## 2 Fault Diagnosis Algorithm Selection
### Introduction of Support Vector Machines  
The composition of the support vector machine is simple, usually composed of three layers, including the input layer, the kernel function layer, and the output layer. Among them, the input layer is mainly used to receive data samples that need to be trained. The selection of kernel functions in the kernel function layer is important which solves the defect of insufficient computing power [19]. As for the output layer, it is mainly affected by the kernel function layer and the influence of the definition of the sample label. Set \(x_{i}\) and \(x_{j}\) as the sample points in the data space, and \(\phi\) represents the mapping from the data space to feature space. The process of transformation is as in Equation (5). \[\left(x_{i},x_{j}\right)\to K\left(x_{i},x_{j}\right)=\phi(x_{i})\phi \left(x_{j}\right) \tag{5}\]  
In SVM, the kernel function directly affects the final result of SVM processing data classification and nonlinear regression. Therefore, the kernel function of SVM must be carefully selected when applying. The usually used kernel functions are shown as follows. 1. Linear kernel function: \(K\left(x_{i},x_{j}\right)=x_{i}^{T}x_{j}\). 2. Polynomial kernel function: \(K\left(x_{i},x_{j}\right)=\left(yx_{i}^{T}x_{j}+r\right)^{p},r>0\). 3. Radial basis kernel function: \(K\left(x_{i},x_{j}\right)=\exp(-r\|x_{i}-x_{j}\|^{2}),r>0\). 4. Two-layer perceptron kernel function: \(K\left(x_{i},x_{j}\right)=\tanh(\gamma x_{i}^{T}x_{j}+r)\). When the samples exist crossover, classification hyperplane for SVM cannot be established and the samples may be incorrectly classified. To classify data using SVM under this situation, it is necessary to introduce the slack variables. With this measure, the classification of the data can be achieved in the case of sample misclassification [20]. The slack variable means a non-negative variable \(\xi_{i}\) for each data sample point \(x_{i}\), and it represents the distance from the wrong sample point to its corresponding boundary hyperplane. In Equation (6), \(\|W\|^{2}\) represents the objective function. Its value will be increased by the introducing of the slack variables in cross samples, which will increase the loss of classification. With the purpose to measure the loss, in this paper, a first-order soft-interval classifier is chosen. By introducing penalty factors into the target optimization function, the optimized function is transformed into the following form. \[\min\left\{\frac{1}{2}\|W\|^{2}+C\sum_{i=1}^{l}\xi_{i}\right\} \tag{6}\]  
\[\text{Restrictions}:\left\{ \begin{array}{c}y_{i}((W\cdot X)+b))\geq p-\xi_ {i}\\ \xi_{i}\geq 0,(i=1,2,\ldots,l)\end{array} \right. \tag{7}\]  
When there is a misclassification, SVM with penalty factors can realize the control of the classification error and classification interval of the data sample. This is the model of C-SVM where \(C\) represents the penalty factor, and it indicates the degree of punishment for the misclassified sample. The greater the \(C\) value, the greater the penalty and the stricter the misclassification of the sample. When using C-SVM for classification, it is necessary to select the appropriate parameter \(C\) firstly. Due to the large value range of \(C\), it is difficult to find a suitable penalty parameter \(C\) in practical application. To overcome this problem, \(v\)-support vector machine was proposed by [PERSON], which replaces \(C\) with parameter \(v\). The value of \(v\) reflects the ratio of the error rate to the number of samples and \(v\in(0,1)\). The original optimization problems of \(v\)-SVM can be expressed as Equations (8) and (9). \[\min\left\{\frac{1}{2}\|W\|^{2}-vp+\frac{1}{l}\sum_{i=1}^{l}\xi_{i}\right\} \tag{8}\]  
\[\text{Restrictions}:\left\{ \begin{array}{c}y_{i}((W\cdot X)+b))\geq p-\xi_{i }\\ \xi_{i}\geq 0,(i=1,2,\ldots,l)\end{array} \right. \tag{9}\]Compared with \(C\)-SVM, the equations here use the parameter \(v\) which needs to be actually selected rather than \(C\). Also, the parameter \(p\) is added.