## Methodology  
### Development of the Support Vector Machine Model  
#### 3.1.1 Support Vector Machine  
The SVM, which is supervised machine-learning algorithms, combine the Vapnik-Chervonenkis Dimension from statistics with Structure Risk Minimization Theory [28]. It has been widely used for DM decision-making and predictions and to determine optimal hyper-planes in high-dimensional space that can be utilized to solve regression problems using division, thus minimizing the regression error rate [29]. The optimal high-dimensional hyper-plane can be represented by a linear regression function \(f(\mathbf{x})=\mathbf{\theta}\cdot\Phi(\mathbf{x})+b\) mapping the input vector \(\mathbf{x}\) into a higher-dimensional feature space \(f(\mathbf{x})\), where \(\mathbf{\theta}\) and \(b\) are the weightings and biases of the regression function, respectively. If there is a number of data \(\mathbf{x}\) and \(y\), the objective function for the minimum values of \(\mathbf{\theta}\) can be defined using Equation (1):  
\[\mathbf{min}\mathbf{\theta}=\mathbf{min}\frac{1}{2}\mathbf{\theta}^{T}\mathbf{\theta} \tag{1}\]  
subject to  
\[\forall i:|y_{i}-(f(x_{i})|\leq\varepsilon\]  
where \(\varepsilon\) is a tolerable bandwidth, as shown in Figure 1.  
Figure 1: Tolerable bandwidth of Support Vector Machine (SVM). However, it is possible that no one linear regression function \(f(\mathbf{x})\) will satisfy the objective function in Equation (1) for all data \(\mathbf{x}\) and \(y\). Thus, the slack variables \(\xi_{i}\) and \(\xi_{i}^{*}\) are introduced for \(x_{i}\) and \(y_{i}\), respectively; this allows regression errors to exist up to the value of \(\xi_{i}\) and \(\xi_{i}^{*}\), but the objective function to remain satisfied, as shown in Figure 2. The slack variables \(\xi_{i}\) and \(\xi_{i}^{*}\) are defined by Equations (2) and (3):  
\[\xi_{i}= \begin{cases}0&0<(y-f(\mathbf{x}))\leq\varepsilon\\ (y-f(\mathbf{x}))-\varepsilon&\text{\emph{otherwise}}\end{cases}  \tag{2}\]  
\[\xi_{i}^{*}=\left\{ \begin{array}{cc}0&0<(f(\mathbf{x})-y)\leq\varepsilon\\ (f(\mathbf{x})-y)-\varepsilon&\text{\emph{otherwise}}\end{array} \right. \tag{3}\]  
Therefore, the objective function Equation (1) can be transformed to Equation (4), taking account of Equations (2) and (3):  
\[min\ J(\mathbf{\theta})=min\frac{1}{2}\mathbf{\theta}^{T}\mathbf{\theta}+C\sum_{i=1}^{n}( \xi_{i}+\ \xi_{i}^{*}) \tag{4}\]  
and subject to  
\[\forall i:y_{i}-(\theta\Phi(x_{i})+b)\leq\varepsilon+\xi_{i}^{*}\]  
\[\forall i:(\theta\Phi(x_{i})+b)-y_{i}\leq\varepsilon+\xi_{i}\]  
\[\forall i:\xi_{i}\geq 0,\xi_{i}^{*}\geq 0\]  
where the constant \(C\) is a positive numerical value used to control the penalty for the observation \(\Phi(x_{i})\) lying outside the tolerable bandwidth, to prevent over-fitting. The value of \(C\) determines the trade-off between the complexity of \(f(\mathbf{x})\) and the amount up to which deviations larger than \(\varepsilon\) are tolerated. A Lagrange function is then determined by introducing non-negative multipliers \(\alpha_{i}\) and \(\alpha_{i}^{\prime}\) for each observation \(\Phi(x_{i})\) as shown in Equation (5):  
\[ \begin{array}{rl}\min_{\alpha}\mathcal{L}(\mathbf{\alpha})&=\frac{1}{2}\sum_{i=1 }^{n}\sum_{j=1}^{n}\big{(}\alpha_{i}-\alpha_{i}^{\prime}\big{)}\left(\alpha_{ j}-\alpha_{j}^{\prime}\right)\Phi(x_{i})^{T}\Phi\big{(}x_{j}\big{)}+ \varepsilon\sum_{i=1}^{n}\big{(}\alpha_{i}+\alpha_{i}^{\prime}\big{)}-\sum_{ i=1}^{n}y_{i}\big{(}\alpha_{i}-\alpha_{i}^{\prime}\big{)}\\ &=\frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{n}\big{(}\alpha_{i}-\alpha_{i}^{\prime} \big{)}\left(\alpha_{j}-\alpha_{j}^{\prime}\right)K\big{(}x_{i},\ x_{j}\big{)}+ \varepsilon\sum_{i=1}^{n}\big{(}\alpha_{i}+\alpha_{i}^{\prime}\big{)}-\sum_{ i=1}^{n}y_{i}\big{(}\alpha_{i}-\alpha_{i}^{\prime}\big{)}\end{array}