# 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 Support Vector Regression  
#### 3.3.1 Principle of Support Vector Regression Algorithm  
As a general method to solve the problem of high-dimensional function estimation, support vector regression is established based on the Vapnik chervonenkis (VC) theory [22]. If the dimension of VC is low, the expected error probability is also low, which indicates that the generalization ability is strong. Support vector regression is an optimization problem. First, define \(\varepsilon\)-insensitive loss function, minimize it, and then find the narrowest one containing most of the training data sets \(\varepsilon\)-insensitive zone. Therefore, according to the geometric characteristics of the loss function and the \(\varepsilon\)-insensitive zone, a multi-objective function is constructed. Given a training set \(\{(x_{1},y_{1}),\ldots,(x_{n},y_{n})\}\subset R^{\text{M}}\times R\), where \(R^{\text{M}}\) is the input space of the input feature \(x_{\text{i}}\) and \(y_{\text{i}}\) is the observed object, the goal is to find a function \(f(x)\) that is within the deviation range of \(\varepsilon\) from the actual target value \(y_{\text{i}}\), and make it as gentle as possible. \(f(x)\) can be expressed in linear form, as shown in Equation (18). \[y=f(x_{\text{i}})=\langle\mathbf{w}_{\text{i}},\mathbf{x}_{\text{i}}\rangle+b \tag{18}\]  
where \(w_{\text{i}}\) represents the weight vector of the linear function whose unit length is at right angles to the hyperplane, and \(b\) represents the threshold. A support vector regression approximates the data set with a linear function. The form of the linear function in the high-dimensional feature space is shown in Equation (19). \[y=f(x)=\sum_{i=1}^{N}w_{\text{i}}(x_{\text{i}})+b \tag{19}\]  
where (\(x_{\text{i}}\)) transfers the input vector to the feature space, that is, non-linear mapping from the input space \(R^{\text{M}}\) to the high-dimensional feature space. Using the \(\varepsilon\)-insensitive loss function in the error risk minimization with regularization, the support vector regression function can be established more conveniently. Only the samples in the \(\varepsilon\)-insensitive band have non-zero relaxation variables. Generally, if the predicted value is within the region, the loss is zero; if the prediction point is outside the insensitive band, the error is the difference between the prediction value and the insensitive band radius \(\varepsilon\). Loss function represents less than \(\varepsilon\). There is no loss for the deviation, and the larger deviation will be subject to linear penalty [23], as shown in Formula (20):  
\[L(\zeta)= \begin{cases}0&if|\zeta|<\varepsilon\\ |\zeta|-\varepsilon&\text{otherwise}\end{cases}  \tag{20}\]  
where the parameter \(\varepsilon\) is equivalent to the approximate accuracy of the training data points. When the data points are within the range of \(\pm\varepsilon\), it will not cause errors. The support vector regression describes the function approximation problem as an optimization problem by finding the narrowest insensitive band centered on the surface and minimizing the prediction error, that is, the distance between the prediction and the expected output. The objective function is shown in Equation (21). \[\min\frac{1}{2}\|w\|^{2} \tag{21}\]  
The goal of the support vector regression algorithm is to find the optimal linear hyperplane, and the optimal hyperplane can be guaranteed by comprehensively considering the regression error and flatness [23].