## 1 Introduction
### Support Vector Machine Algorithm
The classification problem can be treated as an optimization problem with constraints and then solved for the objective function:  
\[ \begin{cases}\begin{gathered} min\frac{1}{||\alpha||^{2}}+C\sum_{i= 1}^{m}\xi_{i}\\ s.t\;\;yi(\alpha xi+b)\geq 1-\xi_{i},\xi\geq 0,1\leq i\leq m\end{gathered}\end{cases}  \tag{1}\]  
where \(C\) is the error term penalty coefficient, typically a constant not less than zero, which determines the robust regression model. \(\xi_{i}\) is the slack variable. The above Equation (1) can be obtained in its dual form by introducing the Lagrange multiplier method to operate on it, i.e., by adding the Lagrange multiplier \(\alpha_{i}\) to all the constraints present, where \(\alpha_{i}\geq 0\) is then obtained by varying the last:  
\[ \begin{cases}\begin{gathered} max\Big{(}-\frac{1}{2}\sum_{i=1}^{n }\sum_{j=1}^{n}\alpha_{i}\alpha_{j}y_{i}y_{j}K(x_{i},y_{i})+\sum_{i=1}^{n} \alpha_{i}\Big{)}\\ s.t\;\;0\leq\partial_{i}\leq C,i=1,2,\dots,n\end{gathered}\end{cases}  \tag{2}\]  
Derived from Equation (2), the decision function can be obtained as follows:  
\[f(x)=sign\Big{(}\sum_{i=1}^{n}\alpha_{i}y_{i}K(x_{i},y_{i})+b^{*}\Big{)} \tag{3}\]  
where \(b^{*}\) is known as the deviation and \(K(x_{i},y_{i})\) as the kernel function, and with the current application of several widely used kernel functions, including the Laplace kernel function, radial basis function, etc., of which the radial basis kernel function requires fewer parameters, and its adaptability to fresh samples, in this paper the kernel function required is the SVM-selected radial basis kernel function. The formula for the radial basis kernel function is as follows:  
\[K(x_{i},y_{i})=exp\left(-\frac{(x-x_{i})^{2}}{g^{2}}\right) \tag{4}\]  
The smaller the value of the parameter \(g\) in Equation (4), the finer the categories will be divided, which can easily lead to overfitting, and the larger the value of the parameter \(g\), the more coarsely the categories will be divided, which cannot accurately separate the data. The penalty factor \(C\) and the kernel parameter \(g\) in the support vector machine have a direct link to the speed as well as the accuracy of the classification of the support vector machine, so when fault detection is performed, the selection of the penalty factor \(C\), and the kernel parameter \(g\) has a great impact on the final fault detection results. The \(C\) parameter adds a penalty for each misclassified data point. If \(C\) is small, the penalty for misclassified points is lower and, therefore, decision boundaries with larger intervals will be selected at the cost of more misclassifications. If \(C\) is larger, the SVM tries to minimize the number of misclassified examples due to the higher penalty, which will result in decision boundaries with smaller intervals. The penalty does not apply to all misclassified examples, it is proportional to the distance from the decision boundary. If \(C\) is large, the SVM tries to minimize the number of misclassified examples due to the higher penalty, which leads to decision boundaries with smaller intervals. The penalty does not apply to all misclassified examples, and it is proportional to the distance from the decision boundary. As \(g\) decreases, the more support vectors there are, the more generalized the regions separating different classes become, whereas very large \(g\) values result in fewer support vectors, leading to overly specific category regions (overfitting). Therefore, the penalty factor in SVM and the parameter \(g\) in the kernel function need to use the improved particle swarm algorithm for parameter optimization to finally obtain the most suitable value for the model.