## 3 Results
### Ultimate Strength Prediction  
Testing and training results for the prediction of strength of the RBNN, ANFIS-SC, ANFIS-FCM and MSTree models are listed in Table 2. The control parameter values of the optimal models are also provided in the second column. Different numbers of parameters and structures were tried for each method and the optimal ones were selected. Gaussian membership functions are used in ANFIS-SC and ANFIS-FCM models. The RBNN, ANFIS-SC and ANFIS-FCM methods can be easily obtained and applied by using new RBNN, genfis2 and genfis3 tools in MATLAB command windows. For the MSTree method, code which is available for free online ([[http://www.cs.rtu.lv/jekabsons/regression.html](http://www.cs.rtu.lv/jekabsons/regression.html)]([http://www.cs.rtu.lv/jekabsons/regression.html](http://www.cs.rtu.lv/jekabsons/regression.html))) is used. In the table, 0.8 and 15 indicate the spread value and the number of hidden layer neuron of the RBNN model, respectively, while 0.1 and 10 show the radii and cluster number of the ANFIS-SC and ANFIS-FCM models. A radii value of 0.1 in ANFIS-FCM corresponds to 15 clusters. This means that the ANFIS-FCM has fewer membership functions and parameters (10 Gaussian membership function each have 2 parameters, or 20 parameters in total) than those of ANFIS-SC. Table 2 implies that RBNN and ANFIS-FCM have almost the same accuracy, and they both are more efficient than the ANFIS-SC and MSTree models with respect to RMSE, MARE and \(R^{2}\). It is interesting that the MSTree approximated training data very well whereas its test results are worse than those of the other models. This implies that this method cannot adequately learn the investigated phenomenon. Different statistical indices were obtained for each of the methods. The main reason for this may be the fact that each method has different assumptions in developing models and their behaviors with the used data are distinct from each other. The rule base of the optimal ANFIS-FCM model is given in Table 3. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multirow{2}{*}{**Control**} & \multicolumn{3}{c}{**Training**} & \multicolumn{3}{c}{**Test**} \\ \cline{3-8}  & & **RMSE** & **MARE** & \(R^{2}\) & **RMSE** & **MARE** & \(R^{2}\) \\ \hline RBNN & 0.8,15 & 0.32 & 11.6 & 0.880 & 0.27 & 10.5 & 0.899 \\ ANFIS-SC & 0.1 & 0.31 & 11.1 & 0.891 & 0.32 & 11.3 & 0.879 \\ ANFIS-FCM & 10 & 0.30 & 11.0 & 0.896 & 0.27 & 10.7 & 0.903 \\ MSTree & & 0.26 & 8.25 & 0.921 & 0.43 & 14.3 & 0.739 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Statistical performance of RBNN, ANFIS-SC, ANFIS-FCM and MSTree models in strength predictions. Table 3 demonstrates that the model has 10 clusters and one rule for each of them. Figure 4 illustrates the strength estimates of the applied models in the forms of time variation and scatterplot. From the figure, it is apparent that the RBNN and ANFIS-FCM models have less scattered assessments than other models, and the M5 Tree model gave the most scattered estimates. This reveals that the investigated phenomenon is nonlinear, and therefore linear M5 Tree cannot adequately simulate strength behavior. The variation of strength versus strain ratio and confinement stiffness ratio for the optimal ANFIS-FCM model is illustrated in Figure 5, in which it is clear that strength takes its maximum value when the strain ratio is also at its maximum. The linear behavior for a confinement-stiffness-ratio greater than 0.3 is observed in Figure 5. This is because of the exponent of the confinement-stiffness-ratio in equations is usually 0.7; therefore, values greater than 0.3\({}^{0.7}\) behave linearly. Figure 4: Strength estimates of the RBNN, ANFIS-SC, ANFIS-FCM and M5 Tree models (dimensionless: \(f_{cc}^{\prime}/f_{co}^{\prime}\) ).