# Prediction of Tunnel Face Stability Using a Naive Bayes Classifier
## 2 Description of the Proposed Method  
Figure 1 illustrates the framework of the proposed method. First, four main features that influence tunnel face stability including the cover depth (C), the diameter of the tunnel (\(D\)), the cohesive strength (\(c\)) and the friction angle (\(\varphi\)) of soil are used to constitute a parameter set, \([C,D,c,\varphi]\). Secondly, different values of parameter sets are chosen using full factorial design within the specified ranges of the features and labeled as stable samples (denoted by \(+1\)) or unstable samples (denoted by \(-1\)) according to strength reduction analyses. The parameter sets and the corresponding labels will then be combined to create a training dataset, with which a naive Bayes classifier will be constructed with strong independence (naive) assumptions between the four features. Finally, the constructed  
classifier can be used to classify an unknown sample, \(\{C^{*},D^{*},c^{*},\varphi^{*}\}\), into the stable category if the stable probability is larger than the unstable probability, or into the unstable category if it is not. Equation (1) gives the safety factor [33; 34; 35; 36] that can be obtained according to the built-in strength reduction analysis embedded in OptumG2. \[F_{s}\ =\ \frac{c}{c_{cr}}\ =\ \frac{\tan\varphi}{\tan\varphi_{cr}}, \tag{1}\]  
where \(c\) and \(\varphi\) denote the inputted strength parameters; \(c_{cr}\) and \(\varphi_{cr}\) denote the critical strength parameters. A factor greater than 1 implies a stable state while a factor less than 1 implies an unstable state that requires ground improvement or additional support pressure to prevent the collapse. With the simulated safety factor, the face-stability of tunnels can be mainly classified into two categories, i.e., \(F_{s}\geq 1\) (stable) and \(F_{s}<1\) (unstable). Bayes' theorem is a useful tool in the area of statistical analysis [37; 38; 39]. Researchers have developed many Bayesian methods to solve various geotechnical problems [40; 41]. The general expression of the [PERSON] theorem is given as follows:  
\[P(A|B)\ =\ \frac{P(B|A)P(A)}{P(B)}, \tag{2}\]  
where \(A\) and \(B\) denote two events, \(P(A|B)\) denotes the conditional probability of event \(A\) in the presence of the occurrence of event \(B\). Generally, \(P(A)\) and \(P(A|B)\) are referred to as prior and posterior probabilities, respectively. \(P(B|A)\) is the likelihood function that characterizes the probability of the  
Figure 1: Framework of the proposed method. unknown model parameters to be identified with given observed data. The event \(A\) can be partitioned into \(N\) mutually exclusive events, \(A_{1},A_{2},\ldots A_{N}\), and the posterior can be rewritten as:  
\[P(A_{n}|B)\ =\ \frac{P(B|A_{n})P(A_{n})}{\sum_{n=\,1}^{N}P(A_{n})P(B|A_{n})}. \tag{3}\]  
Relating the event \(A\) to the stability of a tunnel face, it can be simply partitioned into two mutually exclusive events, i.e., the stable and the unstable state. The event \(B\) accordingly contains the features influencing the face-stability of tunnels, including the cover depth, the diameter of the tunnel, the cohesive strength, and the frictional angle of soil. Naive Bayesian is a conditional probability model that can be used to formulate classifiers. It is based on the assumption that the value of a feature is independent of the value of the others. Accordingly, the posterior probability of an event \(A_{n}\) can be expressed as follows:  
\[P(A_{n}|B_{1},B_{2},\ldots B_{n}), \tag{4}\]  
where the vector \(\mathbf{B}\ =\ (B_{1},B_{2},\ldots B_{n})\) represents some \(n\) features that are assumed to be independent with each other, \(A_{n}\) represents the \(n\)th possible outcome of class.