## 4 Results and Discussion
### Comparison Results  
To evaluate the prediction performance of the proposed model, three models are applied to the comparisons using the same dataset, i.e., the AE-based DRN, the BPNN, and least squares support vector regression (LSSVR). The BPNN with one hidden layer and the LSSVR with the kernel function (none hidden layer) are typical shallow learning frameworks in the peer work. All of the models have the same input-output structures (19 inputs and 1 output), and the hidden layer and hidden node (except for the LSSVR) are specified by the experimental method. For the AE-based DRN, the experimental design is the same as the \"DRN for manufacturing quality prediction\" shown in Table 2. The optimal model structure is set as the hidden layer \(l=2\), with 25 nodes in each hidden layer for Case 1, and it is set as the hidden layer \(l=2\), with 25 nodes in each hidden layer for Case 2. For the BPNN, an empirical formula is applied to determine the hidden node, i.e., \(\textit{hidden nodes}=\sqrt{\textit{inputnodes}+\textit{outputnodes}}+b(b \in[0,10])\). Therefore, the optimal hidden node for Case 1 and Case 2 is calculated as 10 and 12, respectively, in terms of the MAPE. In addition, other main computing parameters are set to a learning rate of 0.05 and a goal of 0.0001 for 200 iterations. For the LSSVR, there are two parameters, i.e., kernel width and penalty coefficient [9], that are determined using the ten-fold cross-validation method in this paper. Hence, the optimal parameters are set as (25.016 and 9.565) for Case 1 and (20.525 and 10.410) for Case 2. The prediction results for the two cases using the three models are displayed in Figures 7 and 8, seperately. Figure 7 reveals the following: (1) There are some unsatisfactory prediction results in the three models for Case 1, which illustrate that the performance is not related to the multi-parameter inputs; rather, it is related to the input feature. Compared with Figure 5, the three comparison models, which consider 19 parameters, exhibit a lower prediction performance. Specifically, there are six (accounting for 6% in the test data) outliers in the AE-based DRN, and seven (accounting for 7%) outliers in both the LSSVR and the BPNN. (2) Deep architecture is superior in performance to feature learning. That is, the performance of the proposed model and the AE-based DRN is better than that of the LSSVR and the BPNN in terms of the number of outliers. Cumulatively, the MDRN model has the best performance for Case 1 among these comparison models, attributed to multi-parameter reduction and deep feature representation, both of which reduce the invalid information interference and improve the prediction capacity. As Figure 8 illustrates, the fluctuations in the three models also exhibit diversity, and are caught in the mutations. The residual analysis shows evidence of 10 outliers (accounting for 5%) in the AE-based DRN and the LSSVR model, and 12 outliers (accounting for 6%) in the BPNN. Compared with Figure 6, the performance of the MDRN model is in a second place with 11 outliers. This result of Case 2 differs with that of Case 1. Therefore, to evaluate the performance of these models, quantitative evaluations are also investigated, and the results are summarized in Table 3.  
Figure 8: _Cont_. Figure 7: Prediction results of Case 1 using different models. (**a**) AE-based DRN with 19-25-25-1, (**b**) least squares support vector regression (LSSVR), and (**c**) BPNN with 19-10-1. As shown in Table 4, the statistical indexes of the two case applications demonstrate the following: (1) In terms of the highest MAPE and RMSE, the shallow model (BPNN with one hidden layer and LSSVR without a hidden layer) has difficulty sufficiently capturing the features and quality of the manufacturing parameters. However, the deep network (MDRN and AE-based DRN) demonstrates an increase capacity for feature learning and regression.