## 3 Experiment and Analysis
### Comparing Related Works  
In this study, we measured the RUL prediction accuracy of testing bearings to evaluate the proposed method and compare with related studies. As a performance evaluation measurement of the RUL prediction, a score metric, mean, and standard deviation (SD) of the percent error were applied. These metrics were used for the IEEE PHM 2012 Prognostic Challenge [31]. The percent error on the \(i\)th bearing is defined as follows:  
\[Er_{i}=100\times\frac{ActRUL_{i}-R\hat{U}L_{i}}{ActRUL_{i}}, \tag{9}\]  
Figure 12: Clustering results of the testing datasets. Figure 13: Comparison of the actual RUL and the predicted RUL: (**a**) Bearing 1_3; (**b**) Bearing 2_7. where \(ActRUL_{i}\) and \(R\hat{UL}_{i}\) are, respectively, the actual RUL and the predicted RUL of the \(i\)th testing bearing. Underestimates and overestimates of the RUL were considered in different manners by the following scoring function. \[A_{i}=\left\{ \begin{array}{ll}exp(-\ln(0.5)\cdot(Er_{i}/5))&if\ Er_{i}\leq 0 \\ exp(+\ln(0.5)\cdot(Er_{i}/20))&if\ Er_{i}>0\end{array} \right. \tag{10}\]  
The score \(A_{i}\) is 1 when the percent error is 0. As the percent error increases, the score decreases. If the failure is predicted later than the actual occurrence (i.e., cases in which \(Er_{i}\leq 0\)), the score is cut more than the earlier predictions. The average score of the RUL prediction for all testing bearings is defined as follows. \[Score=\frac{1}{11}\sum_{i=1}^{11}A_{i} \tag{11}\]  
To verify the superiority of the proposed method, we compared the prediction result with the other four similar studies that predicted bearing RULs using the same dataset. Table 2 represents the RUL prediction results of our related studies and proposed method. For each testing bearing, the actual RUL and predicted RUL of the proposed method are displayed in columns 3 and 4, respectively. The percent errors of the proposed method are shown in the final column, and the four comparative studies are shown in columns 5 through 8. The mean and SD of the percent errors and the score metric are displayed in the last three rows. The first similar study was published by [PERSON] et al. [PERSON] et al. (2012), the winner of the IEEE PHM 2012 Prognostic Challenge. This study predicts the RUL using frequency analysis-based anomaly detection, degradation feature extrapolation, and survival time ratios. However, the method has the disadvantage of defining the anomaly detection time point based on subjective criteria used to calculate the bearing survival time ratios. The means as well as the deviations of the percent errors are large in the table. The second study was published by [PERSON] et al. [PERSON] et al. (2016) using the same bearing dataset. They used wavelet packet-EMD for feature extraction and SOM for constructing the HI (i.e., CV). The method shows improved errors over previous studies, but requires the extraction of approximately one hundred features to estimate bearing performance. The third study, published by [PERSON] et al. [PERSON] et al. (2016), proposed a new HI (i.e., WMQE) to predict the RUL of bearings. WMQE was constructed by fusing a select few weighted features based on correlation clustering among the 28 features extracted from the bearings. The study showed the best performance among existing studies. The most recent similar study is an RNN-based RUL prediction method published by [PERSON] et al. [PERSON] et al. (2016). The proposed method was also constructed by selecting and fusing multiple features extracted from the time, frequency, and time-frequency domains. To improve the accuracy of RUL prediction, a deep learning approach was adopted. This method demonstrated its superiority over SOM-based HIs. However, it obtained higher percent errors and lower scores than [PERSON] et al. [PERSON] et al. (2016). The proposed method shows overall low percent errors and low deviation, indicating that the model worked accurately and reliably on every tested bearing.