# Scouring Depth Assessment Downstream of Weirs Using Hybrid Intelligence Models
## 2 Materials and Methods
### Uncertainty Analysis  
The uncertainty associated with the input parameters and model structure for estimating the scour depth downstream of weir is quantified. The following procedure is used to measure the model structure uncertainty:  
1. All of the results obtained by the best estimator models (e.g., ANFIS-BBO, ANFIS-TLBO, ANFIS-Cultural, ANFIS-IWO) are considered for each computed scouring depth. 2. A normal distribution function is assigned to each predicted set. 3. To quantify the variability of predicted scouring depth, many samples (1000 generation), corresponding each predicted scouring depth, are generated by Monte Carlo simulation using the probability density function (PDF) obtained in step ii. 4. Using the scouring depths generated in step iii, the 95% confidence interval band, which is the interval between the 2.5% and 97.5% percentiles can be obtained. 5. To quantify the uncertainty associated with the model structure in the prediction of scouring depth, the R factor can be computed as follows [62; 63]:
6. \[R\ factor=\frac{X_{m}}{X_{s}}\] (9)  
In the above relation, \(X_{s}\) represents the standard deviation which computed for observed scouring depth and \(X_{m}\) measured by:  
\[Xm=\sum_{i=1}^{n}\left(UL_{L}^{i}-LL_{L}^{i}\right)/n \tag{10}\]  
The term n is referred to number of total observed data and \(UL_{L}^{i}\) and \(LL_{L}^{i}\) represent the values of the Upper (97.5%) and lower (2.5%) limits of the \(i\)th value, respectively. To evaluate the input parameters uncertainty, the scour depths measured from different types of  
Table 3 demonstrates the structure of input combinations proposed in the current study, where \(M1\) consists of all three parameters \(\left(\frac{d_{0}}{h_{t}},\frac{z}{h_{t}},\frac{U_{0}}{U_{C}}\right)\), \(M2\) removes \(\frac{z}{h_{t}}\) from input variables and \(M3\) uses only \(\frac{U_{0}}{U_{C}}\) as the input variable. Meanwhile, by considering five ANFIS algorithms and the three combinations, 15 different predictive scenarios are proposed to predict maximum scour depth. Table 4 demonstrates the outcome of the predictive model performance based on the _RMSE_, _MAE_, _CC_ and _WI_ criteria in the training and testing phases for the proposed models. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Parameters**} & \(\frac{u_{0}}{U_{C}}\) & \(\frac{d_{0}}{h_{t}}\) & \(\frac{z}{h_{t}}\) & \(\frac{d_{s}}{h_{t}}\) \\ \hline \(\frac{U_{0}}{U_{C}}\) & \(1\) & & & \\ \(\frac{d_{0}}{h_{t}}\) & \(-0.3604\) & \(1\) & & \\ \(\frac{z}{h_{t}}\) & \(-0.4891\)From Table 4, it is clear that the \(ANFIS-IWO-M1\) (\(MAE_{testing}=0.108\), \(RMSE_{testing}=0.148\)), which contains all of the input parameters, provides the highest performance prediction compared to other models. Moreover, \(ANFIS-M1\) (\(MAE_{testing}=0.133\), \(RMSE_{testing}=0.192\)) shows very close competition with the best model and ranked as the second-best predictive model, while \(ANFIS-M3\) (\(MAE_{testing}=0.679\), \(RMSE_{testing}=3.400\)) is selected as the worst model to predict the scouring depth. Considering the aspect of input combination, the prediction accuracy of all five algorithms in both the training and testing phase shows a decreasing trend from \(M1\) to \(M3\). At the same time, the \(M1\) exhibits absolute superiority over both training and testing phases. \(IWO\) algorithm with performance indices of (\(MAE_{testing}=0.108\sim 0.239\), \(RMSE_{testing}=0.148\sim 0.400\)) demonstrates the highest performance for three combinations compared to other alternative optimization models. Besides, both training and testing outcomes show similar performance behavior. For further comparison between the optimization model results, several graphical evaluation methods are utilized to investigate prediction performance between the observed and predicted values.