## 3 Data Explanation
### Model Development
\begin{table}
\begin{tabular}{l l c c c} \hline \hline \multirow{2}{*}{**Model**} & \multirow{2}{*}{**Tuning Method**} & \multicolumn{4}{c}{**Optimal Parameters**} \\ \cline{3-5}  & & **Learning\_Rate** & **Maximum\_Depth** & **n\_Estimator** \\ \hline XGBoost & Manually adjusted & 0.217 & 3 & 199 \\ XGBoost-HHO & Harris hawk optimisation algorithm & 0.109 & 3 & 140 \\ XGBoost-JSO & Jellyfish search optimiser & 0.102 & 3 & 131 \\ XGBoost-DA & Dragonfly optimisation algorithm & 0.131 & 3 & 170 \\ XGBoost-FA & Firefly optimisation algorithm & 0.174 & 3 & 165 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Hyperparameter tuning using optimisation algorithms. displayed in these diagrams. As seen from the figures, all four models illustrated smooth convergence, with the error loss decreasing rapidly during the early iterations and stabilising as the models approach their optimal states. XGBoost-JSO and XGBoost-HHO exhibited faster convergence compared to XGBoost-DA and XGBoost-FA, with lower error loss values achieved within fewer iterations. XGBoost-JSO reached a stable solution after approximately 200 iterations, indicating its efficiency in minimising error, while XGBoost-HHO converged similarly around 150 iterations. XGBoost-DA and XGBoost-FA also demonstrated effective convergence, with a more gradual decrease in error loss before stabilising around 100 iterations. These results prove that the selected optimisation algorithms (JSO, HHO, DA, and FA) successfully enhance the XGBoost model's ability to learn from data. The smooth convergence and the final, minimal error loss values highlight the models' reliability and ability to generalise to unseen data. Figure 11: Convergence plot of XGBoost-HHO model. Figure 10: Convergence plot of XGBoost-DA model. The results of the performance of the developed models, including training and testing datasets, based on the values of MAE, R\({}^{2}\), RMSE, VAF, and Bias, are given in Table 3. As found from the table, the XGBoost model demonstrated strong and consistent performance, with an MAE of 2.653, R\({}^{2}\) of 0.958, and RMSE of 3.111 during the training phase. These metrics translated well into the testing phase, where the model maintained an MAE of 1.759, R\({}^{2}\) of 0.952, and RMSE of 2.186, indicating good generalisation to unseen data. The XGBoost-HHO showed a slight improvement in variance explanation during the training phase, with an R\({}^{2}\) of 0.961 and a VAF of 95.53%, though this came at the cost of a marginally higher MAE and RMSE compared to the standard XGBoost. However, in the testing phase, XGBoost-HHO exhibited a significant reduction in MAE and RMSE, achieving values of 1.298 and 1.624, respectively, which suggests improved predictive accuracy and generalisation over the standard XGBoost. In the training phase, XGBoost-JSO achieved a notably low MAE of 1.355, an impressive R\({}^{2}\) value of 0.988, and a RMSE of 1.562, with a VAF of 98.737%. The Bias was also very low, at 1.355. These metrics indicate that the XGBoost-JSO model not only fit the training data exceptionally well but also explained almost 99% of the variance in the training data, showcasing its high accuracy and minimal error. In the testing phase, XGBoost-JSO continued to perform at a high level with an MAE of 0.668, an R\({}^{2}\) of 0.987, and an RMSE of 0.823, achieving a VAF of 98.66%. The Bias remained consistently low at 0.668. These results reveal that XGBoost-JSO not only maintained its  
Figure 12: Convergence plot of XGBoost-JSO model. Figure 13: Convergence plot of XGBoost-FA model. accuracy but also improved its generalisation capabilities on a new, unseen dataset. The reduction in MAE and RMSE from the training to the testing phase was relatively small, indicating that the model did not overfit and provided reliable predictions across different datasets.