least square support vector machine and relevance
TRANSCRIPT
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: [email protected]
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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