1 1 .24 130 .16 1 1 1 .33 130 .22 1 1 1 .06 127 .18 1 1 1 .06 133 .04 1 1 1 .27 146 .18 1 1 1 .27 130 .18 1 1 1 .08 127 .05 1 0 1 .24 133 .16 1 1 1 .07 150 .1 1 1 1 .1 101 .13 1 1 1 .02 133 .03 1 1 1 .1 90 .2 1 1 1 .19 105 .21 1 1 1 .02 164 .18 1 1 1 .15 157 .14 0 0 0 .15 136 .14 0 0 0 .15 152 .14 0 0 0 .2 212 .27 1 1 1 .06 195 .06 1 1 1 .12 148 .16 1 1 1 .21 134 .24 0 0 0 .21 145 .24 0 0 0 .21 142 .24 0 0 0 .2 193 .27 1 1 1 .12 115 .12 1 1 1 .11 153 .15 0 0 0 .15 130 .14 0 0 0 .11 177 .15 1 1 1 .11 199 .15 1 1 1 .44 116 .42 0 0 0 .13 115 .13 1 1 1 .36 158 .42 1 1 1 .22 162 .25 0 0 0 .22 171 .25 0 0 0 .16 79 .19 0 0 0 .21 144 .19 0 0 0 .1 179 .12 1 1 1 .46 210 .5 0 0 0 and testing for MODEL II respectively. MODEL I gives the following equation for prediction of s of soil during an earthquake: () 134 T ðx x Þðx x Þ i i s ¼ w exp ð18Þ 0:02 i¼1 22 J. Karthikeyan and P. Samui Figure 3. The values of w for MODEL I (RVM). Figure3 shows the value of w for MODEL I. For MODEL II, the design value of s is 0.3. MODEL II uses 12 training data as relevance vector. The performance of training and testing dataset has been determined by same way as in MODEL I. The training and testing performances for MODEL II are 98.50% and 98.24%, respective- ly. Tables 3 and 4 show the performances of training and testing dataset, respectively. Two and one data have been misclassiﬁed by training and testing, respectively. Therefore, the performances of MODEL I and MODEL II are the same. The follow- ing equation has been developed for the prediction of s of soil during an earthquake. () 134 T ðx x Þðx x Þ i i s ¼ w exp : ð19Þ 0:18 i¼1 The value of w has been shown in ﬁgure 4. The performance of the RVM model is slightly better than the LSSVM model. The RVM model uses approximately 8 to 10% (MODEL I ¼ 9.70% and MODEL II ¼ 8.95%) of training data as relevance vectors. This relevance vector is only used for Figure 4. The values of w for MODEL II (RVM). Statistical learning algorithms for prediction of liquefaction susceptibility 23 the ﬁnal prediction. So, there is a real advantage gained in terms of sparsity. Sparse- ness is desirable in RVM for several reasons, namely (Figueiredo 2003): Sparseness leads to a structural simpliﬁcation of the estimated function. Obtaining a sparse estimate corresponds to performing feature/variable selection. The generalization ability improves with the degree of sparseness. Sparseness means that a signiﬁcant number of the weights are zero (or effectively zero), which has the consequence of producing compact, computationally efﬁcient models, which in addition are simple and, therefore, produce smooth functions. The developed RVM uses only one parameter (s), but the developed LSSVM model uses two parameters (g and s). Conclusion This study has described two statistical learning algorithms (LSSVM and RVM) for the prediction of the liquefaction susceptibility of soil based on V . The developed LSSVM and RVM models give promising results. The performance of RVM is slightly better than the LSSVM model. The developed RVM model also gives sparse solution. Both models (LSSVM and RVM) have shown good generalization capabil- ity. Users can use the developed equations for prediction of the liquefaction suscepti- bility of soil. Both the methods (LSSVM and RVM) can be used as an accurate and quick tool for the prediction of the liquefaction susceptibility of soil. In summary, it can be concluded that the developed statistical learning algorithms (LSSVM and RVM) are robust models for the prediction of the liquefaction susceptibility of the soil due to an earthquake. References ANDRUS,R.D.and STOKOE, K.H. 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"Geomatics, Natural Hazards and Risk" – Taylor & Francis
Published: Mar 1, 2014
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