Algorithm of Fuzzy Support Vector Machine based on a Piecewise Linear Fuzzy Weight Method

Algorithm of Fuzzy Support Vector Machine based on a Piecewise Linear Fuzzy Weight Method

Yong-bin Yuan, Sheng Lan, Xu Yu, Miao Yu
DOI: 10.4018/IJCINI.2018040105
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This article describes how fuzzy support vector machines (FSVMs) function well with good anti-noise performance, which receives the attention of many experts. However, the traditional center-distance fuzzy weight assignment method assigns support vectors with a small value of a membership degree and this weakens the role of support vectors in classification. In this article, a piecewise linear fuzzy weight computing method is proposed, in which boundary samples are assigned with a larger value of membership degree and samples far from the mean vector are assigned a smaller value of membership degree. The proposed method has a good classification performance, because the influence of noise samples is weakened and meanwhile the support vectors are paid much more attention. The experiments on the UCI database and MNIST data set fully verify the effectiveness of the proposed algorithm.
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1. Introduction

Fuzzy Support Vector Machine (FSVM) theory was firstly proposed by Lin, etc. (Lin & Wang, 2002). Its main idea is to introduce the concept of fuzzy weights to the original sample, enhancing the noise resistance of this algorithm. Currently, this algorithm is widely applied in the fields of network intrusion detection (Lun, University, & Beijing, 2005; Yang, Yu, Xie, & Zhang, 2011), face recognition (Leng & Wang, 2008; Liu & Chen, 2007), text classification (Wang & Chiang, 2007, 2009) and credit risk evaluation (Wang, Wang, & Lai, 2005).

For the fuzzy support vector machine, the original sample IJCINI.2018040105.m01 is usually expressed as IJCINI.2018040105.m02, where IJCINI.2018040105.m03 stands for the fuzzy weight of samples. Fuzzy support vector machine weakens the effect of noise samples for classification results through introducing fuzzy weight. Fuzzy support vector machine algorithm can be achieved by solving the following optimization problem:


Different from the traditional support vector machine algorithm, in the fuzzy support vector machine model, the misclassification penalty IJCINI.2018040105.m05 of samples is influenced by fuzzy weight IJCINI.2018040105.m06 and so the effects of noise samples for classification can be reduced by setting reasonable fuzzy weights. The key of fuzzy support vector machine algorithm is to assign the fuzzy weights, and currently the commonly applied approach is the center distance fuzzy weight assignment method. The specific calculation formula is as follows:

(2) where IJCINI.2018040105.m08 and IJCINI.2018040105.m09 denote the sample center of the two kinds of sample, and IJCINI.2018040105.m10 is used to avoid IJCINI.2018040105.m11. IJCINI.2018040105.m12 and IJCINI.2018040105.m13 denote the radius of the two kinds of samples, and they can be defined as follows:


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