## 4 Validation and Analysis
### Data Preparation
#### 4.1.3 Denoising and Normalization  
In order to improve the data quality, the raw TBM trusting data are denoised by a wavelet filter. More specifically, the DB3 wavelet is selected to decompose the original data into two layers, and then the data are reconstructed by using soft threshold segmentation. The original TBM operational data and denoised data with wavelet filter are shown in Figure 7.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \(\rho\) & \(V\) & \(F\) & \(T\) & \(RSP\) & \(\mathbf{AM}\) & \(\mathbf{AD}\) & \(p\) & \(\mathbf{I}\) & \(\mathbf{MT}\) & \(\mathbf{MF}\) \\ \hline \(V\  
Due to the outliers in the original signal, the traditional min-max normalization method performs poorly. To calculate the normal value range of the data, [PERSON]'s boxplot data ranges [42; 43; 44] was used in this paper.  
In this study, the TBM operational data objects are curves. Let \(\mathrm{B}=\left\{\beta:D\to R^{d}\middle|\beta\right.\) is absolutely continuous) be the space of absolutely continuous parametrized curves from d-dimensional Euclidean space. Define \(SO(d)=\left\{R\in R^{d\times d}\middle|R^{T}R=RR^{T}=I_{d},\det(R)=\mp 1\right\}\) as the rotation group, and \(\Gamma=\left\{\gamma:D\to D\middle|\gamma\right.\) is an orientation-preserving diffeomorphism) as the reparameterization group.  
For a curve \(\beta\in\mathrm{B}\), its square-root velocity function (SRVF) \(q:D\to R^{d}\) is defined using a mapping \(Q:\mathrm{B}\to L^{2}\Big{(}D,\ R^{d}\Big{)}\) as  
\[q=Q(\beta)=\frac{\dot{\beta}}{\sqrt{\dot{\beta}}} \tag{22}\]  
where \(\left|\cdot\right|\) is the Euclidean norm in \(R^{d}\) and \(\dot{\beta}\) is the time derivative of \(\beta\).  
Shape distance is defined in [44] as  
\[D_{s}([q_{1}],\ [q_{2}])=\min_{\gamma\in\Gamma,\ R\in SO(d)}cos^{-1}(\left\langle q _{1},\ R(q_{2},\gamma)\right\rangle) \tag{23}\]  
where \([q]=\left\{R(q,\gamma)|(\gamma,R)\in\Gamma\times SO(d)\right\}\) is the orbit of \(q\), \(\cdot\), \(\cdot\) is the \(L^{2}\) inner product. Then the median of a sample of SRVFs \(\left\{q_{1},\ \cdots,\ q_{n}\right\}\) is  
\[[\overline{q}]=arg_{\left[q\right]\in C/\left(\Gamma\times SO(d)\right)}\sum_ {i=1}^{n}D_{s}([q],[q_{i}]) \tag{24}\]  
The data are ordered according to their distances from the median, then two shape quartiles \(\left(\
u_{Q_{1}},\ \
u_{Q_{3}}\right)\) are acquired based on 50% central data. Given the two quartiles, the shape interquartile range (IQR) is defined as the sum of the shape distances from each quartile to the median:  
Then, the maximum and minimum of the normal value range are defined as:  
\[IQR_{s}=\left\|v_{Q_{1}}\right\|+\left\|v_{Q_{3}}\right\| \tag{25}\]  
Figure 7: Original and denoised TBM advance rate.  
\[vw_{1}=v_{Q_{1}}+k_{s}\times IQR_{s}\times\frac{v_{Q_{1}}}{\|v_{Q_{1}}\|} \tag{26}\]  
\[vw_{3}=v_{Q_{3}}+k_{s}\times IQR_{s}\times\frac{v_{Q_{3}}}{\|v_{Q_{3}}\|} \tag{27}\]  
The choice of \(k_{s}\) represents the tolerance for outliers. In this paper, the normal value range can include mild outliers, thus \(k_{s}=3\) is selected. Then, the eight parameters chosen as input variables are scaled according to their value range to avoid situations where partial features dominate the model.