## 3 Model Validation and Engineering Examples
### Overview of the Subway Tunnel Project
[39] argued that statistical error metrics, often referred to as performance metrics or evaluation measures, play a crucial role in the evaluation and fine-tuning of ML models. These metrics provide quantitative insight into model performance and help data scientists and researchers make informed decisions about model selection, optimization, and overall effectiveness. We also used the coefficient of determination (R2), mean squared error (MSE), mean absolute error (MAE), mean absolute relative error (MARE), mean square relative error (MSRE), root mean squared relative error (RMSRE), mean bias error (MBE), maximum absolute relative error (erMAX), and standard deviation (SD). The mean absolute error (MAE), mean absolute percentage error (MAPE), and root mean square error (RMSE) are widely used in model performance evaluation, mainly because they can comprehensively evaluate prediction errors from multiple perspectives and have strong representativeness and applicability. The MAE directly measures the size of the average deviation between the predicted value and the true value, which is simple and  
Figure 8: A comparison of the measured data at monitoring sections 1–17. straightforward to compute, and the units are consistent with the original data, making it easy to interpret and compare. The MAE assigns the same weight to each prediction error, which accurately reflects the average level of the overall error without being magnified by the presence of a few outliers; thus, the MAE is suitable for most error assessment scenarios with smooth datasets. The MAPE quantifies the prediction error in percentage form without unit limitations, which facilitates performance comparisons between different datasets or models. It can intuitively reflect the relative error between the predicted value and the true value and is more suitable for evaluating the prediction accuracy of datasets of different sizes, especially in data scenarios with large amplitude differences, such as economics and market analysis. The MAPE is able to emphasize the predictive performance of the model in smaller value intervals, but it is sensitive to zero values, which needs to be considered in some scenarios. The RMSE amplifies the effect of large errors by squaring them and is therefore more advantageous in applications where outliers have a significant impact on the quality of the prediction. The RMSE is able to highlight the degree of dispersion of the predicted values, providing a stronger assessment of the overall prediction accuracy of the model. In addition, owing to the good mathematical nature of the RMSE, it is more readily used in optimization problems, while its theoretical consistency with the least squares method further solidifies its importance. Therefore, we select three representative error evaluation indices, the MAE, MAPE, and RMSE, to judge the model prediction results. These are defined as follows:  
\[\text{MAE}=\frac{1}{n}\sum_{t=1}^{n}|y_{t}-\hat{y}_{t}| \tag{17}\]  
\[\text{MAPE}=\frac{1}{n}\sum_{t=1}^{n}|\frac{y_{t}-\hat{y}_{t}}{y_{t}}| \tag{18}\]  
\[\text{RMSE}=\sqrt{\frac{1}{n}\sum_{t=1}^{n}{(y_{t}-\hat{y}_{t})^{2}}} \tag{19}\]  
where \(\hat{y}_{t}\) is the observed value of the actual surface sedimentation sequence, \(y_{t}\), and is the predicted value of the sedimentation sequence.