# Prediction of Tunnel Face Stability Using a Naive Bayes Classifier
## 4 Estimation of Model Parameters  
With the given values of the features, as well as the classifications determined by the simulated safety factors, model parameters associated with the probability density functions of the features can be identified using the so-called maximum likelihood principles. To achieve this, let \(D\) denote the observed data, and let \(\theta\) denote the parameters of an unknown model to be estimated and calibrated. The well-known likelihood function is generally expressed as \(l(\theta|D)\), which is always converted to the log-likelihood function \(L(\theta|D)~{}=~{}ln(l(\theta|D))\) for facilitating the evaluation of its maximization, based on which the equation can be easily solved to yield the optimal value of \(\theta\). Suppose the values of the four features influencing the stability of a tunnel face follow the normal distribution. The means and standard deviations of these values (denoted as \(\mu\) and \(\sigma\)) need to be estimated to identify the probability density function. Let \(C_{1},C_{2},\ldots,C_{n}\) denote the \(n\) inputted values of the buried depth; its likelihood function can be expressed as:  
\[l(\theta|D)~{}=~{}\prod\
olimits_{i~{}=~{}1}^{n}f(C_{i}|\theta)~{}=~{}\prod \
olimits_{i~{}=~{}1}^{n}\frac{1}{\sqrt{2\pi}\sigma}exp\Bigg{[}-\frac{(C_{i}- \mu)^{2}}{2\sigma^{2}}\Bigg{]}. \tag{15}\]  
The log-likelihood of which can be rewritten as:  
\[L(\theta|D)~{}=~{}\sum\
olimits_{i~{}=~{}1}^{n}\Bigg{(}ln\Bigg{(}\frac{1}{ \sqrt{2\pi}\sigma}\Bigg{)}-\frac{(C_{i}-\mu)^{2}}{2\sigma^{2}}\Bigg{)}. \tag{16}\]  
The optimal values of \(\mu\) and \(\sigma\) can be determined by equating the gradients of the above equation with respect to \(\mu\) and \(\sigma\), respectively, to zero, as shown in Equations (17) and (18). \[\mu^{*}~{}=~{}\frac{1}{n}\sum\
olimits_{i~{}=~{}1}^{n}C_{i}, \tag{17}\]  
\[\sigma^{*}~{}=~{}\sqrt{\frac{1}{n}\sum\
olimits_{i~{}=~{}1}^{n}(C_{i}-\mu^{*} )^{2}}, \tag{18}\]  
where \(\mu^{*}\) and \(\sigma^{*}\) denote the maximum likelihood estimations of \(\mu\) and \(\sigma\), respectively. According to the above equations, the optimal values of the model parameters relating to inputted values of the cover depth are \(\mu^{*}~{}=~{}15.78\) m, and \(\sigma^{*}~{}=~{}8.69\) m, respectively, as listed in Table 2, in which the maximum likelihood estimations of the model parameters of the other three features were similarly identified. Figure 4: Simulated safety factors of the 750 samples. The PDFs of the buried depth associated with the stable state and the unstable state are of the type:  
\[P(C|Stable) = \frac{1}{\sqrt{2\pi\cdot 8.689^{2}}}e^{(\frac{-(C-15.78)^{2}}{2\cdot 8.689^{2}})}, \tag{19}\] \[P(C|Unstable) = \frac{1}{\sqrt{2\pi\cdot 8.52^{2}}}e^{(\frac{-(C-17.61)^{2}}{2 \cdot 8.53^{2}})}. \tag{20}\]  
The other PDFs that characterize the distribution of the features can be identified in the same way, as plotted and shown in Figure 5.  
Results in Figure 5 show that the PDF of the cover depth of the stable samples is very close to that of the unstable samples. This is mainly because of the so-called arch effect, which can serve to support most of the vertical stress acting above the tunnel face, resulting in a decrease in the horizontal stress that induces the collapse of the tunnel face. Conversely, the PDF of the diameter of the stable samples differs substantially with that of the unstable samples, which indicates that the tunnel diameter is the most crucial influencing factor. The difference between the PDFs of the cohesive strength for the stable and the unstable samples is similar to that of the friction angle because the values of these two features are always divided by the same reduction in strength reduction analyses.