A New Method to Determine Data Membership and Find Noise and Outlier Data Using Fuzzy Support Vector Machine

Support Vector Machine (SVM) is one of the important classification techniques, has been recently attracted by many of the researchers. However, there are some limitations for this approach. Determining the hyperplane that distinguishes classes with the maximum margin and calculating the position of each point (train data) in SVM linear classifier can be interpreted as computing a data membership with certainty. A question may be raised here: how much the level of the certainty of this classification, based on hyperplane, can be trusted. In the standard SVM classification, the significance of error for different train data is considered equal and every datum is assumed to belong to just one class. However, in many cases some of train data, including outlier and vague data with no defined model, cannot be strictly considered as a member of a certain class. That means, a train datum may does not exactly belong to one class and its features may show 90 percent membership of one class and 10 percent of another. In such cases, by using fuzzy SVM based on fuzzy logic, we can determine the significance of data in the train phase and finally determine relative class membership of data.
The method proposed by Lin and Wang is a basic method that introduces a membership function for fuzzy support vector machine. Their membership function is based on the distance between a point and the center of its corresponding class.
In this paper, we introduce a new method for giving membership to train data based on their distance from distinctive hyperplane. In this method, SVM classification together with primary train data membership are used to introduce a fuzzy membership function for the whole space using symmetrical triangular fuzzy numbers. Based on this method, fuzzy membership function value of new data is selected with minimum difference from primary membership of train data and with the maximum level of fuzzification. In the first step, we define the problem as a nonlinear optimization problem. Then we introduce an efficient algorithm using critical points and obtain final membership function of train data. According to the proposed algorithm, the more distant data from the hyperplane will have a higher membership degree. If a datum exists on the hyperplane, it belongs to both classes with the same membership degree. Moreover, by comparing the primary membership degree of train data and calculated final distribution, we compute the level of noise for train data. Finally, we give a numerical example for illustration the efficiency of the proposed method and comparing its results with the results of the Lin and Wang approach.