## 3 Results and Discussion
### Determination of the Activation Functions for the Hidden and Output Layers  
Commonly used activation functions of hidden and output layers include logsig, tansig, and purelin. Therefore, the optimal activation function combination was determined by comparing the network performance under all combinations of the three activation functions. Since the \(\text{R}^{2}\) values of the two outputs were negative when the output-layer activation function was logsig, combinations with logsig as the output-layer activation function were excluded. The training results of the remaining combinations are shown in Figure 10 and Table 17.  
Network performance is better for the logsig/purelin combination. Except that the training set \(\overline{\text{R}^{2}}\) (0.9641) is slightly smaller than that of the tansig/purelin combination (0.9646), the other network performance indices are the optimal values. Thus, the optimal activation function combination of the hidden and output layers was set to logsig/purelin.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{
\begin{tabular}{c} **No. of** \\ **Hidden-Layer** \\ **Neurons** \\ \end{tabular} } & \multicolumn{5}{c}{**Average Value of Network Performance Indices for Two Outputs**} \\ \cline{2-7}  & \multicolumn{2}{c}{**Performance Indices for the Training Set**} & \multicolumn{2}{c}{**Performance Indices for the Test Set**} \\ \cline{2-7}  & **RMSE** & **MAE** & **R\({}^{2}\)** & **RMSE** & **MAE** & **R\({}^{2}\)** \\ \hline
4 & 2.0104 & 1.4775 & 0.9313 & 2.2744 & 1.6981 & 0.9141 \\
5 & 1.6113 & 1.2020 & 0.9549 & 2.2306 & 1.6582 & 0.9112 \\
6 & 1.7768 & 1.2297 & 0.9466 & 2.2881 & 1.6919 & 0.9196 \\
7 & 1.5325 & 1.1333 & 0.9601 & 2.1607 & 1.6319 & 0.9231 \\
8 & 1.4273 & 0.9935 & 0.9641 & 2.0728 & 1.7391 & 0.9290 \\
9 & 1.5477 & 1.1539 & 0.9593 & 2.4943 & 1.8335 & 0.9004 \\
10 & 1.7167 & 1.1945 & 0.9500 & 2.6538 & 1.9693 & 0.8913 \\
11 & 1.4944 & 1.0783 & 0.9602 & 2.6942 & 1.9353 & 0.8850 \\
12 & 1.5744 & 1.1079 & 0.9579 & 2.6539 & 2.0620 & 0.8738 \\
13 & 1.5520 & 1.1576 & 0.9591 & 2.2583 & 1.8748 & 0.9218 \\ \hline \hline \end{tabular}
\end{table}
Table 16: Optimal network performance under each number of hidden-layer neurons (\(\text{R}^{2}\), RMSE, and MAE denote the coefficient of determination, root mean square error, and mean absolute error between experimental and BPNN-predicted values, respectively; \({}^{-}\) denotes average value of network performance indices for missing index and reseeding index).  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{
\begin{tabular}{c} **Hidden-** \\ **Layer** \\ **Activation** \\ **Function** \\ \end{tabular} } & \multicolumn{5}{c}{**Average Value of Network Performance Indices for Two Outputs**} \\ \cline{2-7}  & **Layer** & **Performance Indices for the Training Set** & **Performance Indices for the Test Set** \\ \cline{2-7}  & **RMSE** & **MAE** & **R\({}^{2}\)** & **RMSE** & **MAE** & **\(\overline{\text{R}^{2}}\)** \\ \hline Logsig & Tansig & 2.2038 & 1.5565 & 0.9179 & 2.8172 & 2.1504 & 0.8766 \\ Logsig & Purelin & 1.4273 & 0.9935 & 0.9641 & 2.0728 & 1.7391 & 0.9290 \\ Tansig & Tansig & 2.0002 & 1.3602 & 0.9323 & 2.6083 & 1.9306 & 0.8949 \\ Tansig & Purelin & 1.4303 & 1.0685 & 0.9646 & 2.1444 & 1.8452 & 0.9281 \\ Purelin & Tansig & 3.0668 & 2.2387 & 0.8409 & 3.4141 & 2.4209 & 0.8192 \\ Purelin & Purelin & 3.8248 & 2.6658 & 0.7512 & 4.2289 & 3.0312 & 0.7071 \\ \hline \hline \end{tabular}
\end{table}
Table 17: Optimal network performance under each activation function combination (\(\text{R}^{2}\), RMSE, and MAE denote the coefficient of determination, root mean square error, and mean absolute error between experimental and BPNN-predicted values, respectively; \({}^{-}\) denotes average value of network performance indices for missing index and reseeding index).