International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 3, March 2013
ISSN 2319 - 4847
A Fuzzy Twin Support Vector Machine
Algorithm
Kai Li 1, Hongyan Ma 2
1
School of Mathematics and Computer Science, Hebei University, Baoding 071002, China
2
Industrial and commercial college, Hebei University, Baoding 071002, China
ABSTRACT
Although twin support vector machine (TSVM) has faster speed than traditional support vector machine for classification
problem, it does not take into account the importance of the training samples on the learning of the decision hyper-plane with
respect to the classification task. In this paper, fuzzy twin support vector machine (FTSVM) is proposed where a fuzzy
membership value is assigned to each training sample. Here, training samples are classified by assigning them to the nearest
one of two nonparallel planes that are close to their respective classes. Moreover, this method only requires solving a smaller
size SVM-type problem as compared to SVMs where the classifier is obtained by solving a quadratic programming problem.
Experiments on several UCI benchmark datasets show that FTSVM is effective and feasible compared with twin support vector
machine(TSVM), fuzzy support vector machine(FSVM) and support vector machine(SVM).
Keywords: Twin Support Vector Machine, Fuzzy weighting, Classification
1. INTRODUCTION
Support vector machine (SVM) is a powerful tool for pattern classification and regression and has drawn many
researchers more and more the attention due to its generalization performance. Its theory basis is from structural risk
minimization (SRM) principle. Support vector machine first maps the input points into a high-dimensional feature
space and then finds an optimal hyper-plane that maximizes the margin between two classes in this feature space.
Maximizing the margin between the two classes is attributed to solve a quadratic programming problem (QPP). In
addition, support vector machine finds the optimal hyper-plane by means of kernel function without any knowledge of
the mapping. The solution of the optimal hyper-plane can be written as a combination of a few input points called
support vectors. Recently, the support vector machine has been successfully applied in many fields like text
categorization, financial applications and etc.
Though SVM possesses better generalization performance compared with many other machine learning algorithms,
it has larger computational complexity due to solving the quadratic programming problem and is sensitive to noise. To
address these problems, a number of novel SVM models were proposed, such as least squares support vector machine
(LS-SVM) and proximal support vector machine (PSVM). Moreover, Lin and Wang reformulated SVM to fuzzy
support vector machine (FSVM) using fuzzy membership to each sample of SVM such that different samples can make
different contributions to the surface [1]. In reality, SVM and proximal support vector machine (PSVM) aim to seek for
one and only one separating plane, but it is difficult to efficiently deal with the complex cases (e.g. XOR problems).
Therefore, Fung and Mangasarian proposed multi-surface proximal support vector machine via using generalized
eigen-values (GEPSVM) [2]. The idea of GEPSVM is to find two nonparallel hyper-planes. Each surface is as close as
possible to the samples of its own class and as far as possible from the samples of the other classes. And its
computational cost is smaller than SVM. Subsequently, Jayadeva et al. proposed the twin support vector machine
(TSVM) in the light of GEPSVM where solves two smaller size SVM-type problems to obtain the hyper-planes [3].
Compared to the traditional SVM, TSVM reduces the time complexity. Then, Kumar et al. proposed a least squares
version of TSVM (LS-TSVM) [4]. In this paper, in order to further enhance the performance of TSVM, we propose a
fuzzy version of TSVM by incorporating a membership value for each sample, called fuzzy twin support vector
machine (FTSVM), based on idea both FSVM and TSVM. In fact, in the primal problem of TSVM, it is sensitive to
noise. However, in our presented algorithm FTSVM, by adding an extra fuzzy value, it ensures the minimal effect of
noise and the better generalization ability. Compared with FSVM, TSVM and SVM, FTSVM has larger superiority in
terms of classification accuracy and computing time.
The paper is organized as follows. Section 2 briefly introduces support vector machine and Twin Support Vector
Machine. Section 3 describes our algorithm FTSVM in detail. Section 4 presents the experimental results. Section 5
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Volume 2, Issue 3, March 2013
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contains concluding remarks.
2. SUPPORT VECTOR MACHINE
Consider
a
binary
data
classification
n
( xi , yi )  R  {1, 1} . Denote by I



problem
with
data
set D  {( x1 , y1 ), ( x2 , y2 ), ( xl , yl ) ,
where
the set of indices i such that yi  1 and by I the set of all indices, i.e.
l1  n
I  I  I . Let matrix A  R and B  R l2 n represent positive and negative training samples, respectively, where l1
and l2 are numbers of samples with +1 class and -1 class.
2.1 Traditional Support Vector Machine
Traditional support vector machine finds an optimal separating hyper-plane between two classes of samples in some
feature space in order to generate a classifier with maximal margin. The optimal separating hyper-plane is written as
follows:
wT x  b  0 .
(1)
In reality, the above described hyper-plane (1) lies in the middle between the bounding hyper-planes given by
wT x  b  1 , wT x  b  1 .
To obtain the hyper-plane, SVM solves the following optimal problem by maximizing margin between +1 class and 1 class:
1
|| w ||2
or
2
s.t. yi ( w, xi  b)  1, i  I
1
min || w ||2
,
w ,b
2
T
s.t. yi ( w  ( xi )  b)  1, i  I
where  is a transformation from primal space to feature space. The final classifier is given
by f ( x )  sign ( w T x i  b ) or f ( x )  sign ( w T x i  b ) . However, samples are non linear separable in most cases. That is
to say that there exists no some separable hyperplane. To allow for the possibility of samples violating
y i ( w T  ( x i )  b )  1, i  I by introducing nonnegative slack variables  i  0 .The optimal hyperplane problem can be
min
w ,b
expressed as the following quadratic programming problem with inequalities
l
1
min
|| w ||2  C   i
,
w , b ,
2
i 1
s.t. yi ( w,  ( xi )   b)  1   i , i  0, i  I
where C is a constant which is a cost trade-off between maximizing the margin and minimizing the classification error
of the training samples.
2.2 Twin Support Vector Machine
The twin support vector machine (TSVM) is a new nonparallel plane classifier for binary data classification. It
generates two nonparallel planes by solving two smaller-sized quadric programming problems such that each plane is
closer to one of the two classes and is as far as possible from the other. That a new sample is assigned to class +1 or -1
depends upon its proximity to the two nonparallel hyper-planes. The linear classifier TSVM aims to obtain following
two nonparallel planes
wT x  b  0, wT x  b  0 .
This leads to the following pair of quadratic optimization problem
1
,
min
|| Aw  e b ||2  c1eT  
s.t.
 ( Bw  e b )  e    ,    0
w , b ,  2
(2)
1
min
|| Bw  e b ||2  c2 eT  
s.t.
( Aw  e b )  e    ,    0 ,
w , b , 2
(3)
where c1 and c2 are the tuning parameters, e and e are of appropriate dimensional vectors whose all elements are
one. The algorithm finds two hyper-planes, one for each class, and classifies samples according to which hyperplane a
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Volume 2, Issue 3, March 2013
ISSN 2319 - 4847
given sample is closest to. For (2), its objective function makes class +1 proximity to the hyper-plane wT x  b  0 ,
which the constraints make class -1 proximity to the hyperplane wT x  b  0 .By introducing the Lagrange multipliers,
the dual problems with (2) and (3) are written in the following:
1
(3)
max eT    T G ( H T H ) 1 G T 
s.t. 0    c1 ,
2
1
max eT    T Q( PT P) 1 QT 
2
0    c2 .
s.t.
(4)
The two nonparallel hyper-planes can be obtained from the solution of (3) and (4)
 T  (wT , b )  ( H T H   I )1 GT 
 T  (wT , b )  ( PT P   I ) 1QT 
.
(5)
In order to deal with the case which H T H or PT P is singular and avoid the possible ill-condition of H T H and PT P ,
formula (5) above artificially introduces a regularization term  I (  0) , where I is an identity matrix of appropriate
dimension.
It is be seen that in the above discussion, the linear TSVM requires matrices of size (n+1)×(n+1), where n is much
smaller in comparison to the number of pattern of class +1 and -1.
For nonlinear case, the separating nonparallel planes are changed by introducing a nonlinear kernel K, namely
K ( xT , C T )u1  b1  0, K ( x T , C T )u 2  b2  0 ,
where C T  [ A B ]T and K is a appropriate kernel. The primal problems of nonlinear TSVM are given as follows:
1
min
|| K ( A, C T )u1  e1b1 ||2  c1eT2  2
s.t .
 ( K ( B , C T )u1  e2b1 )  e2   2 ,  2  0 ,
u1 , b1 , 2 2
1
min
|| K ( B , C T )u 2  e2 b2 ||2  c 2 e1T 1
s.t .
K ( A, C T )u 2  e1b2  e2  1 , 1  0 .
u1 , b1 , 2 2
2.3 v-Twin Support Vector Machine
Similar to v-SVM, introducing two new parameter v1 and v2 instead of the trade-off factors C1 and C2, Peng
proposed v-twin support vector machine (v-TSVM) and rewrite the primal optimization problems as follows [5]:
1
1
min (wT xi  b )2  v1   j
st. .  (wT x j  b )   j ,   0, j  0, j I .
w ,b , j 2 

l
iI
j I
min
w ,b ,i
1
1
(wT x j  b )2  v2   i

2 jI
l iI 
st. .
wT xi  b   i ,   0,i  0,i I .
To understand the roles of   for all  j  0, j  I  (or  I  0, j  I  ), the negative (positive) samples are separated
by the positive (or negative) hyperplane with the margin   / ( wT w ) (or   / ( wT w ) ). At the same time, the adaptive
quality effectively overcomes the above shortcomings in the TSVM. By introducing Lagrangian multipliers, two dual
QPPs are obtained in the following:
1
1
min
  j1 j 2 z Tj1 (  zi ziT ) 1 z j 2 s.t. 0   j  l  ,   j  v1 , j  I  .
2 j1, j 2I 
iI
iI
min
1
  i1 i 2 ziT1 (  z j z Tj ) 1 zi 2
2 i1,i 2I 
jI
0  i 
s.t.
1
,   i  v2 i  I  .
l  i I 
To compute   , the samples xi , i  I  (or xi , i  I  ) with 0    1 (or 0    1 ) are chosen, which means  i  0
i
j


l
l
(or  j  0 ) and wT xi  b    (or wT xi  b    ).
3. FUZZY TWIN SUPPORT VECTOR MACHINE
3.1 Linear Fuzzy Twin Support Vector Machine
It may be seen that no matter twin support vector machine and v-twin support vector machine, they do not consider the
effects of training samples on the optimal separating hyper-plane. In this paper, we propose fuzzy twin support vector
machine (FTSVM) by introducing importance of sample. According to weighted sample on the non-parallel
hyperplane, the quadratic programming problems are given as follows:
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Volume 2, Issue 3, March 2013
ISSN 2319 - 4847
min
w(1) , b (1) , ( 2 ) , 1
1
1
|| Aw (1)  eb (1) ||2  1 1  s 2T  ( 2)
2
l2
s.t .  ( Bw(1)  b (1) )  1   (2) ,  (2)  0, 1  0 ,
(6)
1
1
(7)
|| Bw(2)  eb (2) ||2  2  2  s1T  (1)
s.t. ( Aw(2)  b (2) )   2   (1) ,  (1)  0, 2  0 ,
2
l1
where v1 , v 2  (0,1] denotes the regularization parameter of positive and negative samples, respectively. s1 , s2  (0,1]
min
w(1) ,b (1) , ( 2 ) ,  2
denotes the fuzzy membership of positive and negative samples, respectively. Objective for FTSVM finds the two nonparallel hyperplane, namely a positive hyperplane and a negative hyperplane. For the sake of brevity, we only consider
the dual problem of optimal problem (6). In order to solve the optimization problem, we construct the following
Lagrange function corresponding to the problem (6)
1
1
(8)
L  || Aw (1)  eb (1) ||2  1 1  s2T  (2)   T ( Bw (1)  b (1)  1   (2 ) )   T  ( 2)  1 ,
2
l2
where Lagrange multipliers  ,  ,  are all greater than zero. According to Karush-Kuhn-Tucker (KKT) conditions
L
(9)
 AT ( Aw (1)  eb (1) )  B T   0 ,
w (1)
L
 e1T ( S1 Aw(1)  eb (1) )  e2T   0 ,
(1)
b
L
 v1  e2T     0 ,
1
L
s
 2     0 ,
(2)

l2
By simple Computation according to above equations, we obtain following equation (13).
[ AT e1T ][ A e1 ][ w (1) b (1) ]T  [ B T e T2 ]  0
Let H  [ A e1 ], U  [ w
(1)
(10)
(11)
(12)
(13)
(1) T
b ] , G  [ B e 2 ] and rewrite (13) as follows
H T HU  G T   0, U   ( H T H ) 1 G T  .
(14)
T
It is well known that H H is always positive semi-definite. However, it may be ill-conditioned in some situation.
Thus, according to ridge regression approaches, we introduce a regularization term  I to U to deal with possible illcondition for H T H , where I is identity matrix with suitable order. Now, (14) becomes
U   ( H T H   I ) 1 G T  .
Applying equations (9) to (12) into the Lagrange function, the primal problem (6) can be transforms into the
following dual problem
1
min  T G ( H T H ) 1 G T 

2
.
(15)
s2 T
s.t. 0e    , e   v1
l2
From the KKT conditions, we obtain
 T ( Bw(1)  b (1)  1   ( 2) )  0,  T  ( 2)  0, 1  0.
Similarly, we obtain the parameters of another hyper-plane ( w( 2) b (2) )T  R , R   (Q T Q ) 1 P T  . The dual problem
of the primal optimal problem (7) is given by
1
min  T P(Q T Q ) 1 P T 

2
,
(16)
s1 T
s.t. 0e    , e   v2
l1
where P  [A e1 ] and Q  [B e 2 ] .
3.2 The Nonlinear Fuzzy Twin Support Vector Machine
In this section, we extend the presented method above to the nonlinear situation using kernel trick and consider the
kernel-based
surfaces
rather
than
planes
in
primal
space,
T
(1)
(1)
T
(2 )
(2)
namely K ( A, C )     0, K ( B , C )     0 ,where C  [ A B ] and K denotes an chosen kernel function.
We construct the optimization problem of FTSVM as follows:
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min
w ,b , , 
min
 ,b , , 
1
1
|| K ( A, C T ) w (1)  eb (1) ||2  1 1  s2T  (2)
2
l2
1
1
|| K ( B, C T ) w (2 )  eb (2 ) ||2  2  2  s1T  (1)
2
l1
s.t.  ( K ( B , C T ) w (1)  eb (1) )  1   (2) ,  (2 )  0, 1  0,
(17)
s.t . ( K ( A, C T ) w (2)  eb (2) )   2   (1) ,  (1)  0,  2  0.
(18)
We construct Lagrange function of primal problem (17) as
1
1
L  || K ( A, CT )w(1)  eb(1) ||2  11  s2T  (2)  1   T (K ( B, CT )w(1)  b(1)  1   (2) )   T  (2) ,
2
l2
where   0,   0,   0 are Lagrange multipliers. According to KKT conditions
L
 K (A, C T )T ( K (A, C T ) w(1)  e1b (1) )  B T   0 ,
w(1)
L
 e1T ( K (A, C T ) w (1)  eb (1) )  e2T   0,
b (1)
L
  v1  e2T     0,
1
L
s
 2      0.
 (2 ) l2
According to similar method above, we obtain equation (19).
[( K ( A, C T )T e1T ][ K ( A, C T ) e1 ][ w (1) b (1) ]T  [( K ( B , C T )T e T2 ]  0 .
T
Let H  [ K ( A, C ) e1 ], U  [ w
(1)
(1) T
(19)
T
b ] , G  [ K ( B , C ) e 2 ] . Then the equation (19) is modified as follows
H T HU  G T   0, U   ( H T H ) 1 G T  .
So, the dual problem of nonlinear FTSVM is given by
min

1 T
 G ( H T H ) 1 G T 
2
s.t.
0  
s2 T
, e   v1 .
l2
(20)
Similarly, we also obtain the following dual problem for optimization problem (18), where P  [ K (A,C T ) e1 ] and
Q  [ K (B,C T ) e 2 ] .
1 T
s
(21)
 P(Q T Q ) 1 P T 
s.t. 0    1 , eT   v2 .
2
l1
Based on above derivation, we give fuzzy twin support vector machine algorithm FTSVM in the following which
include linear FTSVM and nonlinear FTSVM.
Step 1 Choose a kernel function and compute membership of each sample for class +1 and class -1 to construct
vector s1 and s2.
Step 2 Compute H and G.
Step 3 Set values of parameters v1 , v 2  (0 ,1) .
min

Step 4 Solve the quadratic programming problems (15) or (20) and (16) or (21) to obtain U and R for two
nonparallel hyperplanes.
Step 5 Compute distance dist+1 between x  R n and x T w (1)  b (1)  0 and distance dist-1 between x  R n and
x T w ( 2 )  b ( 2 )  0 , respectively.
Step 6 Compare dist+1 with dist-1, if dist+1> dist-1 then x is assigned to class +1 else class -1.
3.3 Fuzzy Membership Function
The design of fuzzy membership function is the key to the fuzzy algorithm using fuzzy technology. In this paper, we
use class center method to generate fuzzy membership. Firstly, we denote the mean of class +1 as class-center x and
the mean of class -1 as class center x , respectively. The radius of each class r+ and r- are the farthest distance between
the each class training points and its class-center, respectively, namely r  max || x  xi || and r  max || x  xi || .
{ xi : yi 1}
{ xi : yi  1}
Fuzzy membership si is a function of the mean and radius of each class
1 || x  xi || /(r   ), if yi  1
si  
,
 1 || x-  xi || /(r-   ), if yi  -1
where   0 is used to avoid the case si  0 .
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Volume 2, Issue 3, March 2013
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4. EXPERIMENTAL RESULTS AND ANALYSIS
In this section, to evaluate the performance with proposed algorithm FTSVM, we investigate its classification
accuracies and computational efficiencies on 7 real-world UCI benchmark datasets [6]. In experiments, we focus on the
comparison between the proposed algorithm FTSVM and some methods which include TSVM, FSVM and SVM. All
the classification methods are implemented in Matlab 7.0 environment on a PC with Intel P4 processor with 1GB
RAM. We compute the fuzzy membership by a function of the distance between the points and its class center.
Table 1 gives the classification accuracy of linear FTSVM with TSVM, FSVM, and SVM using 5-fold crossvalidation method. From Table 1, we can see that the accuracy of linear FTSVM is significantly better than linear
TSVM on all 7 UCI datasets. We also report the training time of the algorithms which is shown in Table 2. It indicates
that FTSVM is faster than the FSVM, because it solves two smaller size problems instead of one large size problem for
all samples. However, there is no statistical different in average training time between FTSVM and FSVM for bupa
dataset. Thus FTSVM is better than FSVM in the accuracy. Table 3 compares the performance of the FTSVM classifier
with that of TSVM, FSVM and SVM for Gaussian kernel. The results in Table 3 are similar with that appeared in
Table 1. That is to say that FTSVM has the better classification accuracy than TSVM in all datasets.
Table 1: Classification accuracy using 5-fold cross-validation
Data Set
FTSVM
TSVM
FSVM
SVM
australian
85.97±5.16
85.79±5.09
85.89±4.79
85.51±4.58
breast-cancer
65.00±4.13
62.83±3.16
64.86±2.48
64.86±4.73
bupa
74.82±3.18
68.40±6.38
74.80±2.52
69.28±3.02
fourclass
64.39±7.28
64.39±5.70
68.54±8.84
73.66±6.32
german
78.14±8.15
71.20±6.35
70.80±8.20
76.90±7.63
heart
85.56±4.45
82.22±6.60
82.59±6.08
81.48±8.58
pima
79.08±5.92
73.02±6.05
76.95±2.45
76.55±2.40
Table 2: Training time (in Seconds)
Data Set
FTSVM FSVM
australian
12.50
133.53
breast-cancer
16.27
150.17
bupa
2.14
2.13
fourclass
7.89
29.44
german
14.98
50.09
heart
0.23
0.45
pima
8.58
42.65
Table 3: Classification accuracy using 5-fold cross-validation with RBF kernel
Data Set
FTSVM
TSVM
FSVM
SVM
australian
86.08±1.43
84.81±2.15
85.56±2.30
85.51±2.16
breast-cancer
65.60±4.32
64.42±3.87
65. 01±2.48
65.42±4.53
bupa
77.80±3.87
71.45±5.49
76.67±2.21
72.78±3.97
fourclass
64.53±5.51
64.45±5.49
64.38±6.18
64.35±6.48
german
78.20±8.15
72.45±6.35
71.68±8.20
73.50±7.63
heart
84.44±4.53
81.89±4.31
83.33±5.00
82.22±6.67
pima
79.51±5.92
73.70±6.05
77.42±2.45
76.55±2.40
In addition, aiming at noise sensitive problem of twin support vector machine, we study and compare the performance
of fuzzy twin support vector machine FTSVM and twin support vector machine. First, we use random method to
produce two class samples, denoted by “×” and “·”, respectively. Then, we add three noise data points (-2, -2), (-2, 0)
and (-1, -1), respectively. Figure 1 shows the classification result on TSVM, where data is not added noise. Figure 2
and Figure 3 show the classification results with noise data on TSVM and FTSVM, respectively. From Figure 2 and
Figure 3, we observe that fuzzy twin support vector machine is almost unaffected by noise data, whereas twin support
vector machine is largely affected by noise. This indicates that the fuzzy memberships of samples play an important
role in the classification.
Volume 2, Issue 3, March 2013
Page 464
International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 3, March 2013
ISSN 2319 - 4847
5. CONCLUSIONS
In this paper, we study fuzzy twin support vector machine (TSVM) by applying fuzzy membership to training samples.
Samples are classified by assigning them to the nearest one of two non parallel planes. Experiments on several UCI
benchmark datasets show that the presented algorithm FTSVM is effective and feasible relative to twin support vector
machine, fuzzy support vector machine and support vector machine. Moreover, we show that presented algorithm
FTSVM is of anti-noise capability. In the future, we further study fuzzy twin support vector machine and expand it to
multi-classification problem.
Acknowledgements
This work is support by Natural Science Foundation of China (No. 61073121) and Nature Science Foundation of Hebei
Province (No. F2012201014).
References
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[3] Jayadeva, Khemchandni Reshma, “Suresh Chandra. Twin support vector machines for pattern classification,”
IEEE Transaction on Pattern Analysis and Machine Intelligence, 29(5), pp. 905-910, 2007.
[4] Kumar M. Arun, Gopal M, “Least squares twin support vector machines for pattern classification,” Expert Systems
with Applications, 36(4), pp. 7535-7543, 2009.
[5] Peng Xinjun, “A v-twin support vector machine (v-TSVM) classifier and its geometric algorithms,” Information
Sciences, 180, pp. 3863-3875, 2010.
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AUTHOR
Kai Li received the B.S. and M.S. degrees in Mathematics Department Electrical Engineering
Department from Hebei University,Baoding, China, in 1982 and 1992, respectively. He received the
Ph.D. degree from Beijing Jiaotong University, Beijing, China, in 2001. He is currently a Professor in
School of Mathematics and Computer Science, Hebei University. His current research interests include
machine learning, data mining, computational intelligence, and pattern recognition.
Hongyan Ma received the B.S. and M.S. degrees in Information and Computational Science and
Applied Mathematics from Hebei University,Baoding, China, in 2000 and 2007, respectively. She is
currently with industrial and commercial college of Hebei University as a teacher. Her current research
interests include machine learning, data mining and information retrieval.
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Page 465