## 4 Experimental Validation and Result Analysis
### RUL Prediction Process and Result Analysis
#### 4.2.3 Validation of the Generalization Ability of the Prediction Model  
To further validate the generalization ability and adaptability of the proposed prediction mode, the XJTU-SY dataset from the literature [54] is employed for validation, with a sampling frequency of 25.6 kHz, a sampling interval of 1 min, and each sampling session lasting for 1.28 s. In this paper, the description of the dataset used for training and testing is as follows: Condition 1: radial force of 12,000 N at 2100 rpm, it includes the training set comprising Bearing1-1 to Bearing1-2, and the testing set is Bearing1-3; Condition 2: radial force of 11,000 N at 2250 rpm, it includes the training set comprising Bearing2-1 to Bearing2-2, and the testing set is Bearing2-5; Condition 3: radial force of 10,000 N at 2400 rpm, it includes the training set comprising Bearing3-1 to Bearing3-2, and the testing set is Bearing3-5. Figure 14 illustrates the RUL prediction results for Bearing1-3, Bearing2-5, and Bearing3-5, the unit is minute, the failure points of the three datasets are 58 min, 120 min, and 6 min, respectively. As shown in Table 9, which presents the \(p\)-values when comparing the proposed RUL prediction method with the BiGRU prediction method and the CNN-LSTM prediction method, it also can be observed that in the datasets Bearing1-3, Bearing2-5, and Bearing3-5, the \(p\)-values between the proposed method and the BiGRU method are 0.0196, 0.0012, and 0.0241, respectively, and the \(p\)-values between the proposed method and CNN-LSTM are 0.0001, 0.0083, and 0.0327, respectively. All these values are less than 0.05; this result further demonstrates that when the proposed prediction model is applied to the XJTU-SY dataset, its prediction performance is equally satisfactory. While Table 10 shows the values of evaluation metrics including RMSE, MAE, and R\({}^{2}\). By combining the information from Figure 14 and Table 10, through comparative observation, it is found that when the proposed method is applied to the XJTU-SY dataset, both RMSE and MAE are smaller than those of the comparative methods, the prediction curve of the proposed method lies within the 95% confidence interval, and the R\({}^{2}\) is larger than that of the comparative methods. It can be observed that under any operating condition, the RUL curve of the proposed model is closer to the actual RUL curve. Although the values of the evaluation metrics do not differ significantly, the RUL curve of the comparative model \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Model** & **Bearings** & **RMSE** & **MAE** & **R\({}^{2}\)** \\ \hline \multirow{3}{*}{Prosed method} & 1\_3 & 0.1033 & 0.0729 & 0.8920 \\  & 2\_6 & 0.1316 & 0.0975 & 0.9725 \\  & 3\_3 & 0.0406 & 0.0296 & 0.9802 \\ \hline \multirow{3}{*}{BiGRU} & 1\_3 & 0.0503 & 0.4354 & 0.8298 \\  & 2\_6 & 0.1368 & 0.1803 & 0.7753 \\  & 3\_3 & 0.0532 & 0.0422 & 0.9660 \\ \hline \multirow{3}{*}{CNN-LSTM} & 1\_3 & 0.0503 & 0.1733 & 0.8821 \\  & 2\_6 & 0.2216 & 0.1004 & 0.4372 \\ \cline{1-1}  & 3\_3 & 0.0554 & 0.0429 & 0.9632 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Evaluation metrics of different methods. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Bearings** & **The Proposed Method vs. BIGRU** & **The Proposed Method vs. CNN-LSTM.** \\ _p-Value_ & _p-Value_ \\ \hline
1-3 & \(p_{1:3}=0.0196\) & \(p_{1:3}=0.0001\) \\
2-6 & \(p_{2:5}=0.0012\) & \(p_{2:5}=0.0083\) \\
3-3 & \(p_{3:5}=0.0241\) & \(p_{3:5}=0.0327\) \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Significance level of the RUL prediction results. exhibits substantial fluctuations, resulting in inferior prediction performance compared to the proposed model. Figure 14: The results of the RUL. Based on the aforementioned analysis, it can be concluded that the proposed model not only effectively predicts the RUL of bearings under different operating conditions but also performs well on various datasets, this result further proves the generalization and applicability of this model.