## 3 Methods
### Support Vector Regression (SVR)  
SVR is a significant application form of support vector machine (SVM), which was first introduced by [PERSON] (b. 1961 in Denmark) and [PERSON] [63, 64]. SVM is based on the idea of mapping the input space into a new feature space with higher dimensions using the kernel function, after which a hyperplane, known as the decision boundary, is constructed with the maximum margin [65, 66]. SVM extension to SVR is realized by introducing an \(\varepsilon\)-insensitive region around the function, which is referred to as the \(\varepsilon\)-tube that best approximates the regression function [67]. Given training vectors \(x_{i}\in\mathbb{R}^{p}\), I = 1, \(\cdots,\ n\), and a vector \(y\in\mathbb{R}^{n}\), \(\varepsilon\)-SVR solves the following original problem:  
\[\min_{w,\ b,\ \xi,\ \xi^{*}}\frac{1}{2}\|w\|^{2}+C\sum_{i=1}^{n}(\xi_{i}+\xi_{i} ^{*}) \tag{2}\]  
\[\text{\emph{subject to}}\left\{\begin{array}{l}y_{i}-\langle w,x_{i}\rangle- b\leq\varepsilon+\xi_{i}\\ \langle w,x_{i}\rangle+b-y_{i}\leq\varepsilon+\xi_{i}^{*}\\ \xi_{i},\xi_{i}^{*}\geq 0,\ i=1,\cdots,n\end{array}\right. \tag{3}\]  
where C > 0 is a regularization parameter that gives more weight to minimizing the flatness or error.  
The principal hyperparameter of SVR is the kernel, as it defines the kernel functions of the model. The radial basis function (RBF) was selected as the kernel function because it has been found to be efficient and accurate for regression problems [68, 69]. The RBF kernel is described as follows:  
\[k\big{(}\textbf{{x}}_{i},\textbf{{x}}_{j}\big{)}=exp\left(-\frac{\left\| \textbf{{x}}_{i}-\textbf{{x}}_{j}\right\|^{2}}{2\sigma^{2}}\right) \tag{4}\]  
The SVR model is easy to establish because only two hyperparameters must be tuned: Penalty parameter C and the kernel coefficient gamma. The optimal values of the two hyperparameters that were optimized using GridSearch and 5-fold cross-validation are presented in Table 5. The remainder of hyperparameters for the SVR model remain defaults.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Description** & **GridSearch Values** & **Searching Results** \\ \hline \multirow{2}{*}{activation} & Activation function for the hidden layer & ‘identity’, ‘logistic’, ‘tanh’, ‘relu’ & ‘logistic’ \\ \hline \multirow{2}{*}{solver} & The solver for weight optimization. & ‘bfgs’, ’sgd’, ‘adam’ & ‘bfgs’ \\ \hline \multirow{2}{*}{alpha} & L2 penalty (regularization term) parameter. & \multirow{2}{*}{1e-4, 1e-2, 0.1, 1} & \multirow{2}{*}{0.1} \\ \cline{2-2} \cline{4-4}  & Learning rate schedule for weight updates. & ‘constant’, ‘invscaling’, ‘adaptive’ & ‘constant’ \\ \hline \hline \end{tabular}
\end{table}
Table 4: Parameter settings to determine the optimal hyperparameters for the artificial neural network (ANN) model.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Description** & **GridSearch Values** & **Searching Results** \\ \hline \multirow{2}{*}{C} & Penalty parameter C of the term. & 0.1, 0.5, 1, 5, 10, 15, 20, 50, 100, 500 & 50 \\ \hline \multirow{2}{*}{gamma} & Kernel coefficient for ‘rbf’, ‘poly’ and ‘sigmoid’. & 0.01, 0.1, 0.2, 0.4, 0.8, 1, 15, 2, 5, 10 & 0.2 \\ \cline{1-1} \cline{2-2}  & & 15, 2, 5, 10 & \\ \hline \hline \end{tabular}
\end{table}
Table 5: Parameter settings to determine the optimal hyperparameters for the support vector regression (SVR) model.