## 2 Methods
### Support Vector Machine (SVM)  
Support vector machine (SVM) belongs to a family of generalized linear classifiers based on the Vapnik-Chervonenkis dimension theory. It was first developed by [PERSON] in 1963 for the linear models Vapnik and Lerner. It was later extended to nonlinear training data in 1995. It is a set of supervised machine learning models with associated kernels for regression (SVR) and for implementing nonlinear classification (SVC). Based on the Vapnik-Chervonenkis dimension theory, it creates a separate optimal hyperplane and maps low-dimensional input vectors to high-dimensional feature spaces to ensure high generalization network capabilities. As shown in Figure 4 below, the optimal hyperplane  
Figure 3: The intelligent prediction process.  
uses less data to train data points as the maximum width between the two classification vectors, determining the width of the error-free separation. When this happens, the ratio of the expected value of the support vector to the training data constrains the expected probability error value, as shown in the following equation.  
\[\mathrm{E}[P_{r}(\mathrm{error})]\leq\frac{E(\mathrm{support\ vectors})}{ \text{training\ samples}} \tag{3}\]  
The SVM provides predictive solutions based on functional formal equations [30].  
\[y(x)=\sum\limits_{n=1}^{N}w_{n}K(x,x_{n})+w_{0} \tag{4}\]  
where \(w_{n}\) is the weight of the model and \(K(x,x_{n})\) is a kernel function.  
The basic concept of support vector classification is to use a learning technique to determine the best hyperplane to separate two different classes in n-dimensional space in a more general way. This is achieved by learning similarities between classes and using them as support vectors rather than understanding their differences through other algorithms. The distance between hyperplanes separating different classes is called a boundary, and the boundary always needs to be maximized for better classification. When looking for the right support vector machine, you need to find the sum by minimizing it, then optimizing it, and expressing it as below.  
\[\begin{split}\max&\left(\sum\limits_{i=1}^{N} \alpha_{i}-\frac{1}{2}\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}y_{i}\alpha_{i }k\left(\overrightarrow{x_{i}},\overrightarrow{y_{j}}\right)y_{i}\alpha_{i} \right)\\ &\sum\limits_{i=1}^{n}\alpha_{i}y_{i}=0(0\leq\alpha_{i}\leq C,i =1,\ldots,n)\end{split} \tag{5}\]  
where \(k\left(\overrightarrow{x_{i}},\overrightarrow{y_{j}}\right)\) is related to a kernel function, and \(C\) is a regularization parameter, which represents the error classification tolerance parameter.  
The main disadvantages of using SVM are increased computation time when dealing with large amounts of data, proper model performance generalization depends on proper kernel and hyperparameter selection, and poor model performance with noisy inputs and overlapping data classes. An adaptive learning rate adjustment strategy was adopted, utilizing the Adam optimizer to improve the stability and efficiency of training. The learning rate was finely adjusted through the visualization analysis of the learning rate and loss function to accelerate model convergence and prevent overfitting.  
Figure 4: Support vector machine classification visualization case.