## 3 Results and Discussion
### Results of Soft Computing Based Models
The statistics of the overall error distribution is listed in Table 7. The values of minimum error, first quartile, median, third quartile, and maximum error are listed and presented in Table 7 and Figure 8 for all applied models. The lower quartile value in the stochastic M5P is \(-0.2508\) which is less than all other discussed models. On the other hand, in upper quartile, the stochastic M5P with Q75 = 0.2928 performed better than the other applied models. As can be seen in Figure 8 and Table 7, the maximum and minimum errors in stochastic M5P model are \(-11.53\) and \(2.675\), respectively, which verify the capability of this model to predict the relative energy dissipation (%) of the spillway. The lower box width in the stochastic M5P model confirms the maximum errors around the zero. Taylor diagrams are plotted in Figure 9 for all applied models. A Taylor diagram was used to illustrate schematically the performance of the applied models [48]. Three statistic parameters including standard deviation, correlation, and root mean square error, evaluated the accuracy of the applied models. Figure 9 indicates that the stochastic M5P model achieves higher correlation with minimum RMSE value. The Taylor diagram also confirms that the stochastic M5P model is performing better than the other applied models. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Method** & **SS** & **df** & **MS** & **F** & **\(p\)-Value** & **F Crit** &
\begin{tabular}{c} **Variation** \\ **Among Groups** \\ \end{tabular} \\ \hline
**Equation (14)** & 2.649 & 1 & 2.649 & 0.0056 & 0.9402 & 3.8454 &
\begin{tabular}{c} Insignificant \\ Insignificant \\ Insignificant \\ Insignificant \\
**M5P** & 1.0522 & 1 & 1.052 & 0.0021 & 0.9631 & 3.8454 &
\begin{tabular}{c} Insignificant \\ Insignificant \\ Insignificant \\ \end{tabular} \\
**Stochastic M5P** & 0.7684 & 1 & 0.7684 & 0.0016 & 0.9686 & 3.8454 &
\begin{tabular}{c} Insignificant \\ Insignificant \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Single-factor ANOVA results. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Models** & **Maximum** & **Minimum** & **Mean** & \ \begin{tabular}{c} **Standard** \\ **Deviation** \\ \end{tabular}  & **Kurtosis** & **Skewness** & \begin{tabular}{c} **Confidence** \\ **Level (95\%)** \\ \end{tabular}  \\ \hline \hline \multicolumn{10}{c}{Training data set} \\ \hline _Observation_ & 92.63 & 0.1240 & 57.68 & 22.17 & \(-0.3790\) & \(-0.6655\) & 0.863 \\ \hline
**Equation (14)** & 114.43 & 11.91 & 57.68 & 21.04 & \(-0.5525\) & 0.2231 & 0.819 \\ \hline RF & 92.12 & 1  
Figure 10 presents contours of energy loss in stepped spillways by means of \(H_{dam}\)/\(d_{c}\) vs. \(N\). With a constant value for N, an increase in \(H_{dam}\)/\(d_{c}\) values, shows an increase in energy loss up to 97.5%. This means that any decline in discharge over the spillway or an increase in spillway height, results in increase of energy loss. In addition, for a constant value of \(H_{dam}\)/\(d_{c}\), an increase in \(N\) results in increase of energy loss especially for \(N<90\). Higher number of steps (\(N>90\)) has no significant effect on energy loss. Transformation of contours of energy loss to vertical type demonstrates this phenomenon. Figure 10 is for spillways with 30 degrees in slope and the other contours of energy loss can be provided for different spillway slopes. For brevity, the other contour lines are not presented here. \begin{table}
\begin{tabular}{c c c c c} \hline
**Statistic** & **Equation (14)** & **RF** & **MSP** & **Stochastic MSP** \\ \hline Minimum & \(-31.79\) & \(-11.97\) & \(-25.78\) & \(-11.53\) \\ \hline
1 st Quartile & \(-3.468\) & \(-0.3842\) & \(-0.3555\) & \(-0.2508\) \\ \hline Median & 2.428 & 0.0230 & 0.2870 & 0.0245 \\ \hline
3 rd Quartile & 5.291 & 0.3545 & 0.8805 & 0.2928 \\ \hline Maximum & 7.758 & 4.253 & 3.294 & 2.675 \\ \hline \end{tabular}
\end{table} 
Table 7: The statistics of the applied models’ error distribution using the test data set.