## 3 Methodologies
### Support Vector Machine (SVM)  
The support vector machine can map linearly inseparable data to high-dimensional data space through a kernel function and then classify them in the high-dimensional space. The basic idea of SVM is to solve for the separation hyperplane that can divide the training dataset with the largest geometric spacing. Use \(wx+b=0\) to indicate separation of hyperplanes. The objective function that satisfies the constraints is:  
\[\left\{ \begin{aligned} &\min_{w,k,\lambda}\frac{1}{2}\|w\|^{2}+C \sum_{i=1}^{m}\xi_{i}\\ &\text{s.t.}\ y_{i}(\mathbf{w}^{\text{T}}\mathbf{x}_{i}+b)\geq 1 -\xi_{i},\ \xi_{i}\geq 0,\quad i=1,2,\ldots m\end{aligned} \right. \tag{4}\]  
Characteristics only occur as a linear combination, which kernels may compactly encode in the double construction of the SVM. A reduced feature dataset is mapped into a greater dimensionality information source using the kernel function in SVM. Assume that even after retraining an SVM with a binary decision criterion, users properly deduce that the SVM model is training patterns. SVM models use kernel functions to solve high-dimensional problems. For issues involving function approximation, radial basis function (RBF) systems are a popular variety of artificial neural systems. The ubiquitous approach and quicker training pace of multilayer perceptron systems set them apart from other machine learning systems. Gaussian kernel, also known as radial inner basis function (RBF), is a common kernel function. It can be described as follows:  
\[k\big{(}x_{i},x_{j}\big{)}=\exp\!\left(-\frac{\|x_{i}-x_{j}\|}{2\delta^{2}}\right) \tag{5}\]  
where \(\delta\) is the kernel parameter. The SVM decision function is:  
\[f(x)=\text{sgn}\!\left[\sum_{i=1}^{n}\alpha_{i}y_{i}k(x_{i},x)+b\right] \tag{6}\]  
The classification of samples can be obtained by substituting the new sample points into the decision function. Defects that are close to the actual output are ignored by the linearity \(\varepsilon\)-insensitive loss function and are considered equal to zero. The separation among the actual quantity y as well as the boundaries are used to calculate the damage. The loss function is used in SVM models to quantify the subjective probability of a particular training dataset. A SVM model's attributes and effectiveness rely on how it determines the experimental deviation of the provided training set. As a result, in SVM algorithms, the error function selection is extremely important. To solve the regression problem via SVM, an \(\varepsilon\)-insensitive loss function is introduced with the basis of SVM classification. Similar to SVM, the fundamental idea of SVR is to obtain an optimal classification hyperplane, so that all training samples have the least distance from the optimal classification surface. In the high-dimensional feature space, the decisive function is established as follows:  
\[f(x)=w\varphi(x)+b \tag{7}\]\(f(x)\) is the predicted value returned by the regression function with \(\varphi(x)\) as the nonlinear mapping function. The Lagrangian duality model postulates a method for determining a boundary or resolving an efficiency issue (the primary issue) by taking a look at an unrelated efficient algorithm (duality issue). In SVM, a hyperplane is a determination border that distinguishes between the two categories. Depending on which side of the feature space a source of data falls, it might belong to a different category. The dataset's combination of multiple characteristics determines the hyperplane's dimensions. Lagrange function is introduced and converted into the dual form to find the hyperplane weight vector \(w\) and offset vector \(b\):  
\[ \begin{split} max\sum_{\alpha,\alpha}^{m}\sum_{i=1}^{m}y_{i}( \hat{\alpha}_{i}-\alpha_{i})-\epsilon(\hat{\alpha}_{i}+\alpha_{i})-\frac{1}{2} \sum_{i=1}^{m}\sum_{j=1}^{m}(\hat{\alpha}_{i}-\alpha_{i})(\hat{\alpha}_{j}- \alpha_{j})x_{i}^{T}x_{j}\\ \text{s.t.}\sum_{i=1}^{m}(\hat{\alpha}_{i}-\alpha_{i})=0,0\leq \alpha_{i},\hat{\alpha}_{i}\leq C\end{split}