# Analysis of Variables Influencing Scour on Large Sand-Bed Rivers Conducted Using Field Data
## 3 Results  
### MNLR  
Multiple nonlinear regression (MNLR) is a mathematical method that establishes a functional dependence between variables to predict the target variable. The optimization process finds the best fit for the model based on the principle of least squares. In Section 2.3, 10 influential variables were chosen as suitable for scour analysis on large sand-bed rivers. In order to model the data between the response \(d_{s}\) and the explanatory influential variables, the following dimensional and non-dimensional models were proposed: \begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**Influential variables** & \(b_{\mathit{ef}}\) & \(y\) & \(v\) & \(v_{c}\) & \(\tau\) & \(\tau_{c}\) & \(Fr\) & \(Fr_{c}\) & \(Fr_{d}\) & \(Re_{p}\) \\ \hline \end{tabular}
\end{table} 
Table 3: Influential variables selected for developing new regression-based models. \[d_{s}=a\cdot\Big{(}b_{ef}\Big{)}^{b}\cdot(y)^{c}\cdot(v)^{d}\cdot(v_{c})^{e}\cdot( \tau)^{f}\cdot(\tau_{c})^{g}\cdot(Fr)^{h}\cdot(Fr_{c})^{i}\cdot(Fr_{d})^{j}\cdot (Re_{p})^{k}, \tag{8}\]  
and non-dimensional model:  
\[\frac{d_{s}}{y}=a\cdot\Big{(}\frac{b_{ef}}{y}\Big{)}^{b}\cdot\Big{(}\frac{v}{v_ {c}}\Big{)}^{d}\cdot\Big{(}\frac{\tau}{\tau_{c}}\Big{)}^{f}\cdot(Fr)^{h}\cdot( Fr_{c})^{i}\cdot(Fr_{d})^{j}\cdot(Re_{p})^{k}. \tag{9}\]  
Both models were fitted to the filtered dataset by using MNLR, and the extracted parameters (\(a\), \(b\), \(c\), \(d\), \(e\), \(f\), \(g\), \(h\), \(i\), \(j\), and \(k\)) were used to form two new regression-based models presented in Table 4.  
The performance of MNLR dimensional and non-dimensional models was validated with a commonly used statistical measure, i.e., the coefficient of determination (R\({}^{2}\)). R\({}^{2}\) shows how closely the scour depth, predicted by the MNLR dimensional and non-dimensional equations, resembles the observed scour depth with values of 0.51 and 0.45, respectively. The same observation was made by [PERSON] and [PERSON] [30], who noticed that dimensional scour data were more accurate than those based on dimensionless data. Based on R\({}^{2}\), the dimensional MNLR model was adopted for further analysis. Too many influencing variables reduce the number of degrees of freedom and also increase R\({}^{2}\). Hereby, the adjusted R\({}^{2}\) is useful because it corrects the original R\({}^{2}\) in such a way that if the number of variables increases, the number of degrees of freedom decreases, as does the adjusted R\({}^{2}\). For the developed model, the adjusted R\({}^{2}\) = 0.50 does not change significantly in comparison with the original R\({}^{2}\), which confirms that the number of selected variables is sufficient and not overused. The dependency between measured scour depth values (\(d_{s}\)) and those predicted by the MNLR dimensional model (\(d_{s,pred}\)) is presented in Figure 3. The gray continuous line shows ideal agreement between measured and predicted values, i.e., \(d_{s,pred}\) = \(d_{s}\). Data points that lie below the line of agreement indicate overprediction (54% of predicted values are greater than actual measured values), while data points that lie above the abscissa represent underprediction (46% of predicted values are lower than measured values). \begin{table}
\begin{tabular}{l c c c c c c c c c c} \hline \hline  & \(\mathbf{a}\) & \(\mathbf{b}\) & \(\mathbf{c}\) & \(\mathbf{d}\) & \(\mathbf{e}\) & \(f\) & \(g\) & \(\mathbf{h}\) & \(\mathbf{i}\) & \(\mathbf{j}\) & \(\mathbf{k}\) \\ \hline dimensional & 3.29 & 0.49 & 1.19 & \(-\)0.91 & \(-\)0.99 & \(-\)0.011 & 0.019 & 0.38 & 0.26 & 0.97 & \(-\)0.44 \\ non-dimensional & 0.002 & 0.48 & & \(-\)0.90 & & \(-\)0.047 & & \(-\)0.33 & \(-\)2.50 & 1.16 & \(-\)0.094 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters obtained applying MNLR technique on both dimensional and non-dimensional models. Figure 3: Scattered plot of measured (\(d_{s}\)) and scour depth predicted (\(d_{s,pred}\)) by MNLR dimensional model.