## 4 Results and Discussion
### Case 1: \(q_{u\,max}\)  
A full quadratic model was estimated by PLS regression after expanding the original data set with the quadratic and cross-product terms of the quadratic expansion. All variables were considered as continuous variables. Figure 2 represents the evolution of the model's coefficients of determination versus the number of components of the PLS model, and Figure 3 corresponds to the predicted response versus the actual response in the leave-one-out cross-validation (where the cross-validation blocks are formed by a single observation). The vertical line in Figure 2 shows that the optimum model has five components and can explain 70% of the model response. The response plot (Figure 3) indicates that the model predicts the removed observations quite well. Although there are differences between the fitted and cross-validated fitted models, none are severe enough to indicate an extreme leverage point. The ANOVA test of the model for \(\text{qu}_{\text{max}}\) with five components is presented in Table 5, which shows the degree of freedom (DF), the sum of squares (SS), the mean square (MS), and the F- and P-tests. As the \(p\)-value is under 0.05, the model is significant [51]. Table 5 also shows the variance of the indicators (X variance), the error (Er), the square error (R\({}^{2}\)), the prediction sum squares (PRESS), and the predicted determination coefficient (R\({}^{2}\) (pred)). The model with five components has a reasonably high R\({}^{2}\) of 78% and R\({}^{2}\)(pred) of 70% (Table 5), providing good indicators for its fitting ability and predictive accuracy. The regression coefficients for the \(\text{qu}_{\text{max}}\) model were used with the predictors to calculate the fitted value of the response variables, and the results are listed in Table 6.  
In a PLS regression, standardized coefficients identify the importance of each predictor in the model and correspond to the standardized X variables and standardized Y variables. The coefficient matrix (dimension p.r, where p = number of predictors and r = number of responses) is calculated from the x-weights and x-loadings [50]. Standardization of the variables can be conducted in different ways. In this study, for the normalization of the variables, the procedure of subtracting the mean and dividing by the standard deviation  
Figure 3: PLS response plot (5 components)—q\({}_{\text{u max}}\). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{5}{c}{**ANOVA Test**} \\ \hline
**Source** & **DF** & **SS** & **MS** & **F-test** & **P-test** \\ \hline Regression & 5 & 65,501.9 & 13,100.4 & 24.25 & 0.000 \\ Residual & 35 & 18,905.7 & 540.2 & & \\ Error & 40 & 84,407.6 & & & \\ \hline \hline \multicolumn{5}{c}{**Performance Statistics for the PLS Model**} \\ \hline
**Components** & **X Variance** & **Er** & **R\({}^{2}\)** & **PRESS** & **R\({}^{2}\) (pred)** \\ \hline
1 & 0.27653 & 53.943.3 & 0.360919 & 70,208.9 & 0.168216 \\
2 & 0.58601 & 46,419.1 & 0.450060 & 58,481.8 & 0.307150 \\
3 & 0.67210 & 26,476.7 & 0.686324 & 31,339 & 0.628718 \\
4 & 0.99928 & 24,830.4 & 0.705828 & 31,357.6 & 0.628498 \\
5 & 1.00000 & 18,905.7 & 0.776019 & 25,195.1 & 0.701507 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Summary of the performance statistics for the PLS model—q\({}_{\text{u max}}\). Figure 2: PLS model selection plot—q\({}_{\text{u max}}\). was used. In this way, multicollinearity resulting from the interaction terms can be minimized [50]. From the magnitude and sign of the standardized coefficients (Table 6), it can be seen that the CNT concentration (x3) and its quadratic relation (x3.x3) are the most important factors influencing the response (q\({}_{\text{u max}}\)). While x3 has a positive effect on q\({}_{\text{u max}}\), x3.x3 has a negative impact. It means that the CNT concentration has a non-linear effect in the response, i.e., there is an optimum value of CNT concentration that maximizes the response.