# 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
#### 4.4.2 PCA Results  
A set of 9 variables, described in 4.3, was subjected to PCA using R software [34]. Firstly, the suitability of data for PCA was assessed. The number of samples was 156, which exceeds the minimum recommended sample size of 150 [18]. The value of the KMO index was 0.772, which exceeds the minimum recommended value of 0.6, and indicates sampling is adequate [18]. Most correlation coefficients (23 out of 36) in the correlation matrix were above 0.3 [18] and [PERSON]'s test of sphericity reached statistical significance (\(p<0.05\)) [27], indicating the factorability of the correlation matrix.  
In the second step, PCA analysis revealed the presence of 3 components with eigenvalues exceeding 1 ([PERSON]'s criterion) [26], which explain 49.13%, 20.01%, and 13.58% of the variance respectively, accounting for 82.72% of total variance. [PERSON] recommends retaining all components above the elbow or break in the plot (Figure 5), as these factors contribute the most to the explanation of the variance in the data set [28]. Based on [PERSON]'s test it has been found that reduction to the 2, 5, or 6 principal components can be considered.  
[PERSON]'s parallel analysis involves comparing the size of the eigenvalues with those obtained from a randomly generated data set of the same size. Only components with eigenvalues that exceed the corresponding values from the random data set are retained [29]. The results of the parallel analysis, shown in Table 3, indicate three components whose eigenvalues exceed the corresponding threshold values obtained using an equally sized matrix of random numbers (9 variables and 162 records). [PERSON]'s parallel analysis was carried out using the Monte Carlo PCA for parallel analysis software [35].  
Figure 5: Screen plot of principal components.  
Although [PERSON]'s criterion and [PERSON]'s parallel analysis suggest a three-component solution, accounting for 82.72% of total variance, we arbitrarily selected a five-component solution based on [PERSON]'s test, in order to retain more information about the process of generating water losses in the HT and facilitate a more accurate model of water losses. The five-component solution explains a total of 94.61% variance, with component contributions of 49.13%, 20.01%, 13.58%, 8.44%, and 4.45%, respectively.  
In the third step, several rotation methods were performed. Finally, the direct oblimin rotation method was chosen as appropriate, enabling the following PC interpretations: \(T_{wcc}\)--which includes water temperature, concrete temperature, and cracks opening \(hP\)--which includes the influence of water level in the piezometers, \(\bar{t}\)--which represents time, \(\overline{H}_{st}\)--which represents water level in the surge tank, and \(\overline{H}_{r}\)--which represents water level in the upper reservoir. Table 4 shows correlation coefficients between selected PCs.  
It can be seen from Table 4 that some components are linearly independent of each other because the correlation coefficients are less than 0.3, and some components have mild linear dependence.