In order to deal with the classification problems of type-2 fuzzy training samples on generalized credibility space. Firstly the type-2 fuzzy training samples are reduced to ordinary fuzzy samples by the mean reduction method. Secondly the definition of strong fuzzy linear separable data for type-2 fuzzy samples on generalized credibility space is introduced. Further, by utilizing fuzzy chance-constrained programming and classic support vector machine, a support vector machine based on type-2 fuzzy training samples and established on generalized credibility space is given. An example shows the efficiency of the support vector machine.
The fuzzy set was proposed by Zadeh (1965) in 1965, it is an interesting extension of classical set. In classical sets the characteristic function can take only one of two values from the {0, 1} , in fuzzy sets the membership function can take any value from the interval [0, 1] . The membership function plays an important role in fuzzy set theory and its applications. But in some actual problems, it is often very difficult to determine the value of membership function (or membership degree for short). To solve this problem, the concept of type-2 fuzzy set was introduced by Zadeh (1975) in 1975. A type-2 fuzzy set is characterized by a special membership function, the value of which for each element of this set is a fuzzy set in [0, 1] . Compared with ordinary fuzzy set, type-2 fuzzy set could describe the objective phenomenon more accurately because of secondary membership function. It has been applied more efficiently to pattern recognition and other machine learning fields successfully (Mitchell, 2005).
Support vector machine built on statistical learning theory is a kind of effective machine learning method, has advantaged superiority in dealing with small samples classification problems, and has become the standard tool in machine learning field now (Cristianini and Shawe-Taylor, 2000). The traditional support vector machine is based on the real valued random samples and established on the probability space, and it is difficult to handle the small samples classification problems with non-real random samples on non-probability space. Naturally, it is a very interesting and valuable research direc-tion to construct the support vector machine based on non-real valued random samples and established on nonprobability space or probability space. On this basis, Lin and Wang (2002) proposed the support vector machine based on a non-real valued random samples-fuzzy samples, constructed the fuzzy support vector machine, which determines membership degree by different weight for each sample. Then Ha
Definition 2.1(Zadeh, 1965): A fuzzy set
represents
Definition 2.2: Suppose
Then
Definition 2.3(Zadeh, 1975): A type-2 fuzzy set in
is characterized by a secondary membership function
where
can also be expressed as
Definition 2.4 (Qin
is called triangular if its secondary membership function
is
for
for
takes the value
with the above condition by (
Definition 2.5(Qin
Definition 2.6(Qin
be a type-2 fuzzy set in
via the mean reduction method, if it satisfies
We denote
Example 2.1(Qin
be a type-2 triangular fuzzy number. According to the definition 2.6, we have the reduction
and the membership function of
Definition 2.7(Qin
of the event {
Theorem 2.1(Qin
Proof: It can be easily got by type-2 fuzzy set theory in the reference.
Consider the fuzzy type-2 training samples set
where
and
is called a positive class; when
is called a negative class. The classification based on the type-2 fuzzy training set
is to find a decision function
such that the positive class and the negative class can be separated with the low classification error and good generalization performance.
Definition 3.1(Ji
if for a given level
there exist
Then the type-2 fuzzy training samples set
is strong type-2 fuzzy linear separable.
The support vector machine for strong linear separable type-2 fuzzy sample set is to solve the fuzzy chance constrained programming:
To solve the above programming,
is equivalent to
We can obtain its dual problem:
if
then
belongs to the positive class; if
then
belongs to the negative class.
We shall apply the support vector machine for twoclass classification with type-2 fuzzy training samples to the diagnosis of Coronary. The data is sourced from the reference (Ji
and
The programming was solved by SVM toolbox of matlab, when parameter
This paper constructs firstly the support vector machine in which the training samples is type-2 fuzzy number input, and the proposed support vector machine is an interesting extension of the traditional support vector machine and the fuzzy support vector machine. We shall focus on the proposed support vector machine’s applications in some practical problems.