## 3 Analysis and Results
### Performance Analysis of Machine Learning Models
#### 3.2.1 Machine Learning Model Performance for Single Axle Stress Prediction  
Table 10 displays the performance of different regression models. The results indicate that gradient boosting is the top performer, with the lowest testing MSE (0.006061), RMSE (0.077851), and a high R\({}^{2}\) value of 0.971168, demonstrating its strong ability to generalize to new data. Likewise, random forest performs well, with low testing MSE (0.016694) and RMSE (0.129204), and a solid R\({}^{2}\) of 0.920587. However, the decision tree model overfits the training data, obtaining desirable outcomes on the training set (MSE: 0, R\({}^{2}\): 1, MAE: 0, RMSE: 0) but struggling with testing data (MSE: 0.020817, R\({}^{2}\): 0.900970, MAE: 0.124350, RMSE: 0.144282). The huge disagreement between its training and testing performance indicates it is memorizing the training data instead of learning general patterns. Both linear and ridge regression illustrate uniform results, with ridge slightly beating linear regression on the testing data. On the other hand, lasso regression displays considerable underfitting, with high errors and a negative R\({}^{2}\) value for testing. LightGBM's performance is insufficient, with large errors in both training and testing. Meanwhile, CatBoost performs moderately. In summary, gradient boosting is the most dependable model. To further confirm the efficiency of gradient boosting, diagnostic plots were studied as shown in Figure 6. The Q-Q plot demonstrated that the residuals follow a normal distribution, verifying the model's correctness. The actual vs. predicted plot also indicated strong alignment, reflecting the model's predictive reliability. Additionally, the externally studentized residuals plot confirmed there was no bias, and the residuals vs. run number plot indicated that the  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Source** & **Sequential _p-Value_ & **Adjusted R\({}^{2}\)** & **Predicted R\({}^{2}\)** & **Performance** \\ \hline Linear & \(<\)0.0001 & 0.883 & 0.8753 & \\
2 FI & 0.044 & 0.8893 & 0.8737 & \\ Quadratic & \(<\)0.0001 & 0.9859 & 0.9838 & Suggested \\ Cubic & \(<\)0.0001 & 0.9964 & 0.9952 & Aliased \\ \hline \end{tabular}
\end{table}
Table 8: Fit summary of the various models for tridem axle with dual tires.  
\begin{table}
**Source** & **Sum of Squares** & **df** & **Mean Square** & **F-Value** & _p-Value_ & **Performance** \\ \hline Model & 27.93 & 9 & 3.1 & 763.87 & \(<\)0.0001 & significant \\ A-A & 2.52 & 1 & 2.52 & 620.06 & \(<\)0.0001 & \\ B-B & 0.0591 & 1 & 0.0591 & 14.54 & 0.0003 & \\ C-C & 0.1624 & 1 & 0.1624 & 39.98 & \(<\)0.0001 & \\ AB & 0.0582 & 1 & 0.0582 & 14.33 & 0.0003 & \\ AC & 0.2107 & 1 & 0.2107 & 51.86 & \(<\)0.0001 & \\ BC & 0.0002 & 1 & 0.0002 & 0.0495 & 0.8244 & \\ A\({}^{2}\) & 2.54 & 1 & 2.54 & 624.05 & \(<\)0.0001 & \\ B\({}^{2}\) & 0.0079 & 1 & 0.0079 & 1.94 & 0.1672 & \\ C\({}^{2}\) & 0.0354 & 1 & 0.0354 & 8.71 & 0.0041 & \\ Residual & 0.3616 & 89 & 0.0041 & & \\ Cor Total & 28.29 & 98 & & & \\ \hline \end{tabular} \
\begin{tabular}{c c c c} \hline Std. Dev. & 0.0637 & & R\({}^{2}\) & 0.9872 \\ \hline Mean & 0.9651 & & Adjusted R\({}^{2}\) & 0.9859 \\ \hline C.V. \% & 6.6 & & Predicted R\({}^{2}\) & 0.9838 \\ \hline \end{tabular}
\begin{tabular}{c c c} \hline  & & Adeq Precision & 92.953 \\ \hline \end{tabular}
\end{table}
Table 9: ANOVA for quadratic model for tridem axle model.  
Figure 6: Residuals plots analyzing gradient boosting regression model for single axle.