# Analysis of Variables Influencing Scour on Large Sand-Bed Rivers Conducted Using Field Data
## 2 Methodology
### Variable Reduction  
Estimation of scour depth requires the detection of the most significant input variables relevant to the complex scour process in large sand-bed rivers. This can be achieved using principal component analysis (PCA), which has already been proven to be an effective decision-making tool for dimensionality reduction when many variables are involved [43]. PCA determines which variable influences scour depth the most by analyzing how much each variable contributes to the variance. Therefore, PCA was applied to the filtered dataset of 348 measurements and a total of 21 variables (including the observed scour depth \(d_{s}\)). PCA interprets data with principal components--linear combinations of the input variables that are orthogonal to each other. In this study, there are 21 principal components as well as 21 input variables. The first principal component explains the largest amount of variability in the dataset; the second principal component accounts for the next largest variance; etc. The principal components are lines in the coordinate system with a corresponding eigenvalue--the sum of the squared distances between the orthogonally projected observed data onto the line and the origin of the coordinate system. Variable reduction in PCA analysis is based on the number of retained principal components without diminishing the total variance of the scour dataset. A commonly used criterion for reducing the number of variables is the Kaiser-Guttman criterion, which considers only those principal components whose eigenvalues are greater than 1. Applying the Kaiser-Guttman criterion for variable reduction would keep the first 7 principal components and thus eliminate only 2 variables. Since the Kaiser-Guttman criterion was previously proven to be inaccurate because it overestimates the number of principal components [44], it was decided to use a more stringent criterion. If we discard the principal components whose eigenvalues are less than 2, then the first three principal components remain. The first three principal components explain 56% of the variance of the entire dataset. The variables that contribute the most to the first three principal components  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Variable**} & \multicolumn{4}{c}{**Before Filtering**} & \multicolumn{4}{c}{**After Filtering**} \\ \cline{3-8} \multicolumn{2}{c}{} & \multicolumn{1}{c}{**Range**} & \multicolumn{1}{c}{**Average**} & \multicolumn{1}{c}{**Standard**} & \multicolumn{1}{c}{**Range**} & \multicolumn{1}{c}{**Average**} & \multicolumn{1}{c}{**Standard**} \\ \hline \multirow{8}{*}{Original} & \(d_{s}\) & 0–10.4 & 1.1 & 1.3 & 0.21–7.8 & 1.5 & 1.3 \\  & \(b_{\text{eff}}\) &  
can be identified as influential (_b_eff, y, d_50, v, v\({}_{\sigma}\), \(\tau\), \(\tau_{\sigma}\), Fr, Fr, Fc, Fr\({}_{d}\), Re\({}_{p}\), B, b, l, and S), while other variables (\(\sigma_{g}\), \(\theta\), \(RI\), pier type, and nose shape) can be excluded from further analysis. The results of the PCA analysis are presented in the form of a loading plot, in which the variable vectors are positioned with respect to the first two principal components (Figure 2a). Variables located near the center of the coordinate system have the smallest impact on the variance of the dataset (within the inner dashed circle) and are therefore eliminated from further analysis. When two variable vectors are perpendicular to each other, they are not correlated at all, while variables whose vectors are close to each other are highly correlated. The contribution of each variable to the first three principal components can be evaluated by the squared cosine values (Figure 2b). Since the principal components are axes of the rotated coordinate system, the squared cosine represents the quality of the variables after rotation. The values in bold are the maximum squared cosine values for each variable. Therefore, only the variables that have the maximum squared cosine for one of the first three principal components are retained for further analysis.