## 3 Results  
In this study, the dataset is divided into training and test sets in a ratio of 80% to 20%. For the neural network, the validation sample was 20% of the training set. The models were evaluated using the key metrics: MSE, MAE, and \(R^{2}\). The scaling of the training and test sets is presented in Table 3.  
Figure 8 shows a scatterplot comparing the training data and testing data for the models trained. Each graph displays a series of data points where the actual values of the dependent variable are compared with the predicted values obtained from the linear regression model. To find the optimal parameters, a grid search was applied to optimize the hyperparameters of the tree-based models and artificial neural networks (ANNs). The optimal combinations of the hyperparameters were determined based on the accuracy of the test set used as the performance metric. A range of hyperparameters for these models was defined based on the dataset size and their characteristics. Within these given ranges, different combinations of parameters were systematically explored to find the optimal set. Table 4 shows the \(R^{2}\) score using the optimized parameters. Analyzing the performance of various machine learning models on training datasets revealed the following key observations. First, most models, including linear regression (LR), polynomial regression (PR), decision tree (DT), random forest (RF), gradient boosting (GB), and extreme gradient boosting (XGBoost), demonstrate high accuracy in both the training and test data, exceeding the threshold value of 0.97. This indicates the high generalization ability of these models, which can be seen in Figures 9 and 10. The neural network (NN) also achieves high accuracy on both datasets, although it was slightly lower compared to the other models. Figure 10 shows the actual and predicted results for the regression models based on the testing data. This graph clearly demonstrates the models' ability to generalize beyond the training set. The models show comparable coefficients of determination \(R^{2}\), indicating that they are effective at capturing the underlying patterns in the data, even when exposed to new, unseen data. Remarkably, the PR model achieved a high \(R^{2}\) value of 0.977, effectively capturing the nonlinear relationships among the variables. The remaining models showed results of about 0.974, which are also very high. These results highlight the ability of both traditional algorithms and advanced machine learning techniques, including neural networks, to achieve high prediction accuracy and maintain consistency across different pieces of data. The second observation is that the stacking regressor shows a comparable performance to the other models on the test dataset but lags slightly on the training set. This may indicate that the stacking regressor is more robust to overfitting and is able to generalize better to new data. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & **LR** & **PR** & **DT** & **RF** & **GB** & **XGBoost** & **NN** & **Stacking Regressor** \\ \hline Training set & 0.97 & 0.98 & 0.98 & 0.99 & 0.99 & 0.98 & 0.97 & 0.97 \\ Test set & 0.97 & 0.98 & 0.97 & 0.98 & 0.98 & 0.98 & 0.97 & 0.97 \\ Average & 0.97 & 0.98 & 0.975 & 0.985 & 0.98 & 0.98 & 0.97 & 0.97 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: \(R^{2}\) score of various algorithms on training and test sets after optimization. \begin{table}
\begin{tabular}{c c} \hline \hline
**Data Type** & **Data Points** \\ \hline Training set & 295,680 \\ Test set & 73,920 \\ Total & 369,600 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The size of the training set and test set. Figure 10: _Cont._  
Figure 9: Comparison of predicted versus actual values on training set. A comparison of the \(R^{2}\) scores before and after the model optimization, shown in Figure 11, is important for assessing the effectiveness of the improvements made to the algorithms.