## 1 Introduction
In terms of prediction models, commonly used deep learning architectures include Convolutional Neural Network (CNN), Recurrent Neural Network (RNN), and their various improvements and variants. [PERSON] et al. [18] proposed a dual-CNN architecture for bearing RUL prediction, and their experimental results demonstrated good prediction accuracy and robustness. While CNNs can be effective for RUL estimation, their inherent structural limitations hinder their ability to capture long-term dependencies in sequential data. To address this issue, researchers have adopted RNNs with recurrent structures for RUL prediction. Among these, Long Short-Term Memory (LSTM) networks and Gated Recurrent Unit (GRU) are the most widely used RNN variants. [PERSON] et al. [19] enhanced feature signals using maximum correlation kurtosis deconvolution and applied multiscale permutation entropy as a degradation indicator, which was then fed into an LSTM model optimized by the sparrow search algorithm for RUL prediction. [PERSON] et al. [20] improved prediction accuracy by integrating multi-sensor information with a GRU network. Additionally, based on CNNs, researchers have proposed Temporal Convolutional Networks(TCNs), which incorporate residual connections and dilated causal convolutions to effectively capture long-range dependencies. [PERSON] et al. [21] extracted and selected multi-domain features, which were highly correlated with the degradation process, segmented the bearing lifecycle, and utilized TCN for prediction, thereby overcoming the limitations of traditional models in handling time series. Moreover, Transformer networks have been widely adopted across various domains due to their exceptional modeling efficiency and strong performance on long-sequence data. In the context of bearing RUL prediction, Transformer-based models have also shown promising results. [PERSON] et al. [22] used cumulative-transformed traditional features as inputs and employed a Transformer network to predict the RUL of rolling bearings. Although the aforementioned studies have achieved promising results in the RUL prediction of rolling bearings, several challenges remain: (1) insufficient feature selection during feature set construction often leads to redundancy; and (2) the limited predictive capability of single models constrains overall prediction performance. To address these issues and improve the accuracy of bearing RUL prediction by eliminating redundant features, this paper proposes a novel method that combines hierarchical-clustering-based feature selection with a hybrid Transformer-GRU model. The main contributions of this paper are as follows:  
1. An adaptive feature selection method combining hierarchical clustering and the elbow method is proposed, which can effectively eliminate redundant information in the feature set, thereby reducing data volume and model complexity. Compared with other studies, this approach does not require manual specification of clustering parameters. 2. A hybrid Transformer-GRU model is developed for predicting the RUL of rolling bearings, leveraging the Transformer encoder's ability to capture relationships across different time steps and GRU's strength in modeling temporal dependencies (trends), thus improving prediction accuracy. 3. The proposed approach provides a novel and effective method for RUL prediction of rolling bearings. The structure of this paper is organized as follows: Section 2 presents the relevant theories, and Section 3 introduces the proposed methods and models, as well as the entire prediction process. Section 4 provides experimental verification of the proposed methods, and Section 5 concludes the paper.