## 5 Conclusions  
To address the issues of insufficient feature selection, feature redundancy, and low prediction accuracy of single models in traditional bearing RUL prediction methods, a novel approach is proposed that combines hierarchical clustering for feature selection and  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Model** & **MAE** & **RMSE** \\ \hline GRU & 0.0885 & 0.1099 \\ TCN & 0.0797 & 0.1025 \\ Transformer & 0.0859 & 0.1123 \\ Proposed Method & 0.0674 & 0.0914 \\ \hline \hline \end{tabular}
\end{table}
Table 10: Evaluation indexes of different models.  
Figure 19: Comparison of different model predictions.  
a Transformer-GRU hybrid model for RUL prediction. The method was validated using the PHM 2012 dataset, and the experimental results provide the following insights:  
1. The proposed adaptive feature reduction method, which integrates hierarchical clustering with the elbow method, effectively eliminates redundant information in the feature set while avoiding the subjectivity associated with manually determining the number of clusters. Compared to the unreduced feature set, the reduced set exhibits lower inter-feature similarity and fewer dimensions, thereby reducing model complexity, enhancing training efficiency, and improving prediction accuracy in the later stages of bearing RUL prediction. These findings demonstrate the necessity of redundant feature removal. Furthermore, comparative experiments with other feature selection methods validate the superiority of the proposed approach.
2. The Transformer encoder and GRU-based hybrid model demonstrate stronger time-series modeling capabilities and better capture temporal dependencies compared to single models. This hybrid model can deeply explore the relationship between the extracted features and the RUL of bearings, leading to more accurate predictions of bearing RUL. Comparative analyses with models from related studies further highlight the effectiveness and superiority of the proposed model.
3. Bearings exhibiting abrupt degradation patterns pose greater challenges in RUL prediction due to the limited availability of degradation data, making them more difficult to predict than gradually degrading bearings. Additionally, early-stage RUL prediction is inherently more difficult, as signal features tend to show minimal variation during this period, hindering the model's predictive accuracy.
4. The research in this paper, as well as the referenced studies, were conducted only on ball bearings, and thus, the findings are applicable solely to this type of rolling bearing. The applicability of the proposed methods to other types of rolling bearings requires further investigation and validation.  
W.L., Conceptualization, software, and supervision; X.D., conceptualization, investigation, writing--original draft preparation, and software; F.C., writing--original draft preparation, software; G.H., writing--review and editing, visualization. All authors have read and agreed to the published version of the manuscript.  
This work is partially supported by the National Natural Science Foundation of China (51775515).  
Not applicable.  
The PHM2012 dataset is available at: [[https://github.com/Lucky-Loek/ieee-phm-2012-data-challenge-dataset](https://github.com/Lucky-Loek/ieee-phm-2012-data-challenge-dataset)]([https://github.com/Lucky-Loek/ieee-phm-2012-data-challenge-dataset](https://github.com/Lucky-Loek/ieee-phm-2012-data-challenge-dataset)), accessed on 8 May 2025 the XJTU-SY dataset is available at: [[https://github.com/WangBiaoXJTU/xjtu-sy-bearing-datasets](https://github.com/WangBiaoXJTU/xjtu-sy-bearing-datasets)]([https://github.com/WangBiaoXJTU/xjtu-sy-bearing-datasets](https://github.com/WangBiaoXJTU/xjtu-sy-bearing-datasets)), accessed on 8 May 2025.  
The authors declare no conflicts of interest.  
The following abbreviations are used in this manuscript:  
RUL Remaining useful life GRU Gated Recurrent Unit MAE Mean absolute error