## 3 Machine Learning Technique  
### Support Vector Machines and Model Development  
SVM is a supervised learning method for regression and classification analysis based on risk minimization, first proposed by [PERSON] (1967). The SVM method can process the classification of data using a kernel trick, implicitly mapping inputs into high-dimensional feature spaces. The SVM method uses an effective separation by a hyperplane, with the largest distance to the nearest training-data point and the lowest generalization error, allowing the SVM to obtain a better generalization. In comparison with other machine learning algorithms, the SVM method has several advantages, such as a unique optimization approach and the effective utilization of high-dimensional feature spaces and computational learning theory [PERSON] (1967); [PERSON] (1967). The mathematical algorithm is summarized below [PERSON] (1967). For a training dataset of n points of the form (\(xi\), \(di\)), \(xi\) is the input vector, di is the target value, and n is the size of the dataset obtained by the mapping (\(\phi\)) of \(xi\) into a high-dimensional feature space. This can be expressed as follows:  
\[\mathrm{f}(x)=\omega\phi(x)\ +\ \mathrm{b} \tag{4}\]  
where \(\omega\) is the weight vector, \(\phi\) is the high-dimensional feature space, and b is the bias of the hyperplane. The basic concept of the SVM is to minimize the structural risks, and \(\omega\) and b can be obtained by minimizing the following risk penalty function:  
\[\mathrm{Minimize}\ \frac{1}{2}\|\omega\|^{2} \tag{5}\]  
subject to  
\[\left\{ \begin{array}{l}d_{i}-\omega\cdot\phi(x)-\ \mathrm{b}\ \leq\ \delta\\ \omega\cdot\phi(x)+\ \mathrm{b}-d_{i}\ \leq\ \delta\end{array} \right. \tag{6}\]  
where \(\delta\) is the maximum deviation. To determine the optimal values of \(\omega\) and b, slack variables \(\xi\), and \(\xi^{*}\) are adopted, and the following new equivalent function is introduced:  
\[\mathrm{C}\frac{1}{n}\sum_{i=1}^{n}(\xi_{i}+\xi_{i}^{*})+\frac{1}{2}\|\omega\| ^{2} \tag{7}\]subject to:  
\[\left\{ \begin{array}{l}d_{i}-\omega\phi(x_{i})-\mathrm{b}_{i}\leq\varepsilon+k_{ i}\\ \omega\phi(x_{i})+\mathrm{b}_{i}-d_{i}\leq\varepsilon+k_{i}^{*}\\ k_{i}\xi_{i}^{*}\geq 0\end{array} \right. \tag{8}\]  
where \(\mathrm{C}_{n}^{\frac{1}{2}}\sum_{i=1}^{n}(\xi_{i}+\xi_{i}^{*})\) is the estimated risk, \(\frac{1}{2}\|\omega\|^{2}\) is the Euclidean norm as a penalty item, and \(\mathrm{C}\) is the penalty constant used to control the penalty error. Using the Lagrange multipliers, \(a_{i}\) and \(a_{i}^{*}\), the SVM function becomes:  
\[f\Big{(}x,a_{i},\ a_{i}^{*}\Big{)}=\sum_{i=1}^{n}\Big{(}a_{i}-a_{i}^{*}\Big{)} K(x,x_{i})+b \tag{9}\]  
Next, the Lagrange multipliers are included in the penalty objective function and the dual function can be obtained as follows:  
Maximize  
\[R\Big{(}a_{i},\ a_{i}^{*}\Big{)}=\sum_{i=1}^{n}d_{i}\Big{(}a_{i}-a_{i}^{*} \Big{)}-\varepsilon(a_{i}-a_{i}^{*}\ )-\frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{n}\Big{(}a_{i}-a_{i}^{*}\Big{)}\ \Big{(}a_{j}-a_{j}^{*}\Big{)}\mathrm{K}(x,x_{i}) \tag{10}\]  
subject to:  
\[\sum_{i=1}^{n}\Big{(}a_{i}-a_{i}^{*}\Big{)}=0\qquad \begin{array}{l}0\leq a_ {i}\leq C\ \ i=1,2,\ldots n\\ 0\leq a_{i}^{*}\leq C\ \ i=1,2,\ldots n\end{array}  \tag{11}\]  
\(K(x_{i},x_{i})\) is the kernel function containing the inner product of \(x_{i}\) and \(x_{j}\), with corresponding feature spaces \(\psi(x_{i})\) and \(\psi(x_{j})\), respectively. Linear, polynomial, sigmoid, and Gaussian kernels are the most commonly used kernel functions. The Gaussian kernel is the most popular because it can reduce the complexity of the inputs effectively through \(C\) and \(\gamma\)[29]. In this study, the Gaussian kernel was applied for building the SVM classifiers. The Gaussian kernel was as follows:  
\[K\Big{(}x_{i},x_{j}\Big{)}=exp\Big{(}-\gamma\Big{(}x_{i},x_{j}\Big{)}^{2}\ \Big{)} \tag{12}\]  
where \(\gamma\) is a constant.