# Modeling of Tunneling Total Loads Based on Symbolic Regression Algorithm
## 4 Case Study
### Modeling Based on Symbolic Regression Algorithm  
In this study, the initial population number is set to 1000, the evolutionary algebra is set to 20, and the dominant number of each generation is set to 20. The data we are using have been normalized, so the range of constants is set as 0-10. The constraints on the complexity of the training model and the operation symbols are critical. When the model complexity is too high or there are too many operation symbols, the model will be very complex and cumbersome, whereas when the model setting is too simple, the risk of invalid solutions increases. In this algorithm, the rationality of the model is controlled by setting the _parsimony_coefficient_ and _function_. Trial calculations show that, for training the tunneling load models in this paper, setting the _parsimony_coefficient_ to be less than the default value of 0.001 increases the model complexity, but does not change \(R^{2}\) significantly. Therefore, the _parsimony_coefficient_ is set to 0.001 in the modeling process. As for the setting of _function_, considering the theoretical relationship between the parameters and the target quantity, the five symbol setting schemes shown in Figure 7 are used to train the model. Five random sampling calculations are then carried out for each scheme and the \(R^{2}\) values are recorded. Through comparative analysis, reasonable model symbols are then determined according to the principle of [PERSON]'s razor.  
Figure 8 shows how \(R^{2}\) changes under different regression symbol settings. In the thrust modeling, when the regression symbol is \"add, mul\", \(R^{2}\) is higher than for other symbols and the fluctuation range is small, that is, the model recognition is relatively stable. Therefore, the regression symbol for thrust modeling is set to \"add, mul\". For torque modeling, the regression symbol \"add, mul\" gives a significantly higher \(R^{2}\) value than \"add\" and \"add, mul, div, sqrt, log\", but a similar value to \"add, mul, div\" and \"add, mul, div, sqrt\". However, the model complexity of the latter two symbols is significantly higher, and the model becomes unstable. Therefore, following the principle of Ockham's razor, it is more reasonable to use the \"add, mul\" regression symbol for torque modeling.  
Figure 7: Regression symbol setting schemes.  
Based on the above settings, the tunneling data from the Yin-Song Project are randomly divided into a training set and a test set at a ratio of 7:3. The total thrust and total torque are modeled based on the training dataset, and the calculated results are given by Equation (8). The \(R^{2}\) of the thrust and torque models in the independent test dataset are 0.95 and 0.81, respectively. The measured and predicted values in the test set are compared in Figure 9. The serial number of the sample points in the figure is the number of sample points according to the construction sequence.  
\[\left\{\begin{array}{ll}\frac{F}{GD^{2}}=(c+\sigma_{t}+\sigma_{c}\cdot 10^{ -2})\frac{c}{G^{2}}\cdot 10^{-2}\\ \frac{T}{GD^{3}}=(\frac{v}{D\omega}+\frac{\sigma_{c}}{G}\cdot 10^{-6})\frac{v}{G} \end{array}\right. \tag{8}\]  
After dimension reduction, the load models are as shown in Equation (9). The statistical results of the prediction error of the model in the total dataset are shown in Figure 10. The thrust prediction error is less than 6% for 95% of the data, and the torque prediction error is less than 20% for 92% of the data. This indicates that the prediction model for tunneling total loads accurately reflects the tunneling total load values in actual engineering projects.  
Figure 8: \(R^{2}\) with different regression symbol settings: (**a**) the total thrust, (**b**) the total torque.  
Figure 9: Comparison between the predicted values and the measured values of the dimensionless load models: (**a**) the total thrust, (**b**) the total torque.