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Table 3. Input parameters combination for prediction of scouring depth downstream of weirs.

Table 3 Input parameters combination for prediction of scouring depth downstream of weirs.

Related Figures (12)

Figure 1. Sketch of the scouring process downstream of a weir. depth, flow velocity, tailwater depth and geometry of weirs [1]. Figure 1 illustrates the graphical description of scour depth downstream of weirs.
Figure 1. Sketch of the scouring process downstream of a weir. depth, flow velocity, tailwater depth and geometry of weirs [1]. Figure 1 illustrates the graphical description of scour depth downstream of weirs.
Table 1. Range of the target and input variables employed in the present study. 2.2. Adaptive Neuro-Fuzzy Inference System Table 1 shows the range of input and output parameters employed in the present study. The adaptive neuro-fuzzy inference system (ANFIS) was first introduced by Jang [38]. The key strengths of ANFIS include its potential to estimate non-linear relations and less dependency on user knowledge [50]. The non-linear relationships between inputs and outputs are yielded through membership functions. Membership functions map each input correspondent to values between 0 and
Table 1. Range of the target and input variables employed in the present study. 2.2. Adaptive Neuro-Fuzzy Inference System Table 1 shows the range of input and output parameters employed in the present study. The adaptive neuro-fuzzy inference system (ANFIS) was first introduced by Jang [38]. The key strengths of ANFIS include its potential to estimate non-linear relations and less dependency on user knowledge [50]. The non-linear relationships between inputs and outputs are yielded through membership functions. Membership functions map each input correspondent to values between 0 and
Table 2. Pearson correlation between the predictive variables and scouring depth downstream of weirs.
Table 2. Pearson correlation between the predictive variables and scouring depth downstream of weirs.
Table 4. The performance indices obtained through different predictive models for predicting the scouring depth downstream of weirs.
Table 4. The performance indices obtained through different predictive models for predicting the scouring depth downstream of weirs.
Table 4. Cont. and observed scour aeptn values and the corresponding linear correlation, 1s aaopted in rigure o. 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 raining = 0.917, CCtesting = 0.932). Itcan 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.
Table 4. Cont. and observed scour aeptn values and the corresponding linear correlation, 1s aaopted in rigure o. 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 raining = 0.917, CCtesting = 0.932). Itcan 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.
Figure 3. Scatter plots of the predicted values of scour depths against the observed ones: (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.
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.
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 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.
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.
Figure 6. Generated prediction uncertainty band against observed values of scouring depth over testing phase based on (a) model structure and (b) input variables. a On To measure the effectiveness of predictive model selection on forecasting the scouring depth, he M1 selected in the previous section is chosen as the reference combination. In the next step, ive ANFIS algorithms predicted values in the nominated combination and are utilized to quantify he 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 he 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. a On To measure the effectiveness of predictive model selection on forecasting the scouring depth, he M1 selected in the previous section is chosen as the reference combination. In the next step, ive ANFIS algorithms predicted values in the nominated combination and are utilized to quantify he 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 he generated uncertainty band with the related observed values for both model structure and input variable in the testing phase.
Table 5. Comparison between the prediction performance achieved by the best predictive model i the present study and empirical formulas proposed by Guan et al. [1]. From Table 5, it is evident that the ANFIS-IWO offers the RMSE of 0.147 and MAE of 0.107, while the Guan 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.
Table 5. Comparison between the prediction performance achieved by the best predictive model i the present study and empirical formulas proposed by Guan et al. [1]. From Table 5, it is evident that the ANFIS-IWO offers the RMSE of 0.147 and MAE of 0.107, while the Guan 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.
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