## 5 Conclusions  
The original bearing vibration signals were denoised, and multiple features were extracted; sensitive feature subsets were selected as inputs for bearing RUL prediction, and an Attention-based Double Bidirectional Long Short-Term Memory network was proposed for RUL prediction, followed by experimental validation and results analysis. The following conclusions have been drawn:  
1. After wavelet threshold denoising, the RMSE (Root Mean Square Error), RMS (Root Mean Square), and MAE (Mean Absolute Error) values of the signal are 0.0153, 0.123, and 0.00008, respectively. These values are all smaller than PSO-VMD method, which proves that this method has a strong denoising effect. The signal free from noise can provide a data foundation for bearing feature extraction and remaining life prediction. Different features exhibit varying degrees of sensitivity to bearing degradation, and not all features are rich in degradation information. By comprehensively evaluating the sensitivity of each bearing signal feature based on monotonicity, trend, predictability, and robustness, the optimal features that are more sensitive to bearing degradation can be screened out, and redundant signal features can be eliminated. 2. The failure threshold is an important parameter in bearing performance evaluation, marking the starting point where a bearing transitions from a normal state to a faulty state, and influencing the prediction of the bearing's RUL. The failure times detected using the 3\(\sigma\) threshold method for the PHM dataset are 1330 min, 673 min, and 323 min, and this method detects the failure time of the bearing earlier than the comparative methods, and it can detect bearing failure earlier than the comparative method. Early detection of bearing failure time provides staff with sufficient time to formulate safety measures, thereby avoiding missed detections. 3. When establishing a network model that combines a Double-way Bidirectional Long Short-Term Memory network with an attention mechanism for predicting the RUL of bearings, when tested with the datasets Bearing1-3, Bearing2-6, and Bearing3-3 from PHM2012, the \(p\)-values between the prediction results of the proposed RUL method and those of the BiGRU prediction method are 0.0188, 0.0012, and 0.0234, respectively; the \(p\)-values between the prediction results of the proposed RUL method \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Model** & **Bearings** & **RMSE** & **MAE** & **R\({}^{2}\)** \\ \hline \multirow{3}{*}{Prosed method} & 1\_3 & 0.0682 & 0.0741 & 0.8743 \\  & 2\_5 & 0.0811 & 0.0591 & 0.9211 \\  & 3\_5 & 0.0866 & 0.0533 & 0.9100 \\ \hline \multirow{3}{*}{BiGRU} & 1\_3 & 0.0881 & 0.0567 & 0.8298 \\  & 2\_5 & 0.0917 & 0.0763 & 0.9090 \\  & 3\_5 & 0.0899 & 0.0422 & 0.8570 \\ \hline \multirow{3}{*}{CNN-LSTM} & 1\_3 & 0.1255 & 0.0907 & 0.8538 \\  & 2\_5 & 0.1666 & 0.0914 & 0.8863 \\ \cline{1-1}  & 3\_5 & 0.2166 & 0.4836 & 0.8782 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Evaluation metrics of different methods. and those of the CNN-LSTM prediction method are also 0.0188, 0.0012, and 0.0234, respectively, with \(p<0.05\). When tested with the datasets Bearing1-3, Bearing2-5, and Bearing3-5 from XJTU-SY, the \(p\)-values between the prediction results of the proposed RUL method and those of the BiGRU prediction method are 0.0196, 0.0012, and 0.0241, respectively; the \(p\)-values between the prediction results of the proposed RUL method and those of the CNN-LSTM prediction method are 0.0001, 0.0083, and 0.0327, respectively, with \(p<0.05\). The prediction results from both datasets can demonstrate significant differences between the proposed method and the comparative methods. Therefore, the RUL prediction performance of the proposed method is higher. Moreover, the prediction curves for both datasets fall within the 95% confidence interval, which demonstrates the reliability of the prediction results obtained using the proposed model.