# Research on Prediction of TBM Performance of Deep-Buried Tunnel Based on Machine Learning
## 3 Prediction Model of TBM Penetration Rate Based on Machine Learning
### Prediction Model of TBM Penetration Rate Based on Multiple Regression (MR)
#### 3.1.2 Empirical Analysis of TBM Penetration Rate Prediction Model Based on Multiple Regression  
Among the mechanical parameters of the TBM, the penetration index (FPI) includes two mechanical parameters which have the greatest impact on TBM performance, namely, cutterhead thrust TF and penetration P. There is a good correlation between them and the uniaxial compressive strength (UCS) and penetration index (FPI) in rock mass parameters, which can be used as a bridge connecting rock mass parameters and TBM mechanical parameters. Therefore, the penetration index (FPI) is used to explore the relationship between TBM mechanical parameters and rock mass parameters.  
The uniaxial compressive strength (UCS) of rock mass and the number of rock volume joints \(J_{\rm v}\) representing the integrity of rock mass will be used as independent variables in the multiple regression analysis of the penetration index (FPI), and the empirical prediction formula based on the penetration index (FPI) will be established, namely Formula (2).  
\[\rm{FPI}=1971.32-703.36\ln J_{\rm v}+3.07 UCS \tag{2}\]  
where \(J_{\rm v}\) is the number of rock volume joints.  
After the multiple regression analysis, the empirical formula shows good accuracy, and the goodness of fit R\({}^{2}\) is 0.76, which means that 76% of the change of the penetration index (FPI) can be explained by the uniaxial compressive strength (UCS) and the number of rock volume joints \(J_{\rm v}\).  
After obtaining the penetration index (FPI), the expression of penetration P (3) can be obtained.  
\[\rm{P}=\frac{TF-f}{FPI\cdot D} \tag{3}\]  
where TF is the cutterhead thrust, \(f\) is the friction force to be deducted from the total thrust, and D is the cutterhead diameter. The weight of the TBM studied in this paper is 1300 t, which is calculated according to \(f\) = 2600 kN.  
Furthermore, according to the definition of penetration P (3), the expression of the penetration rate PR (4) can be obtained:  
\[\rm{PR}=\frac{P\cdot 60\cdot RPM}{1000}=\frac{(TF-f)\cdot 60\cdot RPM}{(1971. 32-703.36\ln J_{\rm v}+3.07 UCS)\cdot 1000} \tag{4}\]  
In this way, the prediction model of the TBM penetration rate based on multiple regression is obtained. The input parameters of the prediction model are cutterhead thrust (TF), rotational speed (RPM), uniaxial compressive strength (UCS) and number of rock volume joint \(J_{\rm v}\). Through the correlation of the penetration index (FPI) with rock mass parameters and mechanical parameters, it is verified that the random forest model is scientific and reasonable in ranking the weight of influencing factors of the penetration rate.  
To evaluate the accuracy of the prediction model of a penetration rate based on multiple regression, the predicted value of the model is compared with the measured penetration rate value in the data set. The comparison results are shown in Figure 3. It can be seen from Figure 3 that the correlation coefficient between the predicted value and the measured value of the TBM penetration rate (PR) is 0.7, and the deviation between the predicted value and the measured value is small in the range of 0.5-1.5 m/h, which may be due to a large number of samples in this range included in the sample set. After the penetration rate is greater than 2 m/h, the deviation between the predicted value and the measured value is large. In general, the predicted value can reflect the variation law of the actual net driving speed PR of the TBM within a certain range, indicating that the empirical prediction formula of the penetration rate (PR) of the TBM determined by the method of multiple regression analysis can be applied to the preliminary prediction of the penetration rate of the TBM in the south of the Qinling tunnel of the Hanjiang-to-Weihe River Diversion Project.