# Hazard Prediction of Water Inrush in Water-Rich Tunnels Based on Random Forest Algorithm
## 2 Construction of Evaluating Indicator System and Grading Standards
### Principles of the RF Algorithm  
The RF algorithm is an ensemble learning method based on decision trees and has the characteristics of high accuracy, high stability, and good robustness. It constructs a large number of decision trees and uses voting mechanisms for tasks such as classification, regression, and feature selection. RF mainly includes classification models and regression models. For classification problems, a voting mechanism is usually used to determine the  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
\begin{tabular}{c} Vertical infiltration \\ zone \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table}
Table 1: Classification of evaluating indicators of water inrush.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**level of Water Inrush** & **Hazard Manifestation** \\ \hline I & The water inrush of the tunnel is less than 1000 m\({}^{3}\)/d or the mud inrush is less than 500 m\({}^{3}\) \\ II & The water inrush of the tunnel is higher than 1000 m\({}^{3}\)/d and less than 3000 m\({}^{3}\)/d \\  & or the mud inrush is higher than 500 m\({}^{3}\) and less than 1000 m\({}^{3}\) \\  & The water inrush of the tunnel is higher than 3000 m\({}^{3}\)/d and less than 10,000 m\({}^{3}\)/d \\  & or the mud inrush is higher than 1000 m\({}^{3}\) and less than 2000 m\({}^{3}\) \\ IV & The water inrush of the tunnel is higher than 10,000 m\({}^{3}\)/d \\  & or the mud inrush is higher than 2000 m\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Hazard level of water inrush in water-rich tunnels.  
final prediction category. For regression problems, the average value is usually used as the final prediction result. The RF algorithm can improve the robustness and generalization ability of models by integrating the results of multiple decision trees and taking into account the opinions of multiple models. The label (hazard levels) established in this article belongs to discrete variables and is suitable for classification models. Therefore, the classification model in the RF algorithm is selected as the application model. The algorithm includes the following steps:  
(1) There is a data set \(S\) with \(N\) samples. Let \(i\) (\(i\) = 1, 2, \(\ldots\), \(N\)) represent its sample serial number, then the information of the \(i\)th sample can be expressed as (\(x_{i}\), \(y_{i}\)), where \(x_{i}\) is an \(M\)-dimensional vector, representing \(M\) features of the sample. \(y_{i}\) is the category to which the sample belongs, so dataset \(S\) can be represented as:  
\[S=\{(x_{i},y_{i}),i=1,2,\ldots,N\} \tag{1}\]  
This is divided into the training sets \(c\) of \(C\) samples and the testing sets \(d\) of \(D\) samples according to a certain ratio, so that \(j\) (\(j\) = 1, 2, \(\ldots\), \(c\)) represents its serial number in the training sets. Let \(k\) (\(k\) = 1, 2, \(\ldots\), \(d\)) represent its serial number in the training sets, where \(N\) = \(c\) + \(d\). With the above representation, training set \(C\) and testing set \(D\) can be represented as:  
\[C =\big{\{}\big{(}x_{j},y_{j}\big{)},j=1,2,\ldots,c\big{\}} \tag{2}\] \[D =\big{\{}(x_{k},y_{k}),k=1,2,\ldots,d\big{\}} \tag{3}\]  
In training sets \(C\), \(t\) samples were selected with the self-sampling method and the training subset was constructed.  
(2) Feature selection is carried out on the training subset--\(M\) features are randomly selected from m features without being placed back, where \(m=\log_{2}M\) (rounded up) is used as the basis for node splitting of the decision tree--and a complete decision tree is generated.  
(3) By repeating steps 1 and 2 above, \(s\) training subsets and \(s\) decision trees can be obtained, and \(s\) decision trees can be combined to form an RF model.  
(4) The testing set \(D\) is input into the RF, so that each decision tree makes a decision on each sample, then the majority voting method is adopted to vote on the decision results. Finally, the classification of all samples in the testing sets is completed.