# Scouring Depth Assessment Downstream of Weirs Using Hybrid Intelligence Models
## 2 Materials and Methods
### Uncertainty Analysis
The scatter plot, which measures the \(ANFIS\) models' accuracy by visualizing the proximity of predicted and observed scour depth values and the corresponding linear correlation, is adopted in Figure 3.  
Figure 3 indicates that all of the hybrid models provide the acceptable ranges of performance prediction in both training and testing phases, where the \(ANFIS-IWO-M1\) model offers the highest correlation (\(CC_{training}=0.917\), \(CC_{testing}=0.932\)). It can be noted that all the \(ANFIS\) algorithms, including standalone and hybrids, have an excellent capability in scour depth prediction, as demonstrated by the close proximity of a large portion of the prediction points to the best line. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**No** & **Predictive Model** &
\begin{tabular}{c} **Input** \\ **Combination** \\ \end{tabular} & **Phase** & \(MAE\) & \(RMSE\) & \(CC\) & \(WI\) \\ \hline
**Model 9** & ANFIS-IWO-M2 & Without \(\frac{z}{h_{t}}\) & Training & 0.144 & 0.210 & 0.656 & 0. 760 \\ \cline{3-8}  & & Testing & 0.202 & 0.340 & 0.482 & 0.573 \\ \cline{3-8}
**Model 10** & ANFIS-TLBO-M2 & Without \(\frac{z}{h_{t}}\) & Training & 0.138 & 0.228 & 0.574 & 0.681 \\ \cline{3-8}  & & Testing & 0.226 & 0.379 & 0.251 & 0.379 \\ \hline
**Model 11** & ANFIS-M3 & Only \(\frac{U_{0}}{U_{C}}\) & Training & 0.153 & 0.208 & 0.665 & 0.775 \\ \cline{3-8}  & & Testing & 0.679 & 3.400 & 0.106 & 0.052 \\ \hline
**Model 12** & ANFIS- & \multirow{2}{*}{Only \(\frac{U_{0}}{U_{C}}\)} & Training & 0.179 & 0.272 & 0.214 & 0.242 \\ \cline{3-8}  & & Testing & 0.236 & 0.394 & 0.002 & 0.226 \\ \hline
**Model 13** & ANFIS-BBO-M3 & Only \(\frac{U_{0}}{U_{C}}\) & Training & 0.180 & 0.278 & 0.078 & 0.023 \\ \cline{3-8}  & & Testing & 0.234 & 0.392 & 0.000 & 0.214 \\ \hline
**Model 14** & ANFIS-IWO-M3 & Only \(\frac{U_{0}}{U_{C}}\) & Training & 0.162 & 0.229 & 0.598 & 0.625 \\ \cline{3-8}  & & Testing & 0.239 & 0.400 & 0.026 & 0.260 \\ \hline
**Model 15** & ANFIS-TLBO-M3 & Only \(\frac{U_{0}}{U_{C}}\) & Training & 0.171 & 0.262 & 0.343 & 0.408 \\ \cline{3-8}  & & Testing & 0.246 & 0.411 & 0.000 & 0.225 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: _Cont._As a statistical comparison method, a boxplot (Figure 4) shows the variability of observed and predicted scouring depth by measuring their 25%, 50%, 75% quartile values and the interquartile range (_IQR_). Figure 4: Boxplots of the predicted against observed scouring depth: (**a**) training phase, (**b**) testing phase. Figure 3: Scatter plots of the predicted values of scour depths against the observed ones: (**a**) training phase, (**b**) testing phase. From Figure 4a, it can understood that \(ANFIS-M1\) and \(ANFIS-Cultural-M1\) models (\(IQR=0.18\)) are closer to the observed values (\(IQR=0.24\)) in the training phase. Similar results are obtained in the testing phase (Figure 4b), where the \(ANFIS-IWO-M1\) and \(ANFIS-Cultural-M1\) (\(IQR=0.16\)) exhibit very similar results to the interquartile range of the observed data (\(IQR=0.26\)). As another comprehensive graphical presentation, a Taylor diagram is employed to make a further comparison between the observed and predictive values by using three statistical parameters, namely the \(CC\) (black dot lines), the \(RMSE\) (red contours) and the standard deviation (black contours) simultaneously (Figure 5). Figure 5: Normalized Taylor diagrams of predicted and observed scour depths values: (**a**) training phase, (**b**) testing phase. The iso-correlation, iso-\(RMSE\), and iso-normalized standard deviation lines, respectively, are the black-dotted, red-solid, and black-solid contours. The best model is the one which has less distance to the observed point, which is showed by the cyan circle. The results demonstrate that in both training and testing phase _ANFIS - IWO-M1_ is the closet point to the observed one which is an indicator of this model's performance in the prediction of scouring depth.