## 6 Results and Discussions  
After developing the models, their performance was compared and evaluated using RMSE and R\({}^{2}\), ranking, and residual plots being analyzed. Based on the training dataset, the ilmenite content predictive models (i.e., SVM, RF, SGB, CART, kNN, cubist, and ANN) were developed as described above. Table 3 calculates the indicators of performance of the soft computing techniques for estimating the content of ilmenite in the training process. Figure 12: Artificial neural network (ANN) model for estimating the content of ilmenite. Performance of the developed soft computing models in Table 3 showed that all the AI models were able to generate reasonably accurate ilmenite content estimation. Most models provided good performance with R\({}^{2}\) over 0.7 where some achieved R\({}^{2}\) over 0.8. Additionally, based on the ranking index, the ANN is the best model on the training dataset. To verify the performances of the stated soft computing techniques, 80 experimental datasets on the set of testing data was used as described above. Model validation is considered as the most critical step in the model building sequence. Table 4 computes the performance on the models using the testing dataset based on performance indicators. From Table 4, there are some differences from Table 3. Whereas the ANN is the best soft computing model on the training dataset, the cubist model with the highest performances, as well as the ranking, became the best model on the testing dataset. At the other end, the kNN model with the lowest performances and ranking on validation set also became the worst model on the testing dataset. Notably, although the ANN model on the training data set was the top scorer, its performance decreased significantly on the test set. It showed the unstable nature of ANN model in estimating the ilmenite content in this study. The RF and SGB models still retained the same stable performance as on the training dataset. Figure 13 shows the accuracy of the developed models on the testing dataset through R\({}^{2}\) values. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Model** & **RMSE** & **R\({}^{2}\)** & **Rank for RMSE** & **Rank for R\({}^{2}\)** & **Total Ranking Score** & **Sort** \\ \hline SVM & 0.092 & 0.780 & 1 & 3 & 4 & 5 \\ CART & 0.082 & 0.817 & 4 & 4 & 8 & 4 \\ kNN & 0.092 & 0.766 & 1 & 1 & 2 & 7 \\ RF & 0.080 & 0.824 & 6 & 6 & 12 & 2 \\ SGB & 0.081 & 0.818 & 5 & 5 & 10 & 3 \\ Cubist & 0.078 & 0.830 & 7 & 7 & 14 & 1 \\ ANN & 0.092 & 0.774 & 1 & 2 & 3 & 6 \\ \hline \end{tabular}
\end{table} 
Table 4: Performance of the soft computing techniques for estimating ilmenite content (testing dataset). \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Model** & **RMSE** & **R\({}^{2}\)** & **Rank for RMSE** & **Rank for R\({}^{2}\)** & **Total Ranking Score** & **Sort** \\ \hline SVM & 0.134 & 0.692 & 2 & 2 & 4 & 6 \\ CART & 0.147 & 0.675 & 1 & 1 & 2 & 7 \\ kNN & 0.122 & 0.747 & 6 & 6 & 12 & 2 \\ RF & 0.127 & 0.737 & 5 & 5 & 10 & 3 \\ SGB & 0.132 & 0.716 & 3 & 3 & 6 & 5 \\ Cubist & 0.128 & 0.720 & 4 & 4 & 8 & 4 \\ ANN & 0.091 & 0.860 & 7 & 7 & 14 & 1 \\ \hline \end{tabular}
\end{table} 
Table 3: Performance of the soft computing techniques for estimating ilmenite content (training dataset). Although the R\({}^{2}\) values of the models were high, they did not guarantee that the models fit the data well. Therefore, the residuals analysis of the models was conducted in order to check the assumptions of independence, normality, and homoscedasticity. The histogram (Figure 14) and the normal probability plots (Figure 15) were used to check whether or not it is justifiable to suppose that the random errors inherent in the developed soft computing models were extracted from a normal distribution. Figure 13: Predicted/measured parameters of the soft computing techniques used herein. Based on histogram plots of the residuals of the models (Figure 14), we can see that most of the residuals of the models are Gaussian or normal distribution. It suggested that the models fit the data well.