## 4 Experimental Analysis
### RUL Prediction  
After extracting the HI of the testing bearing using MSCAE, it can be found that the extracted HI has obvious degradation characteristics with time, which indicates that the HI is a time series. Therefore, LSTM is used to perform the RUL prediction of bearings. For Bearing1_1, assuming that the last 100 HI are unknown and the first 2703 HI are known, the known 2703 HI points are used as training data for the LSTM to predict the unknown HI points, and when the predicted value of the LSTM is less than the threshold 0, it means that the predicted current moment bearing has been damaged, and the time interval between the current moment and the starting moment of the prediction is the predicted RUL value. Similarly, for Bearing1_3, assuming that the last 100 HI points are unknown, the training data for the LSTM are the first 2275 HI points. The parameters of the LSTM are configured as shown in Table 3. The number of neurons in the input, hidden, and output layers  
Figure 11: Four models to extract HI of Bearing2_6. (**a**) MSCAE. (**b**) CAE (kernel_size = 3). (**c**) CAE (kernel_size = 7). (**d**) CAE (kernel_size = 11). Figure 10: The HI of four testing bearings. of the LSTM are 360.29 and 1, respectively. The internal parameters of the LSTM are optimized using the Adam optimizer, and the learning rate is set to 0.07. To demonstrate the superiority of the HI extraction method proposed in this paper on RUL prediction, two state-of-the-art deep learning-based HI construction methods are used for comparison. One is the recurrent convolutional neural network (RCNN), which was proposed in the literature [33], and the other is the CNN proposed in the literature [25]. The RCNN is an end-to-end HI extraction framework consisting of a convolutional neural network with residual structure and an LSTM serially connected with the specific structural parameters described in the literature [33]. The CNN-based HI construction method proposed in the literature [25] is divided into two steps, first using two convolutional layers for feature extraction and later using a fully connected layer for HI construction. It is worth noting that both methods, as in this paper, use the original vibration signal as the input to the model without a manual feature extraction step. All three methods are trained using the same training data, and the RUL prediction ability of the extracted HI is verified on Bearing1_1 and Bearing1_3. For convenience, the HI extracted by the proposed MSCAE, RCNN and CNN are MSCAE-HI, CRNN-HI and CNN-HI. The RUL prediction results of HI extracted by the three models are shown in Figures 12-14. From Figure 13, it can be found that RUL prediction using CRNN-HI requires artificially set thresholds, and the choice of thresholds directly affects the RUL prediction results. If the set threshold is large, it will cause the prediction to stop early, and conversely, it will cause the prediction value of RUL to be larger than the real one, or even the phenomenon that it cannot converge to the set threshold. In this paper, the failure threshold of CRNN-HI is set to 0.2. In Figure 13a, the HI of Bearing1_1 extracted using the CRNN method shows an increasing trend at the end of degradation, which deviates from the actual degradation threshold and can affect the prediction ability of the LSTM. In Figure 13b, there are a few outliers in the HI of Bearing1_3, and there are HI points less than the threshold value when the bearing first starts running. It can be found in Figure 14 that there is a significant divergence of CNN-HI at the late degradation stage of the bearing, resulting in the predicted HI value being more likely to be close to the failure threshold when using CNN-HI for RUL prediction, resulting in the predicted RUL often being smaller than the true RUL value. In Figure 12, the MSCAE-HI constructed in this paper has less fluctuation and a smoother degradation trend, which can effectively reflect the real degradation trend of the bearing.