# Improvement of ANFIS Model for Prediction of Compressive Strength of Manufactured Sand Concrete
## 4 Results and Discussions
### Prediction Capability: Improvement of Single ANFIS  
In this section, the prediction capability of the proposed models is analyzed thanks to statistical error criteria such as R, RMSE and MAE. A linear fitting line, defined as \"Predicted = Slope *Actual + C\" is also used for the prediction capability analysis of these models. The statistical results of the ANFIS model optimized by TLBO and using PCA, denoted as ANFIS-TLBO/P, including the values of R, RMSE, MAE, error mean, error standard deviation and slope are summarized in Table 3, for the training and testing data sets. The correlation between actual and predicted compressive strength of ANFIS-TLBO/P was also plotted in Figure 3. For the training part, it can be noted that the correlation coefficient of ANFIS-TLBO/P was R = 0.92 with a slope of 0.86. For the testing part, the correlation coefficient was R = 0.96 with a slope of 0.94. In addition, the comparison of these criteria between ANFIS-TLBO/P and other models is also shown in Table 4. Let ANFIS/R and ANFIS/P are defined as the single ANFIS model using raw data and the processed data using PCA technique, respectively. Besides, the ANN model using PCA technique is designed as ANN/P, whereas ANFIS-TLBO/R is the ANFIS optimized with TLBO without the PCA treatment. In comparison with other techniques for the testing dataset, it can be seen that ANFIS-TLBO/P had the highest values of R and slope. Indeed, the value of R for ANFIS/R, ANFIS/P and ANFIS-TLBO/R was 0.86, 0.88 and 0.93, respectively. On the other hand, the value of slope for these three models was 0.78, 0.76 and 0.89, respectively.  
Figure 1: Optimization of population size with respect to (**a**) Correlation Coefficient (R) and (**b**) Root Mean Squared Error (RMSE).  
Figure 2: Convergence of optimization cost with respect to (**a**) R and (**b**) RMSE using a population size of 40.  
Considering now RMSE and MAE, it can be seen that the values of RMSE, MAE, error mean and error standard deviation of ANFIS-TLBO/P were 6.62, 4.77, \(-\)0.08 and 6.64, respectively for the training part; and 4.93, 4.09, 0.26 and 4.95, respectively, for the testing part. In comparison with other models with respect to the testing part, ANFIS-TLBO/P produced the highest performance, meaning the lowest values of RMSE, MAE and error standard deviation. In particular, the values of these criteria were 8.58, 6.24 and 8.63 for ANFIS/R; 7.65, 5.02 and 7.59 for ANFIS-TLBO/R; and 6.46, 5.28 and 6.49 for ANFIS/P, respectively. In addition, ANN/P was also applied to check the performance of the ANFIS-TLBO/P model and the results show that ANFIS-TLBO/P outperforms ANN/P (Table 4).  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Model** & **Data Used** & **Designation** & **R** & **RMSE** & **MAE** & **Error Std** & **Slope** \\ \hline Individual ANFIS & raw & ANFIS/R & 0.86 & 8.58 & 6.24 & 8.63 & 0.78 \\ \hline Individual ANFIS & pre-processed & ANFIS/P & 0.93 & 6.46 & 5.28 & 6.49 & 0.89 \\ \hline Individual ANN & pre-processed & ANN/P & 0.90 & 7.67 & 5.06 & 7.67 & 0.77 \\ \hline ANFIS+TLBO & raw & ANFIS-TLBO/R & 0.88 & 7.65 & 5.02 & 7.59 & 0.76 \\ \hline ANFIS+TLBO & pre-processed & ANFIS-TLBO/P & **0.96** & **4.93** & **4.09** & **4.95** & **0.94** \\ \hline \hline \end{tabular}
\end{table}
Table 4: Comparison of prediction capability of different models using raw and pre-processed data, respectively. Correlation Coefficient (R), Root Mean Squared Error (RMSE).  
Figure 3: Regression graphs of ANFIS-TLBO/P model for (**a**) training and (**b**) testing dataset, respectively.  
For conclusion, firstly, PCA technique helped increasing the prediction performance of the AI models by reducing noise in the input space. Secondly, TLBO technique well optimized the parameters of individual the models. Finally, out of all tested models, ANFIS-TLBO/P was the best model for the prediction of MSC compressive strength, in taking into account sand's characteristics.