## 4 Results and Discussion  
### Proposed MDRN Results  
Firstly, take Case 1 as an example to introduce the modeling process in detail. As mentioned in the Methodologies section, the instinct dimension of the multi-parameter in the manufacturing system (Table 1) is firstly evaluated by the MLE, and the result is 3, i.e., \(m=3\). Then, the experiments of the model development are implemented based on Table 2, and the performance (MAPE) of the proposed MDRN model is shown in Figure 4.  
For the Isomap algorithm, some statistical data are summarized, namely: \(K=8\), the range of the MAPE is between 2.444% and 2.639% (mean value 2.560%); \(K=10\), the range of the MAPE is between 2.345% and 2.686% (mean value 2.513%); \(K=12\), the range of the MAPE is between 1.943% and 2.624% (mean value 2.365%); and \(K=14\), the range of the MAPE is between 2.183% and 2.671% (mean value 2.429%). These results indicate that the prediction performance improves as \(K\) increases, but they also prove that a large value of \(K\) may lead to performance degradation. Therefore, \(K=12\) is an appropriate selection in this paper. For the DRN model shown in Figure 4, the prediction performance of the deep structure is better (slight or significant) than that of the \"shallow\" framework (one hidden layer), except for a few special samples. In addition, the deep structures with different \(K\) values exhibit the diversity in deep representations, that is, when \(K=8\), the best deep structure is \(50\times 50\) with 2.444% (hidden layer l = 2, 50 nodes in each hidden layer); when \(K=10\), the best deep structure is \(40\times 40\) with 2.345% (hidden layer l = 2, 40 nodes in each hidden layer); when \(K=12\), the best deep structure is \(30\times 30\) with 1.943% (hidden layer l = 2, 30 nodes in each hidden layer); and when \(K=14\), the best deep structure is \(10\times 10\) with 2.183% (hidden layer l = 2, 30 nodes in each hidden layer). Therefore, the deep structure \(30\times 30\) \begin{table}
\begin{tabular}{c c} \hline \hline
**Model** & **Experimental Design** \\ \hline \(K\)-Isomap for dimension reduction & \(K=[8,10,12,14]\) \\  & Structure parameters: \(l=[1,2,3]\), hidden nodes in \\ DRN for manufacturing quality prediction & each hidden layer = [10, 15, 20, 25, 30, 40, 50]. \\  & Computing parameters: dropout 0.5, learning rate 1, \\  & batch-size 10, and iteration 200. \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Experimental designing of each approach. with \(K=12\) was chosen as the optimal pattern for Case 1 prediction (marked as 3-30-30-1), and the results and residual analysis are plotted in Figure 5.  
As Figure 5a shows, the prediction trends are, to some extent, in accordance with the measurement trends, but there are also some significant differences between the two values. As shown in Figure 5b, the residual errors cause some fluctuation in the testing processes within the range of [\(-0.1\), \(0.1\)]. There are only five prediction outliers in a triangle, because the interval around the residual errors does not contain zero. This implies that the five residual errors that are caused by the unfortunate fitting, and have a confidence interval of over 95%, accounted for 5% of the testing data. Furthermore, the quantitative evaluation results, MAPE = 1.943% and RMSE = 0.022, also exhibit a high accuracy. Hence, the thorough processes of the testing are successful, and the results are acceptable. Figure 4: Testing results of different MDRN models with different \(K\): (**a**) \(K=8\), (**b**) \(K=10\), (**c**) \(K=12\), and (**d**) \(K=14\). Figure 5: Prediction results of the MDRN model with 3-30-30-1 of Case 1: (**a**) plots the comparison with the measured and the predicted values, and (**b**) shows the residual error distribution.