## 4 Experiment and Analysis
### Results and Analysis
#### 4.3.2 Comparison of KDE Prediction Results  
To verify that the method proposed in this paper can more effectively predict the random uncertainty characteristics of the RUL, based on the above contents, the probability density prediction results of bearing A_4 are obtained by TCNQR. Taking three operation moments (75 min, 90 min and 105 min) of bearing A_4 as an example, the probability density curve of bearing RUL is obtained. The real RUL values corresponding to the above there operation moments are 47 min, 32 min and 17 min. The prediction results are shown in the Figure 14. Figure 14d-f are the KDE prediction results of 75 min, 90 min and 105 min, respectively. As can be seen from the above figure, the density of the predicted value of the proposed method is concentrated in a range smaller than the real value (at three different moments), which meets the demand for early warning in practical engineering applications. Compared with PCA method, the prediction curve of De-redundant method is more concentrated and closer to the real RUL value. Of course, the peak value of the probability density curve of the RUL prediction result of the proposed method is the closest to the real RUL value, indicating that the probability density forecasting method of bearing RUL based on the TCNQR model and KDE is better than other methods. In order to verify the effectiveness of the proposed method, the RUL interval prediction of three bearings under different working conditions was carried out, the \(PICP\) value and \(MPIW\) value of each model at the confidence level of 90% were recorded, and the results were compared in Table 4. \begin{table}
\begin{tabular}{c} **\({}_{SDAE}\) - DI** \\ **Set** \\ **(TCNQR)** \\ \end{tabular} \\ \hline
30 min &
\begin{tabular}{c} PICP \\ MPIW \\ \end{tabular} & 84.93\% & 87.17\% & 90.71\% & 91.49\% & 93.43\% & 95.16\% \\
60 min &
\begin{tabular}{c} MPIW \\ \end{tabular} & 64.22\% & 63.98\% & 63.06\% & 60.72\% & 60.17\% & 58.44\% \\ \hline
60 min &
\begin{tabular}{c} PICP \\ MPIW \\ \end{tabular} & 87.12\% & 89.74\% & 90.13\% & 92.35\% & 93.39\% & 95.02\% \\
\begin{tabular}{c} MPIW \\ \end{tabular} & 62.91\% & 62.13\% & 59.28\% & 58.90\% & 56.13\% & 53.06\% \\ \hline
100 min &
\begin{tabular}{c} PICP \\ MPIW \\ \end{tabular} & 87.03\% & 90.19\% & 89.28\% & 93.04\% & 95.87\% & 96.25\% \\
\begin{tabular}{c} MPIW \\ \end{tabular} & 62.15\% & 59.74\% & 59.87\% & 55.39\% & 58.62\% & 41.06\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Reliability and clarity comparison of prediction results at three different moments. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{3}{*}{**Working Condition**} & \multirow{3}{*}{**DI Set**} & \multicolumn{4}{c}{**The Prediction Results**} \\ \cline{3-6}  & & **MAE (LSTMQR)** & **RMSE (LSTMQR)** & **MAE (TCNQR)** & **RMSE (TCNQR)** \\ \hline \multirow{3}{*}{Condition A 2100} & Initial feature set & 2.928 & 3.207 & 2.801 & 3.034 \\  & PCA feature set & 2.407 & 2.683 & 2.473 & 2.650 \\  & \(F_{SDAE}\) & 1.706 & 1.976 & 1.391 & 1.589 \\  & \(\frac{\mathit{\mathit{DEMICF}}}{\mathit{\mathit{DIF}}}\) & 1.411 & 1.694 & 1.317 & 1.533 \\  & \(\mathit{\mathit{Opt}_{DI}}\) & 0.729 & 0.903 & 0.584 & 0.783 \\ \hline \multirow{3}{*}{Condition B 2250} & Initial feature set & 2.643 & 2.916 & 2.509 & 2.733 \\  & PCA feature set & 2.296 & 2.602 & 2.217 & 2.406 \\  & \(F_{SDAE}\) & 1.781 & 1.940 & 1.445 & 1.690 \\  & \(\frac{\mathit{\mathit{DEMICF}}}{\mathit{\mathit{DIF}}}\) & 1.803 & 2.117 & 1.283 & 1.466 \\  & \(\mathit{\mathit{Opt}_{DI}}\) & 0.657 & 0.851 & 0.579 & 0.747 \\ \hline \multirow{3}{*}{Condition C 2400} & Initial feature set & 2.727 & 2.985 & 2.648 & 2.811 \\  & PCA feature set & 2.279 & 2.549 & 2.335 & 2.560 \\ \cline{1-1}  & \(\mathit{\mathit{FDDAE}}\) & 1.502 & 1.737 & 1.277 & 1.492 \\ \cline{1-1}  & \(\mathit{\mathit{DEMICF}}\) & 1.412 & 1.689 & 1.182 & 1.369 \\ \cline{1-1}  & \(\mathit{\mathit{Opt}_{DI}}\) & 0.701 & 0.884 & 0.458 & 0.619 \\ \hline \hline \end{tabular}
\end{table}