# A New Method of Fuzzy Support Vector Machine Algorithm for Intrusion Detection
## 3 Methodology Based on SVDD
### Analysis  
The definition of fuzzy membership function is the key of FSVM algorithm. There are many definitions of fuzzy membership functions, but there are no general guidelines. Traditional fuzzy membership functions based on the distance between the sample and its cluster center are not effective  
Figure 1: The overview of the proposed algorithm. to distinguish noises or outliers from SVs. Sometimes the distance is not the only criterion for judging whether it is normal. As shown in Figure 2, point A on the left, in Figure 1(a), has a high probability of being a valid sample. Point A on the right, in Figure 1(b), has a high probability of being a noise or outlier. It is not enough to just rely on linear functions of distance. The relative position relation between samples should be considered. That is, fuzzy membership function must consider the affinity between samples. As shown in Figure 3, the distance between B and the cluster center is not the same as the distance between C and the cluster center. Since these two points have the same distance to the classification hyperplane, they contribute the same to the hyperplane. Compared with point B and point C, point D is further away from the cluster center, but closer to the classification hyperplane. Point D is the point that contributes the most to the hyperplane. So we have to take that into account when we design membership functions. The optimal classification hyperplane and its nearby support vectors are far away from the cluster center. The closer the sample is to the cluster center, the greater the weight coefficient. The noises and SVs distributed on the boundary are far from the cluster center. They are all given less weight coefficients. This is quite different from the objective situation. SVs should be given greater weight coefficients. The optimal hyperplane of standard SVM is determined by SVs. The geometric principle of SVM is to use two hyperplanes with maximum spacing to separate the training samples as far as possible in the original space or feature space. Therefore, using the hyperplane in class to replace the center can better approximate the actual situation. The solution of the optimal classification hyperplane of SVM is usually converted to solving quadratic programming problem. However, the solving complexity of the quadratic programming problem will increase significantly with the sample increase. When the sample size is large, the traditional fuzzy support vector machine needs large memory to store and calculate the kernel function matrix. Therefore, by selecting the boundary vector containing SVs in advance, the number  
Figure 3: The membership function design based on the position of samples and the hypersphere. Figure 2: The difference of the affinity among samples at two different classes. by the maximum sum of the distance [69] from two centers to the hyperplane. As shown in Figure 4, in order to maximize the sum of the distances, it should satisfy \(d{=}\|{\mathbf{a}}^{+}{-}A\|{+}\|{\mathbf{a}}^{-}B\|{\leq}\|{\mathbf{a}}^{+}{-}O\|{+}\|{\mathbf{a }}^{-}O\|\)\(=\|{\mathbf{a}}^{+}{-}a^{-}\|\). That is, the distance is maximized \(d{=}\|{\mathbf{a}}^{+}{-}a^{-}\|\) when hyperplane and vector \({\mathbf{a}}^{+}{-}a^{-}\) are perpendicular to each other. For the given data, the relative positions of the two hyperspheres can be intersected, tangent and separated. The two separate cases and tangent case can be classified as one case. Although in the case of intersection, the radial basis kernel function can always choose parameters to make the two hyperspheres separated, this will cause an overfitting phenomenon. Therefore, our paper summarizes the above three cases as two cases of separation and intersection. As shown in the Figures 5 and 6, the normal vector to the hyperplane is \({\mathbf{w}}={\mathbf{a}}^{+}{-}\)\(a^{-}\) according to the principle of maximum distance.