# Using Optimisation Meta-Heuristics for the Roughness Estimation Problem in River Flow Analysis
## 6 Experimental Results
This is possibly due to (i) the small budget of computational time allowed for the task, and (ii) the trajectory-based nature of \((1+1)\)-ES which does not require the 'warmup time' typical of population-based meta-heuristics. Therefore, we believe that even for different data in input, \((1+1)\)-ES can be reasonably chosen as the algorithm to go with our methodology. However, this study does not take into account situations where the number of cross-sections, so the dimensionality of the space, is very large. In such situations, two viable strategies are: (i) reduce the dimensionality by subsampling the cross-sections, (ii) increase the computational budget and give a try also to the population-based meta-heuristics DE and CMA-ES. Figure 8: Empirical probability density functions of the final fitness values registered by the executions of each algorithm estimated using the Kernel Density Estimation methodology [55]. (**a**) Case study 1—River Paglia; (**b**) Case study 2—River Aniene. Finally, for the sake of completeness, we analysed the standard deviation of the estimated [PERSON]'s values across the 25 different executions of the (1 + 1)-ES algorithm. These standard deviation values, for five selected cross-sections, are provided in Tables 3 and 4. By recalling that the selected range for the roughness coefficients is \([0.02,0.2]\), Tables 3 and 4 show that the standard deviations never overstep the 5% of the selected range length. This means that the different executions of the considered algorithm converge to very similar [PERSON]'s values, thus confirming the robustness of the algorithm. Similar behaviour has been observed also for the other four meta-heuristics. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Cross Section**} & \multicolumn{2}{c}{**Std Dev. of the Manning’s Values for River Aniene**} \\ \cline{2-4}  & **Left Channel** & **Main Channel** & **Right Channel** \\ \hline
1 & 0.006855 & 0.006383 & 0.006739 \\
10 & 0.007071 & 0.006672 & 0.006709 \\
20 & 0.006489 & 0.006073 & 0.006354 \\
30 & 0.006764 & 0.006162 & 0.006629 \\
40 & 0.008475 & 0.005951 & 0.007752 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Standard deviations of the Manning’s values obtained by the 25 executions of the (1 + 1)-ES algorithm for the case of study ‘river Aniene’. For the sake of readability, five cross-sections were reported, though the magnitude of the standard deviations is roughly the same for all the other cross-sections. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Cross Section**} & \multicolumn{2}{c}{**Std Dev. of the [PERSON]’s Values for River Paglia**} \\ \cline{2-4}  & **Left Channel** & **Main Channel** & **Right Channel** \\ \hline
1 & 0.007826 & 0.007999 & 0.009895 \\
5 & 0.008051 & 0.007276 & 0.008452 \\
10 & 0.007449 & 0.008903 & 0.006296 \\
15 & 0.003794 & 0.008512 & 0.008731 \\
20 & 0.006102 & 0.008623 & 0.008709 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Standard deviations of the [PERSON]’s values obtained by the 25 executions of the (1 + 1)-ES algorithm for the case of study ‘river Paglia’. For the sake of readability, five cross-sections were reported, though the magnitude of the standard deviations is roughly the same for all the other cross-sections. Figure 9: Average convergence graphs of the algorithms’ best-so-far (abscissa is in log-scale). (a) Case study 1—River Paglia; (b) Case study 2—River Aniene.