## 4 Results and Discussion
### Proposed MDRN Results
Following the modeling process of Case 1, the testing results of Case 2 are plotted in Figure 6. Note that the optimal MDRN model for Case 2 is constructed by \(m=3\), \(K=14\), and has a deep structure of \(25\times 25\) (hidden layer \(l=2\), with 25 nodes in each hidden layer), i.e., 3-25-25-1. The results shown in Figure 6 can be summarized as follows: (1) The instinct dimension, \(m\), is related to the data feature, not the length of data. The calculation result, that is, \(m=3\), is the same, because it uses the same type of manufacturing parameters (\(\mathbf{x_{1}}\)-\(\mathbf{x_{19}}\)). (2) Because of its deep feature learning capacity, the DRN with two hidden layers achieves better prediction results (MAPE = 1.861% and RMSE = 0.023) than that of the model with a single hidden layer. (3) There are eleven outliers (accounting for 5.5% of the testing data) in the residual error analysis, which illustrate that the MDRN approach has a stable ability for feature learning and regression, because of the 5% of outliers in the Case 1 prediction. Table 3 summarizes the quantitative results in the training and the testing process for the robust analysis. From Table 3, one can find that all performances in both the training and testing processes are similar, indicating that the proposed model has a generalization ability and robustness. It is attributed to the \"dropout\", which weakens the joint adaptability between neurons (that is, the parameters of the whole network neurons are only partially updated because of the different neurons eliminated each time). According to the quantitative and qualitative analysis above, the proposed MDRN model, which combines the Isomap for multi-parameter reduction and the DRN for the relationship between the low-dimension information and manufacturing quality learning, predict the manufacturing quality within an acceptable margin of error. In addition, the performance of the model with three hidden layers is not greater than that of the model with two hidden layers. This phenomenon is different from the DL theory, in which the deeper the hidden layer is, the better the performance is. That is,  
Figure 6: Prediction results of the MDRN model with 3-30-30-1 of Case 2: (**a**) plots the comparison with the measured and the predicted values, and (**b**) shows the residual error distribution. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multicolumn{5}{c}{**Performance (Training)**} \\ \cline{2-6}  & **MAPE (\%)** & **RMSE** & **TS\({}_{1}\) (\%)** & **TS\({}_{5}\) (\%)** & **TS\({}_{10}\) (\%)** \\ \hline
1 & 1.575 & 0.020 & 32.6 & 100 & 100 \\
2 & 1.414 & 0.020 & 44.4 & 100 & 100 \\ \hline \hline \multirow{2}{*}{**Case**} & \multicolumn{5}{c}{**Performance (Testing)**} \\ \cline{2-6}  & MAPE (\%) & RMSE & TS\({}_{1}\) (\%) & TS\({}_{5}\) (\%) & TS\({}_{10}\) (\%) \\ \hline
1 & 1.943 & 0.022 & 27 & 95 & 100 \\
2 & 1.861 & 0.023 & 35.5 & 95.5 & 99.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Quantitative results in training and testing process using the proposed model. the deep architecture is closely related to the internal features of the real data. Furthermore, the proposed model has a generalization ability and robustness. Therefore, suitable deep architecture and the lack of blind pursuit for more hidden layers can avoid computational burdens and enhance the prediction performance.