# Scouring Depth Assessment Downstream of Weirs Using Hybrid Intelligence Models
## 2 Materials and Methods
### Uncertainty Analysis
The outcome of this diagram also confirms the previous graphical methods results which nominated the _ANFIS - IWO-M1_ as a model with the highest accuracy. Prediction using AI methods always come with some uncertainties, which their assessment and analyzing is a crucial matter and have significant impact on outcome results. In this study, two major types of uncertainty which include model structure and input variable are evaluated. To measure the effectiveness of predictive model selection on forecasting the scouring depth, the _M1_ selected in the previous section is chosen as the reference combination. In the next step, five ANFIS algorithms predicted values in the nominated combination and are utilized to quantify the corresponding uncertainty. As for the uncertainty related to the selection of input parameters, _ANFIS - IWO_ was the best predictive model employed as the reference algorithm. Figure 6 illustrates the generated uncertainty band with the related observed values for both model structure and input variable in the testing phase. Figure 6: Generated prediction uncertainty band against observed values of scouring depth over testing phase based on (**a**) model structure and (**b**) input variables. From Figure 6, it can be understood that the uncertainty related to input parameter selection (\(R\)-\(factor=1.72\)) has a higher impact on the outcome of scouring depth prediction than the model structure (\(R\)-\(factor=1.15\)). To verify the prediction performance obtained in the present study, the results of the best metaheuristic algorithm result (ANFIS-IWO) is compared with those obtained by [PERSON] et al. [1] through the empirical formulas presented in Equations (3) and (4). In this way, the error metrics obtained by those models are outlined in Table 5.  
From Table 5, it is evident that the ANFIS-IWO offers the _RMSE_ of 0.147 and _MAE_ of 0.107, while the [PERSON] et al. [1] formula attains the _RMSE_ of 0.447 and _MAE_ of 0.395. Hence, the ANFIS-IWO model enhances the _RMSE_ and _MAE_ indices by 67.1% and 72.9%, respectively. To demonstrate the mathematical formulation of the best predictive model (ANFIS-IWO), its fuzzy if-then rules developed in prediction modeling of the scouring depth downstream of weirs and the validity range of each membership function are presented in the Supplementary Materials (Table S2 and Figure S1, respectively). To make a trade-off between the complexity of the predictive model and accuracy obtained in prediction modeling, another version of the ANFIS-IWO with five rules (ANFIS-IWO-5r) is compared with that obtained by ten rules (ANFIS-IWO-10r) which is mentioned earlier as the best predictive model. The fuzzy if-then rules and validity ranges of the membership functions of the ANFIS-IWO-5r are presented in the Supplementary Materials (Table S3, and Figure S2, respectively). The results indicate that the ANFIS-IWO-5r provides the _RMSE_ of 0.1557, _MAE_ of 0.1, and _CC_ of 0.829 in the training phase. Furthermore, this model offers the _RMSE_ of 0.239, _MAE_ of 0.144, and _CC_ of 0.838 in the testing phase. In general, the performance prediction is reduced (_RMSE_: -51%, _MAE_: -27%, and _CC_: -9.7%) by decreasing the number of membership functions. Although numerical, statistical, and artificial intelligence approaches have become more common in hydraulic engineering, open questions have remained about the influences of experimental conditions on obtained results. These approaches are restricted due to several simplifications in geometry properties, and hydraulic conditions which directly affect the flow conditions. Besides, most of these approaches use one- or two-dimensional flow properties which cause under or overestimation of results [64; 65]. Therefore, a deeper understanding of these factors on generated results is required. Furthermore, prediction of scouring depth, as a hydraulic sciences problem, is limited by several uncertainties, in which the turbulent nature of water as a dynamic fluid seems to be the most crucial modelling problem.