## 2 Materials and Methods
### Machine Learning Regression
#### 2.3.3 Support Vector Regression (SVR)  
SVR is a versatile machine learning regression technique developed by [PERSON] [67] and is one of the derivations of Support Vector Machines (SVMs), the other being Support Vector Classification (SVC). The technique has gained widespread applications as a benchmark method in solving complex linear and non-linear regression [47; 68; 69; 68]. The mathematical principle underlying SVR is based on the concept of finding an optimal hyperplane of the form of Equation (8), where, \(f(x_{i},\omega)\) is a high-dimensional kernel input feature space, '\(\omega\)' is the weight coefficient, 'T' is the coefficient of variable transformation, and 'b' is the model error that best fits the given data points in a continuous space by minimizing the prediction error and maximizing the margin between the hyperplane and the closest data points known as the support vectors.  
\[f(x_{i},\omega)=\{\omega\times T(x_{i})\}+b, \tag{8}\]  
The formulation of SVR involves solving an optimization problem to find the hyperplane parameters that minimize the prediction error while also satisfying a set of constraints. For a given dataset of sample size 'n', the objective function of an SVR is typically formulated as shown in Equation (9) and subject to the constraints given in Equation (6), where 'y\({}_{i}\)' and 'f(x\({}_{i}\))' are the actual and predicted values of the output variable for the \(i\)th data point 'xi', '\(\epsilon\)' is the marginal error, 'C' is a regularization parameter used for harmonizing the flatness and errors of the model, and '\(\xi_{i}\)' and '\(\xi_{i}\)'' are positive slack variables used for governing the distance between the actual and boundary values.  
\[\begin{split} L\big{(}\omega,\xi_{i},\xi_{i}^{*}\big{)}=min \bigg{(}\frac{1}{2}\|\omega\|^{2}+C\sum\limits_{i=1}^{n}\big{(}\xi_{i}+\xi_{i} ^{*}\big{)}\bigg{)}\\ \text{Subject to }\begin{cases}y_{i}-f(x_{i},\omega)\leq\varepsilon +\xi_{i}\\ f(x_{i},\omega)-y_{i}\leq\varepsilon+\xi_{i}^{*}\\ \xi_{i}^{*},\xi_{i}^{*},x_{i}\geq 0\end{cases},\end{split} \tag{9}\]  
The loss function is further evaluated using Lagrangian multipliers \(\alpha_{i}\) and \(\alpha_{i}^{*}\) along with Karush-Kuhn-Tucker conditions to obtain the optimal solution. With this step, the equation of hyperplane gets transformed into Equation (10), where \(K(x_{i},x)\) is a kernel function that is used to map the input parameters to the feature space.  
\[f(x)=\sum\limits_{i=1}^{n}(\alpha_{i}-\alpha_{i}^{*})+K(x_{i},x)+b \tag{10}\]  
The task of finding the optimal hyperplane is accomplished by using problem-specific kernel functions. SVR supports a variety of kernel types, such as linear, polynomial, Gaussian or Radial Basis Function (RBF), and sigmoid, but the choice of kernel relies largely on the specific problem and characteristics of the data [70; 71]. In this study, a preliminary evaluation of the SVR with different kernel types was performed using the collected dataset and the Gaussian kernel expressed as in Equation (11) yielded the best results and was chosen for further analysis and improvement of the SVR model.  
\[K(x_{i},x)=exp\Bigg{(}\frac{-\|x_{i},y_{i}\|^{2}}{2\sigma^{2}}\Bigg{)} \tag{11}\]  
Figure 8 shows a conceptual depiction of a non-linear SVR.