## 3 Analysis and Results
### Performance Analysis of Machine Learning Models
#### 3.2.2 Machine Learning Model Performance for Tandem Axle Stress Prediction  
The results shown in Table 11 illustrate the performance of different regression models, with gradient boosting showing as the best model. It shows the lowest testing MSE (0.001118), MAE (0.023510), and RMSE (0.033436), with a near-perfect R\({}^{2}\) value of 0.999367, demonstrating excellent accuracy and generalization. Random forest likewise works well, with low testing errors (MSE: 0.005714, R\({}^{2}\): 0.996763) and negligible overfitting. Decision tree, however, displays evidence of overfitting, attaining excellent training results (MSE: 0, R\({}^{2}\): 1) but inadequate testing performance (MSE: 0.012414, R\({}^{2}\): 0.992966). Linear and ridge regression produce consistent results with high R\({}^{2}\) values (0.990983 and 0.990604, respectively), although ridge slightly follows linear. Furthermore, lasso regression struggles with severe underfitting, indicated in its large testing errors (MSE: 1.702942, R\({}^{2}\): 0.035162). LightGBM and CatBoost perform well during training but demonstrate poor generalization ability on testing data. Therefore, gradient boosting is the most dependable model. In addition, the gradient boosting model is further validated by diagnostic plots, as shown in Figure 8. The Q-Q plot indicates that the residuals are regularly distributed, confirming the assumption of normality. The actual vs. predicted values plot illustrates a good correlation between predicted and actual values, presenting high predictive accuracy. The externally studentized residuals vs. predicted values plot indicates no bias, confirming the absence of systematic mistakes in the model. Finally, the residuals vs. run number plot indicates that the model's predictions are independent of the data point sequence, ensuring stability. Overall, gradient boosting stands out as the most dependable model, supported by consistent performance across all evaluation metrics. However, the SHAP value analysis in Figure 9 displays that the thickness of the pavement has the most significant impact on model predictions, as confirmed by its high average SHAP value. Otherwise, X spacing and Y spacing show lesser average affects, with Y spacing demonstrating a more consistent influence.  
Figure 7: SHAP value analysis for single axle model features.  
\begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & **Training** & **Testing** & **Training** & **Testing** & **Training** & **Testing** & **Training** & **RMSE** & **Testing** \\  & **MSE** & **MSE** & **R2** & **R2** & **MAE** & **MAE** & **MAE** & **RMSE** & **RMSE** \\ \hline Linear & & & & & & & & & \\ Regression & 0.017564 & 0.015914 & 0.985709 & 0.990983 & 0.112486 & 0.100295 & 0.132530 & 0.126152 \\ \hline Ridge & & & & & & & & \\ Regression & 0.017797 & 0.016584 & 0.985520 & 0.990604 & 0.110984 & 0.101056 & 0.133406 & 0.128781 \\ \hline Lasso & & & & & & & \\ Regression & 1.184310 & 1.702942 & 0.036404 & 0.035162 & 0.890889 & 1.113038 & 1.088260 & 1.304968 \\ \hline Decision Tree & 0.000000 & 0.012414 & 1.000000 & 0.992966 & 0.000000 & 0.083175 & 0.000000 & 0.111420 \\ \hline Random Forest & & & & & & & \\ \hline Gradient Boosting & 0.000021 & 0.001118 & 0.999983 & 0.999367 & 0.003692 & 0.023510 & 0.004535 & 0.033436 \\ \hline LightGBM & 0.059909 & 0.098171 & 0.951256 & 0.944379 & 0.179682 & 0.257142 & 0.244764 & 0.313323 \\ \hline CatBoost & 0.000005 & 0.042487 & 0.999996 & 0.975928 & 0.001702 & 0.111855 & 0.002162 & 0.206124 \\ \hline \hline \end{tabular}
\end{table}
Table 11: Performance of different regression models for tandem axle.  
Figure 8: Residuals plots analyzing gradient boosting regression model for tandem axle.