## 4 Results and Comparison
### Comparison with Previous Studies  
There are multiple challenges in analyzing the PRONOSTIA datasets, such as limited historical run-to-failure examples, no information about failure modes, and a wide range of failure times. Since the publication of these datasets, the RUL problem has attracted strong interest from many researchers [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42].  
Due to the difficulty to solve this challenging problem, most of the researchers did not present the comprehensive solutions with RUL prediction results, while they focused on extracting bearing prognostic features and constructing HI using different feature extraction and fusion methods. For example, [PERSON] et al. extracted prognostic features (i.e., SD-IHC and SD-HIS) from the wavelet packet decomposition (WPD). A cumulative function then transformed time series of the prognostic features into descriptors depicting bearing degradation status [26]. In the method proposed by [PERSON] et al., multiple features were fused through feature selection using the pairwise symmetrical uncertainty  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Test Instance (Bearing)** & **1-3** & **1-4** & **1-5** & **1-6** & **1-7** \\ \hline Learning Instance (Bearing) & 1-1 & 1-2 & 1-2 & 1-2 & 1-2 \\ Comparing PC Feature No. & 1 st & 1 st & 2 rd & 2 rd & 2 rd \\ Similarity (\(d_{min}\)) & 0.1137 & - & 0.3750 & 0.1963 & 0.2504 \\ Last Time (time cycles) & 1800 & 1137 & 2303 & 2300 & 1500 \\ \(RUL_{a}\) (time cycles) & 573 & 34 & 161 & 146 & 757 \\ \(RUL_{e}\) (time cycles) & 642 & 23 & 160 & 45 & 822 \\ Relative Error & \(-12.1\%\) & 32.4\% & 0.92\% & 69.2\% & \(-8.61\%\) \\ Score & 0.1877 & 0.3229 & 0.9688 & 0.0909 & 0.3032 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Summary of RUL estimation results for Bearing1-3, 1-4, 1-5, 1-6, and 1-7.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(M\) & **1** & **3** & **5** & **7** & **9** \\ \hline Bearing1-3 & 0.2231 & 0.2243 & 0.2127 & 0.1877 & 0.1581 \\ Bearing1-5 & 0.6502 & 0.6082 & 0.7072 & 0.9688 & 0.1742 \\ Bearing1-6 & 0.0909 & 0.0909 & 0.0909 & 0.0909 & 0.0909 \\ Bearing1-7 & 0.3213 & 0.3011 & 0.3045 & 0.3032 & 0.311 \\ Overall (Average) & 0.3214 & 0.3061 & 0.3288 & 0.3877 & 0.1836 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Scores of RUL estimation scores with \(M=1\), 3, 5, 7, and 9.  
measure and PCA. The empirical mode decomposition (EMD) algorithm was then used to identify a monotonic trend of degradation [27]. [PERSON] et al. employed the WPD technique to extract a set of prognostic features. The prognostic features showing the best degradation trends were iteratively accumulated to construct one fused prognostic feature [28]. [PERSON] et al. trained an adaptive neuro-fuzzy inference system (ANFIS) model for only one-step-ahead prediction of degradation state [29]. In the [PERSON] et al.'s study, the most sensitive prognostic feature was selected using the Chi-Square test to build the General Log-Linear Weibull (GLL-Weibull) models [30]. [PERSON] et al. proposed a hierarchical method by integrating deep neural networks (DNN) and artificial neural networks (ANN) [31]. [PERSON] et al. proposed a CNN-based deep learning approach. Multiple spectrograms were used as inputs to the CNN to map to their corresponding RUL prediction targets [32]. In the two studies [31,32], they mixed all learning and test bearing examples and trained/tested prediction model with one-holdout validation strategy. Apparently, their experiments did not aim to deal with the challenging problem of shortage of learning examples.