# Integration of Functional Link Neural Networks into a Parameter Estimation Methodology
## 4 Case Study  
The printing data proposed by [PERSON] and [PERSON] [38] are discussed in this study for comparative analysis; these data had been used by [PERSON] and [PERSON] [8] and [PERSON] and [PERSON] [11] as well. Three experimental parameters, \(x_{1}\), \(x_{2}\), and \(x_{3}\) (speed, pressure, and distance), of a printing machine are treated as input variables to examine the ability to apply colored inks to package levels (\(y\)). These three control factors are assumed to be examined in three levels (\(-1\), \(0\), \(+1\)), so that there are 27 runs in total. Based on the general full factorial design in the design of experiments, it contains 27 experimental runs considering all combinations of three levels of three factors. The order of the experiment was set in the standard order, and three repeated experiments were performed for each run. Experimental data ([PERSON] and [PERSON] [38]) lists the experimental configurations, which include process mean, standard deviation, and variability, with their corresponding design points. A variety of criteria have been used to analyze RD solutions. Among them, the expected quality loss (EQL) is widely used as a critical optimization criterion. The expectation of the loss function can be expressed as  
\[\text{EQL}=\theta\Big{[}\big{(}\hat{\mu}(\mathbf{x})-\tau\big{)}^{2}+\hat{ \sigma}^{2}(\mathbf{x})\Big{]} \tag{13}\]  
where \(\theta\) signifies a positive loss coefficient, \(\theta=1\), and \(\hat{\mu}(\mathbf{x})\), \(\tau\), and \(\hat{\sigma}(\mathbf{x})\) are the estimated mean function, desirable target value, and estimated standard deviation function, respectively. In this example, the target value is \(\tau=500\). As this model does not exhibit the unrealistic constraint of forcing the estimated mean response to a specific target value, it avoids misleading the zero-bias logic. The main objective of minimizing process bias and variability to obtain efficient solutions has allowed a slight bias between the estimated mean and its assigned target. For this reason, the EQL is selected as an identification and comparison tool to evaluate optimal solutions obtained from each model. MATLAB is used in this study to perform the estimated regression functions of mean and standard deviation using the proposed dual-response approach and conventional LSM-based RSM, respectively. The correlation coefficients of the estimated response functions based on [PERSON] and Myers\({}^{\prime}\)[8] dual-response approach are listed in Table 1.  
Table 2 lists the proposed NN-functional-link-based dual-response RD estimation model after the training procedure. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Treatment Combinations**} & \multicolumn{3}{c}{**Coefficients**} \\ \cline{2-3}  & \multicolumn{1}{c}{**Mean \(\overset{\wedge}{\mu}_{\text{LSM}}\)**} & **Standard Deviation \(\overset{\wedge}{\sigma}_{\text{LSM}}\)** \\ \hline Constant & 327.630 & 34.883 \\ \(x_{1}\) & 177.000 & 11.527 \\ \(x_{2}\) & 109.430 & 15.323 \\ \(x_{3}\) & 131.460 & 29.190 \\ \(x_{1}^{2}\) & 32.000 & 4.204 \\ \(x_{2}^{2}\) & \(-\)22.389 & \(-\)1.316 \\ \(x_{3}^{3}\) & \(-\)29.056 & 16.778 \\ \(x_{1}x_{2}\) & 66.028 & 7.720 \\ \(x_{1}x_{3}\) & 75.472 & 5.109 \\ \(x_{2}x_{3}\) & 43.583 & 14.082 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Coefficients of the estimated response functions using LSM. The weights and biases of the NN for the estimated mean and standard deviation functions are listed in Tables 3 and 4, respectively. In these tables, \(\mathbf{W}_{in\_hid,}^{mean}\), \(\left(\mathbf{w}_{hid,out}^{mean}\right)^{T}\), \(\mathbf{b}_{hid}^{mean}\), and \(b_{out}^{mean}\) 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 mean formula, respectively.