## 5 Results and Discussion
According to the statistical analysis, the linear regression equations of the testing datasets of the ANFIS, MLP-ANN, RBF-ANN, and SGB models were respectively expressed as follows:  
\[y=1.0003x-0.0071;\hskip 28.452756 ptR^{2}=0.8754, \tag{26}\]  
\[y=0.988x+0.0223;\hskip 28.452756 ptR^{2}=0.943, \tag{27}\]  
\[y=0.993x+0.0232;\hskip 28.452756 ptR^{2}=0.9204, \tag{28}\]  
\[y=1.0044x-0.0167;\hskip 28.452756 ptR^{2}=0.9868. \tag{29}\]  
The regression equations express the accuracy or the deviation of the calculated \(C_{p}\) of the ionanofluids from actual values. As these equations are more similar to the bisector line equation, they gave more accurate predictions. The relative deviations (%) between the predicted and actual \(C_{p}\) of the ionanofluids for these approaches are depicted in Figure S3. The MREs (%) of the testing and training data of the SGB model were 0.93789% and 0.81109%, respectively. Furthermore, these values were 1.9799% and 1.8353%, 3.5439% and 3.5403%, and 1.768% and 1.6416% for the MLP-ANN, ANFIS, and RBF-ANN, respectively. It is clear that the SGB predicted the \(C_{p}\) better than the other models. The RBF-ANN and MLP-ANN showed similar results, but the RBF-ANN was slightly better than the MLP-ANN. Table 3 summarizes the statistical techniques, including the \(R^{2}\), MSE, MRE, RMSE, and STD for training, testing, and total datasets for each model. The determined indexes in the training  
Figure 4: ANFIS performance during the training stage using the particle swarm optimization (PSO) approach (with ten clusters). phase show that the proposed models were trained to acceptable degrees of accuracy. After the evaluation of the training phase, it was important to identify the performance of models regarding the determination of the \(C_{p}\) of ionanofluids for unseen conditions. Therefore, the statistical indexes in the testing phase were investigated to ensure the generalization of the model. The generalization of models was confirmed by the determination of the error values. The low error values show that the SGB model had an interesting accuracy for unseen points. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**SGB Tree** & \multicolumn{2}{c}{**ANFIS**} \\ \hline
**Type** & **Value/Comment** & **Type** & **Value/Comment** \\ \hline Learning rate & 0.1 & Membership function & Gaussian \\ Number of additive terms & 300 & No. of membership function (MF) parameters & 120 \\ Number of data points used for training & 429 & No. of clusters & 10 \\ Number of data points used for testing & 142 & Number of data points used for training & 429 \\ Minimum \(n\) in the child node & 1 & Number of data points used for testing & 142 \\  & & Optimization method & PSO \\  & & Population size & 85 \\  & & Iteration & 1000 \\  & & C\({}_{1}\) & 1 \\  & & C\({}_{2}\) & 2 \\ \hline
**MLP-ANN** & \multicolumn{2}{c}{**RBF-ANN**} \\ \hline No. input neuron layer & 5 & No. input neuron layer & 5 \\ No. hidden neuron layer & 12 & No. hidden neuron layer & 100 \\ No. output neuron layer & 1 & No. output neuron layer & 1 \\ Hidden layer activation function & Logsig & Hidden layer activation function & RBF \\ Output layer activation function & Purelin & Output layer activation function & Purelin \\ Number of data points used for training & 429 & Number of data points used for training & 429 \\ Number of data points used for testing & 142 & Number of data points used for testing & 142 \\ Maximum iterations & 1000 & Maximum iterations & 100 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: More details of trained models for the prediction of the \(C_{p}\)s of the ILs. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Model** & **Dataset** & \(\mathbf{R^{2}}\) & **MRE (\%)** & **MSE** & **RMSE** & **STD** \\ \hline \multirow{4}{*}{ANFIS} & Training & 0.8499 & 3.5403 & 0.0298 & 0.1726 & 0.1520 \\  & Testing & 0.8754 & 3.5439 & 0.0320 & 0.1788 & 0.1587 \\  & Total & 0.8575 & 3.5412 & 0.0303 & 0.1788 & 0.1536 \\ \hline \  
Figure 5: \(Cont.\)