# Modeling of Water Losses in Hydraulic Tunnels under Pressure Based on Stepwise Regression Method
## 4 Case Study
### Model of Total Water Losses in the HT Based on Principal Components  
Based on the created pool of regressors, the development of the HT total water losses model was performed in accordance with the strategy described in Section 3 and shown in Figure 1, block 3. The process of modeling was based on the measurements described in Section 4.3. From the total number of records (180), 156 were used for modeling (LDS--learning dataset), and 24 records were used to test the model (TDS--test dataset) (Figure 1(3a)). The TDS, consisted of records from the last four measurement sessions (76-79) of total water losses, in period 2015-2017, i.e., from the last period of HT monitoring. The automated software procedure, in accordance with the stepwise paradigm and numerous stepwise regression method variations described in Section 3.2, resulted in several MLR models of HTs total water losses which are stored in appropriate subsets of MLR models from \(\rho_{\text{MLR1}}\) to \(\rho_{\text{MLR6}}\) (Figure 1(3b)). The numerous models obtained from the iterative procedures are not shown due to their size. The most accurate model, MLR\({}_{\text{Best}}\), which satisfies the condition of non-multicollinearity, was chosen as appropriate (Figure 1(3d)), from the previous selected set of non-multicollinear regression models (MLR\({}_{\chi}\), \(\chi=1\), m) (Figure 1(3c)). The obtained model was the results of the stepwise selection method and the AICC criteria for entry/removal of regressors. AICC is based on the likelihood of the training set given the model and is adjusted to penalize overly complex models [36]. The coefficient of multiple determination (\(R^{2}\)), adjusted coefficient of multiple determination (\(R^{2}\)) and RMSE were selected as measures of selected model accuracy. The values of these indicators were \(R^{2}=0.94014\), \(R^{2}_{adj}=0.87386\) and RMSE\({}_{\text{LDS}}=1.8842\) L/s. The final adopted regression model, with all parameters and results of regression analysis, according to Section 3.3, is shown in Table 6. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Regressors**} & \multicolumn{2}{c}{**Unstandardized**} & \multicolumn{2}{c}{**Standard.**} & \multirow{2}{*}{**t**} & \multirow{2}{*}{**\(p\) Value**} & \multirow{2}{*}{**95\% Confidence Interval for B**} & \multirow{2}{*}{**Correlations**} & \multicolumn{2}{c}{**Collinearity**} \\  & & & & & & & & & & **Statistics** \\ \hline  & **B** & **Std. Error** & \(\beta\) & & & **Lower Bound** & **Upper Bound** & **Partial** & **Part** & **Toler.** & **VIF** \\ \hline Constant & 10.209 & 0.321 & & 31.76 & 0.000 & 9.575 & 10.844 & & & & \\ \(\overline{H}_{st}\) & 4.425 & 0.160 & 0.855 & 27.64 & 0.000 & 4.109 & 4.741 & 0.903 & 0.814 & 0.907 & 1.103 \\ \(T_{wcc}\) & \(-1.787\) & 0.177 & \(-0.345\) & \(-10.09\) & 0.000 & \(-2.136\) & \(-1.438\) & \(-0.610\) & \(-0.298\) & 0.743 & 1.347 \\ \(hP\) & \(-1.151\) & 0.184 & \(-0.222\) & \(-6.25\) & 0.000 & \(-1.514\) & \(-0.787\) & \(-0.430\) & \(-0.184\) & 0.686 & 1.458 \\ \(T_{wcc}\overline{H}_{st}\) & \(-1.169\) & 0.155 & \(-0.266\) & \(-7.55\) & 0.000 & \(-1.475\) & \(-0.863\) & \(-0.499\) & \(-0.222\) & 0.698 & 1.433 \\ \({c^{-1}}^{-}\) & \(-0.663\) & 0.109 & \(-0.185\) & \(-6.10\) & 0.000 & \(-0.878\) & \(-0.449\) & \(-0.422\) & 0.180 & 0.939 & 1.064 \\ \({H_{st}}^{2}\) & 0.451 & 0.147 & 0.100 & 3.08 & 0.002 & 0.162 & 0.740 & 0.228 & 0.091 & 0.817 & 1.223 \\ \(T_{wcc}\)2 & 0.408 & 0.171 & 0.094 & 2.39 & 0.018 & 0.071 & 0.746 & 0.179 & 0.070 & 0.560 & 1.787 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Parameters of the final adopted MLR model of HTs total water losses.