## 3 Results and Discussion  
### Case 1  
During the ANFIS model's formulation, the number of MFs for each input, type of MF, number of learning epochs, initial step size and step size increase rates were varied. In case 1, variations of different types of MF were used, such as generalized bell-shaped, triangular, trapezoidal, pi-shaped, Gaussian's and Gaussian combination MF. The type of MF has not altered the results in a significant manner. Modeling with a different number of MFs for each input (dependent variable) was also performed. Increasing the number of MF number, computational time and number of IF-THEN rules formed by the ANFIS also increase. Input MFs were generated using the grid partition clustering method that uniformly partitioned the input variable ranges and created a single-output Sugeno fuzzy system. Other clustering methods, such as subtractive clustering, have achieved lower modeling precision rates. A Gaussian MF was adopted and the ANFIS structure (Figure 6) was trained for 100  
Figure 5: Case 1 and case 2 flowchart representations.  
epochs using back-propagation and a hybrid algorithm to determine the premise and consequent parameters, with an initial step size equal to 0.1. and a step size increase rate of 1.10. Results are stated in Table 1 for the LOSO cross-validation. The ANFIS structure trained with three MF and the hybrid learning algorithm computed the lowest training RMSE (\(=8.37\times 10^{-7}\)) but a relatively high testing RMSE (\(=0.8594\)). Lowering the number of epochs did not result in improving the generalization ability of the discussed ANFIS that tended to overfit the training set.  
The computational time of the ANFIS with back-propagation learning and two MFs achieved the best results with the input parameters stated in the above paragraph. The discussed ANFIS was about 5.6 times faster compared to the ANFIS structure with the hybrid learning algorithm and three MFs, 4.8. times faster than the ANFIS with back-propagation learning and three MFs and had a similar computation time to the ANFIS structure with hybrid learning and two MFs.  
Based on the set of results in Table 2, further experiments (Figure 7) were performed on the ANFIS structure with the same ANFIS input parameters and back-propagation learning algorithm and two Gaussian MFs. To produce the results in Figure 7, the 10-fold cross-validation was used. Two Gaussian MFs were used due to their simplicity and high performance (low computation). In further experiments, six different initial step sizes (0.001, 0.005, 0.01, 0.05, 0.1 and 0.5) and six different step size increase values (1.00001, 1.001, 1.1, 1.2, 1.3 and 1.4) were used for the ANFIS model construction. The ANFIS structure with the 0.1 initial step size and 1.001 step size increase achieved the best test RMSE (\(=0.4195\)) as can be seen in Figure 7a. The train RMSE of the mentioned structure was 0.1254 as depicted in Figure 7b.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{5}{c}{**ANFIS Information**} \\ \hline Learning method & hybrid & hybrid & Back-propagation & Back-propagation \\ Output MF & linear & linear & linear & linear \\ Number of MF per input & 2 & 3 & 2 & 3 \\ Number of nodes & 34 & 78 & 34 & 78 \\ Number of linear parameters & 32 & 108 & 32 & 108 \\ Number of nonlinear parameters & 18 & 27 & 18 & 27 \\ Number of fuzzy rules & 8 & 27 & 8 & 27 \\ \hline Training RMSE & 0.0603 & \(8.37\times 10^{-7}\) & 0.1643 & 0.1769 \\ \hline Test RMSE & 0.8125 & 0.8594 & 0.3450 & 0.5507 \\ \hline Training R\({}^{2}\) & 0.9985 & 1.000 & 0.9855 & 0.9549 \\ \hline Test R\({}^{2}\) & 0.6775 & 0.4051 & 0.6917 & 0.4870 \\ \hline Time (sec) & 8.04 & 42.85 & 7.60 & 36.44 \\ \hline \hline \end{tabular}
\end{table}
Table 2: ANFIS information.  
Figure 6: ANFIS network with two membership functions (MFs) (**a**) and three MFs (**b**).