# Modeling of Water Losses in Hydraulic Tunnels under Pressure Based on Stepwise Regression Method
## 5 Discussion  
The adopted model includes 94.61% variance of water losses from the tunnel. Based on the _Tolerance_ and _VIF_ columns (Table 6), it can be seen that the adopted model has no problems with multicollinearity (for each of the regressors: Tolerance > 0.1 and VIF < 10). The _Standardized Coefficient_ column (\(\beta\)) shows the contribution of individual regressors to the entire model. The linear impact of the hydrostatic pressure on the water level in the surge tank (\(\overline{H}_{st}\)) dominantly affects water losses (0.855). This dominant effect is followed by the thermal effects (temperatures of water and concrete) coupled with cracks (\(-0.345\)), water height in piezometers (\(-0.222\)), interaction effects of temperature, size of crack openings, the water level in the surge tank (\(T_{wcc}\overline{H}_{st}\)) (\(-0.266\)), and the exponential influence of time since the last repair (\(-0.185\)). Standardized beta values indicate the number of standard deviations that scores in the dependent variable would change if there was one standard deviation unit change in the predictor [17]. In model (10), if we could increase the water level in the surge tank (\(\overline{H}_{st}\)) by one standard deviation (which is 10.64 m a.s.l., from the descriptive statistic in Table 1) the total water loss (\(q\)) would be likely to increase by 0.855 standard deviation units (which is 4.42548 L/s, for \(\sigma=5.176\) L/s, according to Table 1). The \(p\)-value column shows that each regressor in the model gives a unique contribution to the model (for each regressor, \(p\)-value associated with the F-statistic is less than 0.05). The squared values of semi-partial correlation coefficients (\(Part\)) indicate the values of individual regressor contributions to the total variance of the dependent variable. These unique contributions to the \(q\) variable variance for the regressors \(\overline{H}_{st}\), \(T_{wcc}\), \(hP\), \(T_{wcc}\), \(\overline{H}_{st}\), \(e^{-t}\), \(\overline{H}_{st}^{2}\), and \(T_{wcc}^{2}\) are 0.6626, 0.0888, 0.0339, 0.0493, 0.0324, 0.0083, and 0.0049, respectively. In other words, the water level in the surge tank contributes 66.26% to the variance of total water losses in HT, temperatures of water. and concrete temperature on the extrados of the concrete lining in the tunnel coupled with cracks, contribute 8.88%, etc. Note that the \(R^{2}\) value for the model (in this case 0.94014, or 94.014% explained variance) does not equal all the squared part correlation values added up (0.8801). This is because the part correlation values represent only the unique contribution of each variable, with any overlap or shared variance removed or partially taken out. The quality of the adopted model (10) of total water loss in HT of PSHPP \"Bajina Basta\" is visualized by the diagram which shows measured and predicted values of total water losses for the LDS (Figure 6). The 3D diagrams in Figure 7 correspond to the physical nature of the tunnel water loss phenomenon, as described above in Section 4.5. Figure 9: Aggregate effect of regressors in the model of total water losses in HT at PSHPP “Bajina Basta” for the period 2005–2015, grouped by their physical context (**a**) and measurement values of predictors Tc and Hst (**b**). Accuracy indicators for the adopted model, are given in Table 7. For the purpose of model comparison, RMSE values for the previous reference model of HT water losses [1] are also given in Table 7, for the same observation period. Using the novel methodology for modeling water losses at HT, we obtained the RMSE\({}_{\text{LDS}}\) of 1.8842 L/s, which is much better than 3.954 L/s obtained by the previous reference model on the same learning dataset. The model in this paper also performs better on the test dataset (2.0161 < 3.5983).