## 4 Experimental Validation and Result Analysis
### RUL Prediction Process and Result Analysis
#### 4.2.2 Prediction Result Analysis  
Figure 13 shows the RUL prediction graphs for Bearing1-3, Bearing 2-6, the unit is minute, and Bearing 3-3; they belong to operating conditions 1, 2, and 3, respectively. It can be observed that the monotonically decreasing trend represents the bearing degradation time, the RUL predicted by the proposed method (A-DBiLSTM) is very close to the actual one; moreover, the RUL curve of the proposed method falls within the 95% confidence interval, and under different operating conditions, the predicted curves of the bearing's RUL are all very close to the actual RUL curve.  
It can also be seen from the figure that whether the BiGRU method or the CNN-LSTM method is used, there is a relatively large deviation between the predicted results and the actual results. In other words, their performance is lower than that of the proposed method. This indicates that the proposed method can provide sufficient preparation time for maintenance decisions, and reduces the economic losses caused by failure downtime.  
As shown in Table 7, it presents the \(p\)-values when comparing the proposed RUL prediction method with the BiGRU prediction method and the CNN-LSTM prediction method. Based on Formulas (48) to (53), the \(p\)-values were calculated; upon analyzing the data in the table, it can be observed that in the datasets Bearing1-3, Bearing2-6, and Bearing3-3, the \(p\)-values between the proposed method and the BiGRU method are 0.0188, 0.0012, and 0.0234, respectively, and the \(p\)-values between the proposed method and CNN-LSTM are 0.0001, 0.0082, and 0.0329, respectively; all these values are less than the significance level \(\gamma\) = 0.05, and \(p\) < 0.05; it indicates that there is a significant difference between the proposed model and the comparison model, leading to the rejection of the null hypothesis, and this indicates that the proposed method outperforms the comparison methods in terms of predictive performance, and the prediction results are reliable on all three datasets. This indicates that under different operating conditions, the proposed network model can reliably predict the RUL of bearings, demonstrating the applicability and generalizability of the proposed method.  
As shown in Table 8, and the values of performance evaluation metrics MSE, MAE, and RMSE were obtained. By comparing the data in the table, it can be seen that the RMSE and MAE values of the proposed method are both smaller than those of the other two comparative methods; smaller MAE and RMSE values indicate better prediction performance of the RUL model. Moreover, the R\({}^{2}\) values of the proposed method are all greater than those of the other two comparative methods, and the closer the R\({}^{2}\) value is to 1, the higher the fitting degree of the network, indicating that the fitting degree of the proposed network model is superior to the other two. Combining this with Figure 13, it can be observed that compared with other methods, the RUL curve of the proposed prediction model has less overall fluctuation, which further proves that the proposed model is more in line with the actual situation.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Bearings**} & **The Proposed Method vs. BIGRU** & **The Proposed Method vs. CNN-LSTM.** \\  & \(p\)**-Value** & \(p\)**-Value** \\ \hline
1-3 & \(p_{1\text{-}3}\) = 0.0188 & \(p_{1\text{-}3}\) = 0.0001 \\
2-6 & \(p_{2\text{-}6}\) = 0.0012 & \(p_{2\text{-}6}\) = 0.0082 \\
3-3 & \(p_{1\text{-}3}\) = 0.0234 & \(p_{3\text{-}3}\) = 0.0329 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Significance level of the RUL prediction results.  
Figure 13: The results of the RUL.