# Modeling of Water Losses in Hydraulic Tunnels under Pressure Based on Stepwise Regression Method
## 4 Case Study
### Correlation Analysis and PCA of Total Water Losses
#### 4.4.3 Defining the Pool of Candidate Regressors  
In accordance with our methodology, after obtaining the principal components, we defined the pool of candidate regressors. In the modeling of total water losses, PCs can be roughly divided into several groups, based on the physical nature of the input variables: the effects of hydrostatic pressure, the effects of temperature and size of crack openings, and the effects of time (aging, which is considered to be irreversible). The effects of hydrostatic pressure variation are usually represented by polynomials, power, and exponential or logarithmic functions, depending on the water level [21]. In our case study, for hydrostatic principal components (\(\overline{H}_{st}\), \(\overline{H}_{r}\), \(hP\)) we arbitrarily opted for polynomial terms of the first and second order. As described in the previous section, \(T_{wcc}\) includes the water temperature, concrete temperature, and size of crack openings, and the authors have chosen to express its potential predictive effect on total water losses in polynomial and exponential form. It can be assumed that water losses increase with the time elapsed since the last tunnel reparation date. This aging effect can also be interpreted in the following way--the effect of the last tunnel reparation on water losses from the tunnel exponentially decays with time. This long-time behavior is modeled by a negative exponential drift regressor, \(-e^{-\bar{t}/k}\), where \(\bar{t}\) is time in months elapsed since the last reparation date, and \(k\) is a scaling factor,  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(T_{wcc}\) & \(hP\) & \(\bar{t}\) & \(\overline{H}_{st}\) & \(\overline{H}_{r}\) \\ \hline \(T_{wcc}\) & & & & \\ \(hP\) & 0.312 & & & \\ \(\bar{t}\) & \(-\)0.085 & 0.161 & & \\ \(\overline{H}_{st}\) & 0.147 & 0.261 & \(-\)0.067 & \\ \(\overline{H}_{r}\) & 0.357 & 0.484 & \(-\)0.229 & 0.436 \\ \hline \hline \end{tabular}
\end{table}
Table 4: PC correlation matrix.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**PC**} & \multirow{2}{*}{**PC Eigenvalues**} & \multicolumn{2}{c}{**Eigenvalues Obtained by Hom’s**} & \multirow{2}{*}{**Decision**} \\  & & & **Parallel Analysis** & \\ \hline
1. & 4.422 & 1.3549 & accept \\
2. & 1.801 & 1.2326 & accept \\
3. & 1.222 & 1.1374 & accept \\
4. & 0.759 & 1.0594 & reject \\
5. & 0.404 & 0.9872 & reject \\... &... &... &... \\
9 & 0.050 & 0.6734 & reject \\ \hline \hline \end{tabular}
\end{table}
Table 3: Results of [PERSON]’s parallel analysis.  
which cannot be determined a priori. Several regressors with different values of \(k\) (1, 2, 4, and 10) were placed in the pool of regressors, with an assumption that the stepwise regression method will make an appropriate choice of regressors during the modeling process of water losses. During the application of PCA rotation methods, we did not achieve a simple structure (Table 4). This shortcoming is dealt with in the modeling stage of our methodology (Figure 1, block 3), with the use of interaction terms in forms \(T_{wcc}\overline{H}_{st}\), \(T_{wcc}\overline{H}_{st}\), and \(T_{wcc}\overline{H}_{st}\)2.  
Footnote 2: [[https://www.kaggle.com/](https://www.kaggle.com/)]([https://www.kaggle.com/](https://www.kaggle.com/))  
Based on previous considerations, an initial set of regressors was created, using the five principal components (\(T_{wcc}\), \(hP\), \(\overline{t}\), \(\overline{H}_{st}\), \(\overline{H}_{r}\)) (Table 5).