## 3 Experiment and Analysis
### RUL Prediction Results  
Once the CWTCNN-HI was obtained, a GPR algorithm was used to predict the RUL of the test bearings. The GPR is a kernel-based machine learning technique that can conveniently specify high dimensional and flexible nonlinear regression [4]. In this study, we assumed that the mean and covariance functions were zero and a linear function, respectively. Hyper-parameters for the covariance function are optimized by maximizing the log-likelihood function. The GPR model predicts future CWTCNN-HI by estimated CWTCNN-HI up to current time. The estimated and predicted CWTCNN-HIs of bearing1_3, one of the testing datasets, are shown in Figure 9. The obtained CWTCNN-HI up to the current time is indicated by a dot. The solid lines represent the estimated and predicted CWTCNN-HIs before and after the current time obtained by using the GPR model, respectively. The dotted lines are the 95 % confidence interval. The figure shows that the health condition of the bearing gradually degrades over time with the increase of HI. The RUL is calculated as the difference between the time at which the predicted HI reaches the threshold and the current time. The RUL prediction results can be affected by the number of samples in the training dataset. The large number of samples in the training data improves the accuracy of the estimated  
Figure 8: CWTCNN-HI of the six training datasets. The RUL prediction results of the other ten testing datasets are shown in Figure 10. Most testing bearings clearly show different degradation patterns, owing to different operating conditions and failure types. For example, bearing1_5 and bearing1_6 rapidly degraded in their early stage and gradually degraded after a certain period. Alternatively, bearing1_7 shows a constantly degrading pattern, like bearing1_3. In this study, the clustering-based failure threshold was adopted to accurately predict the RUL of testing bearings. The failure threshold can be provided by an application expert or obtained experimentally. In addition, the different thresholds are applied depending on the operating conditions and characteristics of the bearings [4; 37]. The bearings with similar operating condition and failure mode exhibit a similar shape of degradation pattern. In this regard, we used the K-means clustering algorithm to cluster testing bearings with similar degradation trends and applied the same threshold for each cluster. The thresholds were chosen empirically to increase the accuracy of the RUL prediction. To determine the number of clusters in the K-means clustering algorithm, the Dunn index [38] was used to evaluate the clustering performance. The Dunn index is calculated as the ratio of the minimum distance between two objects in different clusters to the maximum distance between two objects in the same cluster as follows:  
\[D=\frac{min_{1\leq i<j\leq n}d(i,\,j)}{max_{1\leq k\leq n}d^{\prime}(k)}, \tag{8}\]  
where \(n\) is the number of clusters, \(d(i,j)\) is the distance between the \(i\)th cluster and \(j\)th cluster, and \(d^{\prime}(k)\) is the intra-cluster distance. A high index value means that the objects are well clustered. Based on the Dunn indexes for different number of clusters, as shown in Figure 11, the testing bearings were grouped into five different clusters. Figure 12 represents the result of clustering. Clusters 3 and 4 include four testing bearings. The remaining three bearings with different degradation trends constitute three clusters. For example, the bearing in cluster 1 shows the pattern of slow and steady degradation. On the other hand, the bearings of cluster 3 show a pattern of rapid initial degradation. The degradation patterns of the bearings vary depending on operational conditions and failure types. Therefore, the clustering-based failure threshold improves RUL prediction performance by applying different threshold to bearings. For each cluster, the failure threshold is set to 0.16, 1.23, 0.85, 0.83, and 0.71, respectively.