## 2 Machine Learning Methods
### Support Vector Regression Model  
The SVR model is adopted to compute the function relationship between independent (parameters--\(P_{ij}\)) and dependent (responses--\(R_{ik}\)) variables whose distributions are not known and concurrently minimize both model complexity and estimation errors. The model complexity and estimation errors are concurrently minimized by implementing SVR in the data set and its performance was outperformed compared to neural network models. A training SVR model used a training data set to learn about the data set and to construct a model about the generalization error by testing the model using an unseen data set. The response value is a function of multivariate parameters as represented in Equation (10), which can be written as Equation (11) to represent a linear regression support vector in terms of weight vector (W), a high dimensional space related to input parameter space, and a bias (b). \[R_{ik}=f\big{(}P_{ij}\big{)} \tag{10}\]  
where,  
\(P_{ij}\)--ith Parameter of ith training data  
\(R_{ik}\)--kth Response of ith training data  
\(i\)--Index for training data (\(i=1,\,2,\,3,\,\dots\,\,\)nt)  
\(j\)--Index for parameter (\(j=1,\,2,\,3,\,\dots\,\,\)np)  
\(k\)--Index for response (\(k=1,\,2,3,\,\dots\,\,\)nr)  
\[f\big{(}P_{ij}\big{)}=W\varphi\big{(}P_{ij}\big{)}+b \tag{11}\]  
The value of weight and bias is estimated by transferring the regression problem into the constraint minimization problem expressed in Equation (12). \[minimize\ \frac{1}{2}\|W^{2}\|+C\sum_{i=1}^{nt}(\xi_{i}+\xi_{i}^{*}) \tag{12}\]  
Figure 1: Proposed methodology. where,  
\(\xi_{i}\), \(\xi_{i}^{*}\)--lower and upper positive slack variables  
\(\frac{1}{2}\|W^{2}\|\)--Flatness value of the function  
\(C\sum_{i=1}^{nt}\bigl{(}\xi_{i}+\xi_{i}^{*}\bigr{)}\)--Empirical error  
\(C\)--Box constraint or penalty parameter value  
Constraints:  
\[R_{ik}-W\varphi\bigl{(}P_{ij}\bigr{)}+b\leq\varepsilon+\xi_{i}^{*} \tag{13}\]  
\[W\varphi\bigl{(}P_{ij}\bigr{)}+b-R_{ik}\leq\varepsilon+\xi_{i} \tag{14}\]  
\[\xi_{i},\xi_{i}^{*}\geq 0 \tag{15}\]  
where  
\(\varepsilon\)--Error tolerance  
The constraint problem is resolved using the following Lagrange multiplier function as given in Equation (16), which should be minimized concerning the conditions mentioned in Equation (17). The list of parameters considered to establish an SVR model is presented in Table 1. A list of kernel functions and their formulas are depicted in Table 2.  
\[f\bigl{(}P_{ij}\bigr{)}=\sum_{i=1}^{nt}\bigl{(}\alpha_{ij}-\alpha_{ij}^{*} \bigr{)}\ K(P_{ij}\cdot P)+b \tag{16}\]  
\[f\big{(}P_{ij}\big{)}=W\varphi\big{(}P_{ij}\big{)}+b \tag{11}\]0  
where  
\(\alpha_{ij}^{*}\) and \(\alpha_{ij}\)--Non-negative Lagrange multipliers  
\(K\bigl{(}P_{ij}\cdot P\bigr{)}\)--Kernel function  
Constraints:  
\[ \begin{array}{c}\sum\limits_{i=1}^{nt}\Bigl{(}\alpha_{ij}^{*}-\alpha_{ij} \Bigr{)}=0\\ 0\leq\alpha_{ij}^{*}\leq C\\ 0\leq\alpha_{ij}\leq C\end{array} \] \begin{table}
\begin{tabular}{c l} \hline \hline
**Parameters** & \multicolumn{1}{c}{**Values**} \\ \hline \hline  & Gaussian \\  & Radial basis function \\ Kernel function (\(k_{f}\)) & Polynomial \\  & Linear \\  & Quadratic \\  & Cubic \\ \hline \hline  & SMO (Sequential Minimal Optimization) \\ Solver (sr) & ISDA (Iterative Single Data Algorithm) \\  & Q1 LP (Quadratic Programming) \\ \hline \hline  & Cross-validation \\  & Holdout validation \\ \hline \hline  & K-fold—Number of folds used for cross-validation (5 to 15) \\ Validation scheme’s parameters & Holdout—\% of data holdout for validation (10\% to 40\%) \\ \hline \hline  & \multicolumn{1}{c}{} \\ \% of training data & 10\% to 30\% \\ \hline \hline  & \multicolumn{1}{c}{} \\ \end{tabular}
\end{table} 
Table 1: List of parameters considered in SVR. data sets) available in the archive into a single objective. The pseudo-code of this method is shown in Algorithm 2. The implementation of the MFO algorithm is represented as a flow diagram in Figure 3. The pseudo-code of the PSO algorithm is presented in Algorithm 3.