## 4 Results and Discussion
### Models Robustness  
Investigation of the robustness of three developed AI algorithms were achieved by performing 1000 Monte Carlo simulations, where an uniform distribution of data index was applied to construct the training and testing dataset for each run. Thereby, 1000 corresponding values of R, RMSE and MAE were obtained. The values of R for testing the PSOANFIS, GAANFIS and SVM over 1000 runs were plotted (Figure 7) to demonstrate the high level of fluctuation of R in function of the choice of dataset.  
In order to estimate the robustness of the AI models, statistical analysis of the criteria was performed. It is worth noticingthat a post-treatment was performed at this stage. The outliers were removed using the quantile at 90% of RMSE as a threshold value. These values were extreme and not representative for the statistical analysis of the results obtained by AI models. Of over 1000 values obtained after the simulations, only 774 relevant values were used to perform the statistical analysis. Firstly, the statistical convergence of R, RMSE and MAE was introduced in order to determine the  
Figure 7: Values of R over 1000 Monte Carlo simulations in case of (**a**) testing PSOANFIS, (**b**) testing GAANFIS and (**c**) testing SVM.  
optimal number of Monte Carlo simulations. It is noticed that the convergence indicator \(\mathrm{I_{C}}\) was introduced in the previous section (see Equation (4)). The values of \(\mathrm{I_{C}}\) with respect to R (Figure 8a), RMSE (Figure 8c) and MAE (Figure 8e) over 774 Monte Carlo simulations are presented. It is observed that the three AI methods exhibited an optimal number of about 300 runs to reach the stationary solution of R, in other words, the PSOANFIS, GAANFIS, SVM were statistically converged after about 300 simulations (Figure 8). As regards to RMSE and MAE, it is shown that the SVM model possessed a smaller number of runs than that of the PSOANFIS and GAANFIS (i.e., 300 compared with 400 runs for PSOANFIS and GAANFIS). Besides, an important fluctuation of the \(\mathrm{I_{C}}\) curves of RMSE and MAE using the PSOANFIS and GAANFIS was observed (i.e., at N smaller than 100). Two conclusions can be deduced: (i) outliers should be removed before performing statistical analysis; (ii) at least 500 Monte Carlo simulations were needed to obtain reliable statistical analysis results, and (ii) the SVM algorithm is the most stable and robust predictor even with the variation of input index in the dataset. Detailed statistical information related to the robustness of the three proposed AI models is summarized in Table 6.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Method** & \(\mathbf{Mean_{R}}\) & \(\mathbf{Std_{R}}\) & \(\mathbf{Mean_{RMSE}}\) & \(\mathbf{Std_{RMSE}}\) & \(\mathbf{Mean_{MAE}}\) & \(\mathbf{Std_{MAE}}\) \\ \hline
**PSOANFIS** & 0.8655 & 0.0784 & 0.5485 & 0.1937 & 0.3782 & 0.0982 \\
**GAANFIS** & 0.8463 & 0.0723 & 0.5769 & 0.1447 & 0.4206 & 0.0889 \\
**SVM** & 0.9246 & 0.0376 & 0.4004 & 0.1158 & 0.2741 & 0.0600 \\ \hline \hline \end{tabular}
\end{table}
Table 6: The robustness of PSOANFIS, GAANFIS and SVM for testing part.  
Figure 8: Statistical results over 774 Monte Carlo simulations using the PSOANFIS, GAANFIS and SVM for \(\mathrm{I_{C}}\) in case of: (**a**) R; (**c**) RMSE and (**e**) MAE along with the probability density distribution of (**b**) R, (**d**) RMSE and (**f**) MAE.  
The probability density distributions of 774 values of R (Figure 8b), RMSE (Figure 8d) and MAE (Figure 8f) are also presented. It is observed that all distributions are highly asymmetric, particularly in case of the SVM model. In conclusion, from overall statistical analysis, the SVM method is the most robust and powerful algorithm to predict MS. The SVM model, after being carefully evaluated herein, could be used for saving time and cost in laboratory experiment.