# Integration of Functional Link Neural Networks into a Parameter Estimation Methodology
## 4 Case Study
Similarly, \(\mathbf{W}_{in\_hid,}^{std}\), \(\left(\mathbf{w}_{hid\_out}^{std}\right)^{T}\), \(\mathbf{b}_{hid}^{std}\), and \(b_{out}^{std}\) represent the weight connection from the input to the hidden layers, the weight connection from the hidden layers to the output, the process bias in the hidden layers, and the process bias in the output layer of the observed standard deviation formula, respectively. According to the estimated regression formulas of the process mean and standard deviation, the response functions of the dual-response models between parameters \(x_{1}\) and \(x_{2}\) for two estimation methods (i.e., LSM and NN) are illustrated in Figures 4 and 5, including statistical indexes such as coefficients of determination \((R^{2})\) and root-mean-square error (RMSE). The rest of parameters \(x_{1}\) and \(x_{3}\), \(x_{2}\) and \(x_{3}\) are additionally provided as figures in Appendix A. The process mean, bias, and variance obtained from the proposed functional-link-NN-based dual-response estimation approach and the conventional LSM-based RSM are compared in Table 5 based on the corresponding optimal input variable settings and the EQL results. Table 5 implies that the proposed RD dual-response estimation \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{3}{c}{**Weight**} & \multicolumn{3}{c}{**Bias**} \\ \hline  & \(\mathbf{W}_{in\_hid}^{std}\) & \(\left(\mathbf{w}_{hid,out}^{std}\right)^{T}\) & \(\mathbf{b}_{hid}^{std}\) & \(b_{out}^{std}\) \\ \hline \(-2.04505\) & \(-3.02946\) & \(-4.90330\) & \(0.86246\) & \(-4.32652\) & \\ \(-0.38368\) & \(-1.22474\) & \(-0.62419\) & \(-2.32404\) & \(-2.08030\) & \(-0.01267\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Weight and bias terms of the NN for the estimated process standard deviation. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Objective** & **Response Function** & **Training Algorithm** & **Structure** & **No. of Epoch** \\ \hline Mean & mse & Trainlm & 3-21-1 & 13 \\ Std & mse & Trainlm & 3-2-1 & 12 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of NN-based estimation method. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{4}{c}{**Weight**} & \multicolumn{3}{c}{**Bias**} \\ \hline  & \(\mathbf{W}_{in\_hid}^{mean}\) & \(\left(\mathbf{w}_{hid,out}^{mean}\right)^{T}\) & \(\mathbf{b}_{hid}^{mean}\) & \(b_{out}^{mean}\) \\ \hline \(0.96075\) & \(0.11736\) & \(1.54028\) & \(2.10096\) & \(3.63174\) & \\ \(0.75123\) & \(0.38223\) & \(0.73934\) & \(1.62200\) & \(0.77913\) & \\ \(-0.28537\) & \(-0  
approach based on the functional link NN produced a significantly smaller EQL value than the conventional LSM-based RSM. In addition, compared with the conventional LSM, the proposed method can provide lower process variance. The process mean and the squared bias with regard to the variability of the standard LSM-based RSM and proposed NN-based estimation model are shown in Figure 6. The criterion spaces of the estimated regression functions are denoted by red stars. The optimal settings are marked as green star in each figure. Figure 4: Graphical representation of the estimated mean response functions between \(x_{1}\) and \(x_{2}\): (**a**) the response surface plot and response function of LSM (\(R^{2}\) = 92.68% and RMSE = 60.40); (**b**) the response surface plot and response function of NN (\(R^{2}\) = 86.37% and RMSE = 82.47). Figure 5: Graphical representation of the estimated standard deviation response functions between \(x_{1}\) and \(x_{2}\): (**a**) the response surface plot and response function of LSM (\(R^{2}\) = 45.41% and RMSE = 34.78); (**b**) the response surface plot and response function of NN (\(R^{2}\) = 44.18% and RMSE = 35.17).