ORI GIN AL PA PER

Least square support vector machine and relevancevector machine for evaluating seismic liquefactionpotential using SPT

Pijush Samui

Received: 6 September 2009 / Accepted: 19 March 2011 / Published online: 1 April 2011� Springer Science+Business Media B.V. 2011

Abstract The determination of liquefaction potential of soil is an imperative task in

earthquake geotechnical engineering. The current research aims at proposing least square

support vector machine (LSSVM) and relevance vector machine (RVM) as novel classi-

fication techniques for the determination of liquefaction potential of soil from actual

standard penetration test (SPT) data. The LSSVM is a statistical learning method that has a

self-contained basis of statistical learning theory and excellent learning performance. RVM

is based on a Bayesian formulation. It can generalize well and provide inferences at low

computational cost. Both models give probabilistic output. A comparative study has been

also done between developed two models and artificial neural network model. The study

shows that RVM is the best model for the prediction of liquefaction potential of soil is

based on SPT data.

Keywords Liquefaction � Relevance vector machine � SPT � Artificial neural network �Least square support vector machine

1 Introduction

One of the major causes of destruction during an earthquake is the failure of soil. The loss

of the shear strength of the soil due to an increase in pore water pressure is the cause of this

failure. This phenomenon is called liquefaction. Determination of liquefaction potential of

soil due to an earthquake is a complex geotechnical engineering problem. Many factors,

including soil parameters and seismic characteristics, influence this problem. A procedure

based on standard penetration test (SPT) and cyclic stress ratio (CSR) has been developed

by Seed and his colleagues (Seed and Idriss 1967; Seed and Idriss 1971; Seed et al. 1983;

Seed et al. 1984) to assess the liquefaction potential of soil. Statistical and probabilistic

methods based on SPT data have been also proposed (Christian and Swiger 1975; Haldar

and Tang 1979; Liao et al. 1988). But all of the above methods have been developed based

P. Samui (&)Centre for Disaster Mitigation and Management, VIT University, Vellore 632014, Indiae-mail: pijush.phd@gmail.com

123

Nat Hazards (2011) 59:811–822DOI 10.1007/s11069-011-9797-5

on some empirical formulae, which are associated with some inherent uncertainties.

Artificial neural network (ANN) based on SPT data has been successfully used for the

prediction of liquefaction potential of soil (Goh 1994). But ANN has some limitations such

as black box approach, arriving at local minima, slow convergence speed, over fitting

problems, and absence of probabilistic output (Park and Rilett 1999; Kecman 2001).

Recently, Lai et al. (2006) used logistic regression to evaluate the liquefaction probability

of a site. But they did not classify between liquefiable and nonliquefiable soil.

Present study examines the potential of least square support vector machine (LSSVM)

and relevance vector machine (RVM) for assessing the liquefaction potential from the

actual SPT field data. This study uses the database collected by Goh (1994). Recently,

LSSVM has been successfully used for classification and function estimation problem

(Suykens and Vandewalle 1999; Lu et al. 2003; Suykens et al. 1999). Here, LSSVM has

been used as a classification tool. LSVM is closely related to Gaussian processes and

regularization networks but uses an optimization approach as in support vector machine

(SVM). Instead of solving a quadratic programming problem as in SVM, LSSVM finds the

solutions of a set of linear equations (Ferreira et al. 2005). Tipping (2001) introduced RVM

that is based on a Bayesian framework. RVM is inspired by the concept of automatic

relevance determination (ARD)(Mackay 1994; Neal 1994). It allows computation of the

prediction intervals taking into account uncertainties of both the parameters and the data

(Tipping 2000). A comparative study has been also presented between two developed

models and ANN model developed by Goh (1994). This study has the following aims:

• To investigate the feasibility of LSSVM and RVM for the prediction of seismic

liquefaction potential of soil is based on SPT data

• To determine probabilistic output

• To make a comparative study between the developed LSSVM, RVM, and ANN model

developed by Goh (1994)

2 Least square support vector machine

This section of the paper serves an introduction of LSSVM. Details of this method can be

found in Suykens et al. (2002). A binary classification problem is considered having a set

of training vectors (D) belonging to two separate classes.

D ¼ x1; y1� �

; . . .; x1; y1� �� �

x 2 Rn; y 2 �1;þ1f g ð1Þ

where x 2 Rn is an n-dimensional data vector with each sample belonging to either of two

classes labeled as y 2 �1;þ1f g, and l is the number of training data. For liquefaction

analysis, x ¼ soil parameter; earthquake parameter½ �. In the current context of classifying

soil condition under earthquake condition, the two classes labeled as (?1, -1) may mean

liquefaction and nonliquefaction. The SVM approach aims at constructing a classifier of

the form:

y xð Þ ¼ signXN

k¼1

akykw x; xkð Þ þ b

" #

ð2Þ

where ak are Lagrange multipliers, which can be either positive or negative, b is a real

constant, and w x; xkð Þ is kernel function. For the case of two classes, one assumes

812 Nat Hazards (2011) 59:811–822

123

wTu xkð Þ þ b� 1; if yk ¼ þ1; Liquefactionð ÞwTu xkð Þ þ b� � 1; if yk ¼ �1; No Liquefactionð Þ

ð3Þ

where w is weight and b is bias,which is equivalent to

yk wTu xkð Þ þ b� �

� 1; k ¼ 1; . . .;N ð4Þ

where u :ð Þ is a nonlinear function that maps the input space into a higher dimensional

space. According to the structural risk minimization principle, the risk bound is minimized

by formulating the following optimization problem:

Minimize:1

2wT wþ c

2

Xl

k¼1

e2k

Subjected to: yk wTu xkð Þ þ b� �

¼ 1� ek; k ¼ 1; . . .;N ð5Þ

where c is the regularization parameter, determining the trade-off between the fitting error

minimization and smoothness, and ek is error variable.

In order to solve the above optimization problem (Eq. 5), the Lagrangian is constructed

as follows:

L w; b; e; að Þ ¼ 1

2wT wþ c

2

X1

k¼1

e2k �

XN

k¼1

ak yk wTu xkð Þ þ b� �

� 1þ ek

� �ð6Þ

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,

ek, and ak (Fletcher1987). Thus, differentiating L (w, b, e, a) with respect to w, b, ek, and ak

and setting the results equal to zero, the following three conditions have been obtained:

oL

ow¼ 0) w ¼

XN

k¼1

akyku xkð Þ

oL

ob¼ 0)

XN

k¼1

akyk ¼ 0

oL

oek¼ 0) ak ¼ cek

oL

oak¼ 0) yk wTu xkð Þ þ b

� �� 1þ ek ¼ 0; k ¼ 1; . . .;N

ð7Þ

Equitation (7) can be written immediately as the solution to the following set of linear

equations (Fletcher 1987)

I 0 0 �ZT

0 0 0 �YT

0 0 cI �IZ Y I 0

2

664

3

775

wbea

2

664

3

775 ¼

0

0

0

1

2

664

3

775 ð8Þ

where Z ¼ u x1ð ÞT y1; . . .;u xNð ÞT yN

� ; Y ¼ y1; . . .; yN½ �; I ¼ 1; . . .; 1½ �; e ¼ e1; . . .; eN½ �;

a ¼ a1; . . .; aN½ �

Nat Hazards (2011) 59:811–822 813

123

The solution is given by

0 �YT

Y Xþ c�1I

�ba

�¼ 0

1

�ð9Þ

where X ¼ ZT Z and the kernel trick can be applied within the X matrix.

Xkl ¼ ykylu xkð ÞTu xlð Þ¼ ykylK xk; xlð Þ; k; l ¼ 1; . . .;N

ð10Þ

The classifier in the dual space takes the form

y xð Þ ¼ signXN

k¼1

akykK x; xkð Þ þ b

" #

ð11Þ

In this study, LSSVM model has been also used to determine the probabilistic output.

The detailed methodology for probabilistic output is given by Suykens et al. (2002).

The main scope of this work to implement the above methodology to predict the

liquefaction potential of soil is based on SPT data. The data set consists of a total of 85

records of 13 earthquakes that occurred in different countries in the period of 1891–1980.

This data set contains the information about the SPT value (N), equivalent dynamic shear

stress (s/r00), total vertical stress (r0), effective vertical stress (r(0), mean grain size (D50),

earthquake magnitude (M), normalized horizontal acceleration at the ground surface (a/g),

and fines content (F). The data set represented 42 sites that liquefied and 43 sites that did

not liquefy. The inputs of LSSVM model are N, s/r00, r0, D50, M, a/g, and F. The data are

normalized against their maximum values (Sincero 2003). To use these data for classifi-

cation purpose, a value of ?1 is assigned to the liquefied sites while a value of -1 is

assigned to the nonliquefied sites. So the output of the model will be either ?1 or -1. In

carrying out the formulation, the data have been divided into two sub-sets such as

(a) A training data set: This is required to construct the model. In this study, 59 out of the

85 data are considered for training.

(b) A testing data set: This is required to estimate the model performance. In this study,

the remaining 26 data are considered for testing.

In this study, same data sets were used for training and testing as used by Goh (1994)

with ANN approach so as to compare the performance of LSSVM and ANN approach. In

case of LSSVM training, Gaussian kernel has been used. A number of experiments were

carried out on SPT data using different combinations of input parameters to assess the

liquefaction potential using LSSVM (see Table 1). Table 1 shows the different model with

different input parameters. In the present study, training and testing of LSSVM have been

carried out by using MATLAB (MathWork Inc 1999).

3 Relevance vector machine

RVM is a probabilistic Bayesian classifier (Tipping 2001). It introduces Gaussian priors on

each parameter or group of parameters, each prior being controlled by its own individual

scale hyperparameter. In this section, a brief introduction on how to construct RVM for

classification is presented. Briefly, RVM is Bayesian approach for training a linear model.

Consider a set of example of input vectors xif gNi¼1 is given along with a corresponding set

814 Nat Hazards (2011) 59:811–822

123

Tab

le1

Per

form

ance

of

LS

SV

Man

dR

VM

for

dif

fere

nt

mod

el

LS

SV

Mm

od

elR

VM

mod

el

Mo

del

Inp

ut

var

iab

les

Ker

nel

cT

rain

ing

per

form

ance

(%)

Tes

tin

gp

erfo

rman

ce(%

)

Ker

nel

Tra

inin

gp

erfo

rman

ce(%

)

Tes

tin

gp

erfo

rman

ce(%

)

Nu

mb

ero

fre

lev

ance

vec

tor

IM

,N

,s/

r00,

FG

auss

ian

,w

idth

(r)

=3

09

08

9.8

38

4.6

2G

auss

ian

,w

idth

(r)

=0

.89

3.2

29

2.3

14

IIM

,N

,a/

g,s/

r0 0

,F

Gau

ssia

n,

wid

th(r

)=

12

10

89

.83

88

.46

Gau

ssia

n,

wid

th(r

)=

1.5

96

.61

96

.15

7

III

r00,

M,

N,a/

g,

s/r0

0,

FG

auss

ian

,w

idth

(r)

=3

01

00

96

.61

88

.46

Gau

ssia

n,

wid

th(r

)=

0.8

94

.92

92

.31

8

IVr0

0,

M,

N,a/

g,

s/r0

0,

F,

D50

Gau

ssia

n,

wid

th(r

)=

25

10

09

8.3

18

8.4

6G

auss

ian

,w

idth

(r)

=0

.89

6.6

19

6.1

51

0

Vr 0

,r0

0,

M,

N,a/

g,

s/r0

0,

D50

Gau

ssia

n,

wid

th(r

)=

40

15

09

6.6

18

8.4

6G

auss

ian

,w

idth

(r)

=0

.99

8.3

19

2.3

16

VI

r 0,r0

0,

M,

N,a/

g,

s/r0

0,

F,

D50

Gau

ssia

n,

wid

th(r

)=

22

10

09

8.3

19

2.3

1G

auss

ian

,w

idth

(r)

=1

.59

8.3

19

6.1

56

Nat Hazards (2011) 59:811–822 815

123

of targets t ¼ tif gNi¼1. For classification problem, ti should be -1 (Liquefaction) for class

C1 and ?1 (Nonliquefaction) for class C2. In this study, x ¼ soil parameter;½earthquake parameter� The RVM constructs a logistic regression model based on a set of

sequence features derived from the input patterns, i.e.,

p C1=xð Þ � r y x; wð Þf g where y x; wð Þ ¼XN

i¼1

wiUi xð Þ þ w0 ð12Þ

where basis function

U xð Þ ¼ U1 xð Þ;U2 xð Þ; . . .;UN xð Þð ÞT¼ 1;K xi; x1ð Þ;K xi; x2ð Þ; . . .;K xi; xNð Þ½ �T

w ¼ w0; . . .;wNð ÞT are a vector of weights, r yf g ¼ 1þ exp �yf gð Þ�1is the logistic sig-

moid link function, and K Xi;Xj

� �N

j¼1are kernel terms. Assuming a Bernoulii distribution

for P t=Xð Þ, the likelihood can be written as:

P t=wð Þ ¼YN

i¼1

r y xi; wð Þf gti 1� r y xi; wð Þf g½ �1�Ti ð13Þ

To form a Bayesian training criterion, we must also impose a prior distribution over the

vector of model parameters or weights, p(w). The RVM adopts a separable Gaussian prior,

with a distinct hyperparameter, ai, for each weight,

p w=að Þ ¼YN

i¼1

N wi=0; a�1i

� �ð14Þ

The optimal parameters of the model are then given by the minimizer of the penalized

negative log-likelihood,

log P t=wð Þp w=að Þf g ¼XN

i¼1

ti log yi þ 1� tið Þ log 1� yið Þ½ � � 1

2wT Aw ð15Þ

where yi ¼ r y xi; wð Þf g and A ¼ diag að Þ is a diagonal matrix with nonzero elements given

by the vector of hyperparameters. Next, Laplace’s method is used to obtain a Gaussian

approximation to the posterior density of the weights,

p w=t; að Þ � N w=l;X�

ð16Þ

where the posterior mean and covariance are given by

l ¼X

UT Bt andX¼ UT BUþ A� ��1 ð17Þ

where B ¼ diag b1; b2; . . .; bNð Þis a diagonal matrix with bn ¼ r y xnð Þf g 1� r y xnð Þf g½ �.The hyperparameter are then updated in order to maximize their marginal likelihood,

according to their efficient update formula

anewi ¼ 1� ai

Pii

l2i

ð18Þ

where li is the ith posterior mean weight,P

ii is the ith diagonal element of the posterior

weight covariance, and the quantity is a measure of the degree to which the associated

816 Nat Hazards (2011) 59:811–822

123

parameter wi is determined by the data. This process is repeated until an appropriate

convergence criterion is met. The outcome of this optimization is that many elements of ago to infinity such that w will have only a few nonzero weights that will be considered as

relevant vector.

The main scope of this work to implement the above methodology to predict the

liquefaction potential of soil is based on SPT data. In this RVM model, same data sets were

used for training and testing as used by LSSVM model. In case of RVM training, Gaussian

kernel has been used. A number of experiments were carried out on SPT data using

different combinations of input parameters to asses the liquefaction potential using RVM

(see Table 1). In the present study, training and testing of RVM have been carried out by

using MATLAB (MathWork Inc 1999).

4 Results and discussion

A simple trial-and-error approach has been used in this study to select design c and

width of Gaussian kernel (r) value for LSSVM model. Table 1 provides the result of

different LSSVM models and corresponding design c and r values. From the Table 1, it

is clear that Model VI gives the best result for testing data set using LSSVM model.

The performance of training and testing has been calculated by using the following

formula:

Training performance ð%Þ or Testing performance ð%Þ

¼ No data predicted correctly by LSSVM or RVM

Total data

� �� 100 ð19Þ

Model VI has an overall success rate of 92.31%, with one error in training data set and

two errors in the testing data set. So the difference between training and testing perfor-

mance of Model VI is very marginal. Therefore, the developed LSSVM model has good

generalization capability. Tables 2 and 3 show the training and testing performance of

Model VI. Figure 1 shows the probability of liquefaction for training and testing data set.

Figure 1 also gives an indication of prediction uncertainty of the developed LSSVM

model.

For RVM model, the design value of r has been chosen by trial-and-error approach

(Samui 2007). Table 1 provides the result of different RVM models and corresponding

design r values. Table 1 also shows that the Model VI gives best result, with one error

in training data set and one error in the testing data set. So the performance of training

and testing data set is almost same. Therefore, the developed RVM model has the ability

to avoid overtraining. From Table 1, it is clear that RVM model outperforms LSSVM

model. Tables 2 and 3 show the training and testing performance of Model VI. Figure 2

shows the probability of liquefaction for training and testing data set. An indication of

prediction uncertainty of the developed RVM model can be obtained from Fig. 2.

Probabilistic results from RVM indicate that all liquefiable soil fall within the 0–50%

probability range and all nonliquefiable soil fall within the 51–100% range. The prob-

abilistic output from RVM model can be used to determine the liquefiable soil. If the

output is greater than 50%, the probability of liquefaction is decreased. If the output is

less than 50%, the probability of liquefaction is increased. In this study, RVM model

employs approximately 7–17% (for Model I = 6.77%, Model II = 11.86%, Model

Nat Hazards (2011) 59:811–822 817

123

Table 2 Performance of training data using Gaussian kernel for Model VI

M r0(kPa) r00(kPa) SPT (N) a/g s/r00 F (%) D50 (mm) Actualclass

Predictedclass byRVM

Predictedclass byLSSVM

7.9 186.4 96.1 20 0.32 0.36 0 0.46 1 1 1

7.9 130.5 81.4 10 0.32 0.32 5 0.28 1 1 1

7.9 111.8 71.16 17 0.28 0.28 3 0.8 1 1 1

7.9 93.2 67.7 13 0.28 0.25 4 0.6 1 1 1

7.9 122.6 93.2 10 0.2 0.16 10 0.25 1 1 1

7.9 141.3 102 1 0.2 0.17 14 0.25 1 1 1

7.9 71.6 69.7 2.2 0.2 0.13 22 0.18 1 1 1

7.9 149.1 80.4 16.5 0.2 0.23 1 0.28 1 1 1

7.9 93.2 63.8 11.9 0.2 0.19 5 0.3 1 1 1

7.9 93.2 73.6 5.7 0.2 0.16 20 0.2 -1 1 -1

7.9 149.1 100.1 2 0.2 0.18 33 0.15 -1 -1 -1

8 89.3 59.8 8 0.2 0.19 10 0.4 1 1 1

8 64.7 35.3 1 0.2 0.24 27 0.2 1 1 1

8 50 45.1 2 0.2 0.15 30 0.15 1 1 1

8 130.5 81.7 10 0.16 0.16 5 0.28 1 1 1

7.3 66.7 35.3 7 0.35 0.39 35 0.13 -1 -1 -1

7.3 141.3 70.6 29 0.35 0.39 2 0.8 -1 -1 -1

7.3 123.6 91.2 19 0.35 0.27 4 0.65 1 1 1

7.3 74.6 47.2 8 0.4 0.38 0 0.45 1 1 1

7.3 100.1 51 8 0.4 0.45 21 0.1 -1 -1 -1

7.3 128.5 63.8 20 0.4 0.45 0 0.45 -1 -1 -1

7.5 130.5 71.6 8 0.16 0.17 2 0.3 1 1 1

7.5 130.5 71.6 12 0.16 0.17 2 0.3 -1 -1 -1

7.5 128.5 79.5 18 0.16 0.15 2 0.3 -1 -1 -1

7.5 186.4 98.1 10 0.16 0.17 2 0.3 1 1 1

7.5 186.4 98.1 16 0.16 0.17 2 0.3 -1 -1 -1

7.5 184.4 105.9 20 0.16 0.15 2 0.3 -1 -1 -1

7.5 80.4 38.3 4 0.16 0.21 10 0.4 1 1 1

7.5 111.8 65.7 27 0.16 0.16 0 0.3 -1 -1 -1

7.5 93.2 68.7 12 0.16 0.13 0 0.36 -1 -1 -1

7.5 84.4 46.1 6 0.16 0.18 0 0.4 1 1 1

7.9 74.6 45.1 5 0.2 0.21 20 0.12 1 1 1

7.9 111.8 72.6 28 0.23 0.22 5 0.25 -1 -1 -1

7.9 74.6 41.2 6 0.23 0.27 5 0.25 1 1 1

7.9 74.6 45.1 16 0.23 0.25 5 0.25 -1 -1 -1

6.7 118.7 67.7 10 0.1 0.09 0 0.6 -1 -1 -1

6.7 61.8 34.3 5 0.12 0.12 5 0.7 1 1 1

6.7 61.8 41.2 7 0.12 0.1 4 0.28 -1 -1 -1

6.7 80.4 47.1 11 0.12 0.11 0 0.4 -1 -1 -1

6.7 80.4 54.9 4 0.12 0.09 10 0.4 -1 -1 -1

6.7 61.8 41.2 13 0.12 0.1 7 1.6 -1 -1 -1

6.7 80.4 41.2 9 0.12 0.13 12 1.2 -1 -1 -1

818 Nat Hazards (2011) 59:811–822

123

III = 13.55%, Model IV = 16.94, Model V = 10.11%, Model VI = 10.11%) of the

training data as relevance vectors. It is worth mentioning here that the relevance vectors

in the RVM model represent prototypical examples. The prototypical examples exhibit

the essential features of the information content of the data and thus are able to trans-

form the input data into the specified targets. So there is real advantage gained in terms

of sparsity. 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.

LSSVM does not exhibit any sparse solution.

A comparative study has been also done between developed LSSVM, RVM, and ANN

model developed by Goh (1994). The comparison has been carried out for testing data set.

From the Table 4, it is clear the RVM model outperforms LSSVM and ANN model. The

performance of LSSVM model is in between RVM and ANN model. The performance of

LSSVM model is better than ANN except Model VI. RVM model requires smaller tuning

parameter (only one parameter i.e., width of Gaussian kernel) compared with ANN and

LSSVM. ANN requires number of hidden layers, number of hidden nodes, learning rate,

momentum term, number of training epochs, transfer functions, and weight initialization

methods. LSSVM requires capacity factor, kernel parameter as their respective tuning

parameters. ANN model uses standardized SPT value {(N1)60} for the prediction of seismic

liquefaction potential of soil, whereas the developed LSSVM and RVM models use SPT

value (N). So the standardization of SPT value is not required for LSSVM and RVM

model.

Table 2 continued

M r0(kPa) r00(kPa) SPT (N) a/g s/r00 F (%) D50 (mm) Actualclass

Predictedclass byRVM

Predictedclass byLSSVM

6.7 103 83.4 9 0.14 0.09 5 0.34 -1 -1 -1

6.7 108.9 70.6 8 0.14 0.11 4 0.36 -1 -1 -1

6.7 59.8 56.9 11 0.14 0.08 5 0.53 -1 -1 -1

6.7 74.6 59.8 6 0.14 0.09 10 0.25 -1 -1 -1

6.7 93.2 68.7 9 0.14 0.1 20 0.15 -1 -1 -1

6.7 111.8 77.5 12 0.14 0.11 3 0.35 -1 -1 -1

6.7 74.6 49.1 4 0.12 0.1 10 0.15 -1 -1 -1

5.5 111.8 48.1 6 0.19 0.1 3 0.2 1 1 1

8.3 56.9 53 10 0.16 0.22 10 0.2 1 1 1

6.6 72.6 86.3 9 0.45 0.29 20 0.1 1 1 1

7.5 72.6 28.4 8 0.14 0.17 3 1 1 1 1

7.5 93.2 34.3 8 0.14 0.14 3 1 -1 -1 -1

7.5 58.9 62.8 14 0.14 0.13 3 1 -1 -1 -1

6.6 107.9 51 31 0.6 0.45 11 0.12 -1 -1 -1

6.6 58.9 51 4 0.6 0.45 25 0.12 1 1 1

6.6 86.7 51 11 0.6 0.45 19 0.1 -1 -1 1

6.6 72.6 46.1 7 0.2 0.21 34 0.09 1 1 1

Nat Hazards (2011) 59:811–822 819

123

Table 3 Performance of testing data using Gaussian kernel for Model VI

M r00(kPa) r00(kPa) SPT (N) a/g s/r0 F (%) D50 (mm) Actual

classPredictedclass byRVM

Predictedclass byLSSVM

7.4 118.7 66.7 10 0.2 0.21 0 0.6 1 1 1

7.4 61.8 38.3 19 0.32 0.31 4 0.28 -1 -1 -1

7.4 61.8 34.3 5 0.32 0.35 5 0.7 1 1 1

7.4 61.8 41.2 7 0.32 0.29 4 0.28 1 1 1

7.4 80.4 47.1 11 0.24 0.25 0 0.4 1 1 1

7.4 97.1 66.7 20 0.24 0.21 0 0.6 -1 -1 -1

7.4 80.4 54.9 4 0.24 0.21 10 0.4 1 1 1

7.4 61.8 41.2 13 0.24 0.22 7 1.6 1 1 -1

7.4 80.4 41.2 8 0.24 0.28 12 1.2 1 1 1

7.4 136.4 77.5 17 0.24 0.24 17 0.35 -1 -1 -1

7.4 103 83.4 9 0.24 0.17 5 0.34 1 1 1

7.4 108.9 70.6 8 0.24 0.21 4 0.36 1 1 1

7.4 5.8 56.9 11 0.24 0.18 5 0.53 1 1 1

7.4 109.9 80.4 23 0.24 0.22 0 0.41 -1 -1 -1

7.4 111.8 77.5 10 0.24 0.2 10 0.3 -1 1 -1

7.4 74.6 59.8 6 0.24 0.18 10 0.25 1 1 1

7.4 130.5 86.3 21 0.24 0.21 5 0.35 -1 -1 -1

7.4 93.2 68.7 9 0.24 0.19 20 0.15 1 1 -1

7.4 83.4 63.8 10 0.24 0.19 26 0.12 -1 -1 -1

7.4 111.8 77.5 12 0.24 0.2 3 0.35 1 1 1

7.4 106.9 71.6 15 0.24 0.21 11 0.3 -1 -1 -1

7.4 124.6 91.2 17 0.24 0.19 12 0.3 -1 -1 -1

7.4 74.6 49.1 4 0.2 0.18 10 0.15 1 1 1

7.4 111.8 66.7 15 0.2 0.2 10 0.18 -1 -1 -1

6.1 105.9 56.9 5 0.1 0.09 13 0.18 -1 -1 -1

6.1 247.2 105.9 4 0.1 0.09 27 0.17 -1 -1 -1

Pro

babi

lity

(%)

No Liquefaction Liquefaction

Fig. 1 Probability from LSSVMmodel

820 Nat Hazards (2011) 59:811–822

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5 Conclusion

This study examines the feasibility of LSSVM and RVM model for the prediction of

liquefaction potential of soil using SPT data. Comparisons indicate that the RVM model is

more reliable than ANN and LSSVM model. LSSVM and RVM have the added advantage

of probabilistic interpretation that yields prediction uncertainty. Proposed LSSVM and

RVM models suggest that standardized SPT value {(N1)60} is not required for the deter-

mination of liquefaction potential of soil. In the development of LSSVM and RVM model

discussed here, significant effort is required to build machine architecture. However, once

developed and trained, the transpired model performed the simulation in a small fraction of

the time required by the physically based model. In summary, this paper has surveyed

RVM that could be viewed as powerful alternative approaches to physically based models

for the determination of liquefaction potential of soil.

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Table 4 Comparison between developed LSSSVM and RVM with ANN model

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