## 2 Methodology
### Least Squares Boosting Ensemble (LSBoost)  
The gradient boosting ensemble method consists of a finite set of weak learners and a meta learner that assigns weights to each learner and combines the predictive results of each one, using voting methods to provide a better predictive performance for regression problems. Boosting is a supervised machine learning method where the data are split into training, validation and testing sets. The algorithm starts by training the individual weak learners, sequentially, that are in the form of decision trees, and fits the residual of errors to achieve a better performance. The LSBoost method utilizes the least squares as the loss criteria. The LSBoost pseudocode is presented in Algorithm 3, as presented by [PERSON] [38]. A total of 36 pilot experiments were executed, to confirm the computational results of the QPSO and LSBBoost algorithms, and compared with the performance of PSO and GA algorithms, which have been extensively reported in the literature, for the milling machine to accomplish the required surface roughness with the minimum machining time, as presented in Table 6.  
The results obtained by the QPSO for the machining parameters show the ability of the QPSO algorithm to achieve minimum machining time in comparison to the experimental results for the required surface roughness with a high extent of accuracy. Table 7 provides the statistical performance indicators of each algorithm in terms of the MAPE, RMSE, MAE and R\({}^{2}\) values. It can be noted that QPSO has achieved the best accuracy in predicting the surface roughness values with RMSE of 2.17% and a high coefficient of determination value of R\({}^{2}\) = 0.95. The high value of R\({}^{2}\) clearly indicates that the predicted values closely match the actual experimental values of the surface roughness and is evident of the superior performance of the QPSO, due to the algorithm's superiority in exploring the search space and avoiding local optima trap. The LSBBoost provides the second best prediction values with an RMSE of 3.74% and R\({}^{2}\) of 0.88. This can be addressed to the gradient ensemble in combining different weak learners into a meta learner that provide the best prediction at each step. The GA algorithm resulted in a relatively comparable performance with LSBBoost algorithm in terms of the coefficient of determination with R\({}^{2}\) = 0.871. However, it has a slightly higher RMSE value of 4.86%, which is higher than the LSBoot algorithm by 1.12%. On the other hand, the PSO algorithm resulted in an RMSE of 4.99% and the lowest R\({}^{2}\) value of 0.84. \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline Iterations & 1000 \\ Particles population & 100 \\ Cognitive acceleration: c\({}_{1}\) & 2 \\ Social coefficient: c\({}_{2}\) & 2 \\ \hline \end{tabular}
\end{table} 
Table 4: PSO parameters. \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline Number of learners & 50 \\ Learning rate & 0.01 \\ Minimum leaf size & 1 \\ \hline \end{tabular}
\end{table} 
Table 3: LSBoost parameters. \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline Maximum number of generation & 100 \\ Number of individuals per generation & 25 \\ Generation gap & 0.5 \\ Crossover & 0.7 \\ Mutation rate & 0.04 \\ \hline \end{tabular}
\end{table} 
Table 5: GA parameters. Figure 1 illustrates a graphical comparison between the actual surface roughness values and the predicted values by QPSO, LSBoost, PSO and GA algorithms. The superior prediction performance of the QPSO can be clearly noted by the close match of each data point on the graph with the actual value of the surface roughness. Figure 2 provides the trendline between actual versus predicted values of each prediction algorithm, to provide the coefficient of determination values R\({}^{2}\). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Exp No.