基于模糊训练集的支持向量机算法1
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Support vector machine for classification based on fuzzy training data
q
Ai-bing Ji
a,b,
*
, Jia-hong Pang
b
, Hong-jie Qiu
b
a
Department of Computer Science and Engineering, Nanjing University of Aeronautics and Astronautics, 210016 Nanjing, PR China
b
College of Medicine, Hebei University, Baoding 071000, Hebei, PR China
article info
Keywords:
Support vector machine
Fuzzy training examples
Possibility measure
Fuzzy linear separable example
Fuzzy chance constraint programming
abstract
Support vector machines (SVM) have been very successful in pattern recognition and function estimation
problems, but in the support vector machines for classification, the training example is non-fuzzy input
and output is y = ±1; In this paper, we introduce the support vector machine which the training examples
are fuzzy input, and give some solving procedure of the Support vector machine with fuzzy training data.
Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction
The support vector machine (SVM) is a training algorithm for
learning classification and regression rules from data, SVMS were
first introduced by Vapnik (1995, 1998) and Cortes and Vapnik
(1995) in the 1990s for classification and have recently become
an area of intense research owing to developments in the tech-
niques and theory coupled with extensions to regression and den-
sity estimation. SVM is based on the structural risk minimization
principle, this principle incorporates capacity control to prevent
over-fitting and thus is a partial solution to the bias-variance
trade-off dilemma.
In the support vector machine for classification, the training
example is non-fuzzy input and the output is y = ±1. Considering
the noisy in the training example set, fuzzy membership was intro-
duced in classification by Chen and Chen (2002), Tsujinishi and Abe
(2003), Kikuchi and Abe (2005) and Lin and Wang (2004) intro-
duced the fuzzy support vector machine, it used the membership
function to express the membership grade of an example to posi-
tive class or negative class. But in nature, it is still a common sup-
port vector machine of Vapnik.
In fact, because the noisy and error of measurement, the train-
ing examples are usually uncertain (Jeng, Chuang, & Su, 2003)or
fuzzy. Then the study of support vector machine with fuzzy train-
ing data is very significant.
In this paper, we first give some preliminary knowledge, then
for fuzzy training data, introduce the concept of fuzzy linear sepa-
rable and approximately fuzzy linear separable. At last, we system-
atically study the support vector machine for two-class
classification with fuzzy training data.
2. Preliminary
Here we focus on SVM for two-class classification, for the train-
ing sample (x
1
,y
1
),(x
2
,y
2
),...,(x
k
,y
k
) 2 R
n
{±1}, y
i
= +1, 1 represent
positive class and negative class respectively. The geometrical
interpretation of support vector classification (SVC) is that the
algorithm searches for the optimal separating hyperplane, SVM is
outlined first for the linearly separable case.
The training data are linearly separable, if there exists a pair
(w,b) such that
w
T
x
i
þ b P 1; for all y
i
¼þ1
w
T
x
i
þ b 6 1; for all y
i
¼1
ð1Þ
with the decision rule given by
f
w;b
ðxÞ¼sgnðw
T
x þ bÞ: ð2Þ
w is termed the weight vector and b is the bias (or b is termed the
threshold). The inequality constraints (1) can be combined to give
y
i
ðw
T
x
i
þ bÞ P 1; ð3Þ
The learning problem is hence reformulated as the convex quadratic
programming (QP) problem
Minimize
w;b
UðwÞ¼
1
2
kwk
2
s:t: y
i
ðw
T
x
i
þ bÞ P 1; i ¼ 1; ...; l:
ð4Þ
This problem has a global optimum, and its dual problem is to max-
imize the objective function
0957-4174/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved.
doi:10.1016/j.eswa.2009.10.038
q
This work was supported by a grant from National Natural Science Foundation
of China (60773062) and Research Foundation of College of Medicine in Hebei
University.
* Corresponding author. Address: Department of Computer Science and Engi-
neering, Nanjing University of Aeronautics and Astronautics, 210016 Nanjing, PR
China.
E-mail address: jabpjh@163.com (A.-b. Ji).
Expert Systems with Applications 37 (2010) 3495–3498
Contents lists available at ScienceDirect
Expert Systems with Applications
journal homepage: www.elsevier.com/locate/eswa
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