# Distributed Kernel Extreme Learning Machines for Aircraft Engine Failure Diagnostics
## 5 Failure Diagnostics for Aircraft Engine
### Performance Degeneration Estimation  
The performance degradation of each component for aircraft engine with different flight cycle numbers derived by NASA [40; 41] from the existing literature is shown in Table 5. The degradation degree of each health parameter is described in the table, including the initial degradation, the degradation after 3000 cycles and after 6000 cycles. The percentage represents the deviation degree from the nominal value. Each separate regression estimator corresponding to the health parameter was established and Table 7 lists performance comparison among KELM algorithm, DKELMs algorithm and n-SVR algorithm in estimating the degeneration degree of each health parameter, including the \(RMSE\), training time and testing time. As can be seen from Table 7, the proposed DKELMs algorithm can achieve similar or even higher prediction accuracy for health parameters while the training time and testing time are much less than original KELM algorithm. In addition, since n-SVR needs to deal with a large number of support vector points for regression problems, the training time is huge and the testing time is also quite long. In summary, among the three algorithms, the proposed DKELMs algorithm performs best when dealing with health parameter degeneration estimation.  
Figure 5 describes the health parameter degradation estimation of DKELMs algorithm with different flight cycle number. The general degradation degree of aircraft engine is assumed to be the linear interpolation with the percentages given in Table 5. In addition, degradation reference values also consider the periodic maintenance which results in the periodic fluctuation of reference values. It is clear that the values of health parameter estimated by DKELMs algorithm are consistent with the degradation reference values on the whole. The prediction result of thrust and surge margin is described in Figure 6. It can be seen that the values of thrust and surge margin estimated by DKELMs algorithm can generally reflect the change of target values. The confidence intervals corresponding to thrust, LPC surge margin and HPC surge marine are \(\pm\)0.0038, \(\pm\)0.0094 and \(\pm\)0.0097 respectively.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Health Parameter** & **Algorithm** & \(RMSE\) & **Training Time/s** & **Testing Time/s** \\ \hline \multirow{3}{*}{LPC efficiency} & KELM & 0.0753 & 16.093 & 1.2098 \\  & DKELMs & 0.0755 & 1.1885 & 0.1631 \\  & n-SVR & 0.0755 & 22.4593 & 0.9052 \\ \hline \multirow{3}{*}{LPC flow capacity} & KELM & 0.0693 & 16.0365 & 1.1827 \\  & DKELMs & 0.0688 & 1.1961 & 0.1655 \\  & n-SVR & 0.0691 & 24.6815 & 0.9342 \\ \hline \multirow{3}{*}{HPC efficiency} & KELM & 0.0489 & 16.4877 & 1.1015 \\  & DKELMs & 0.0469 & 2.3328 & 0.3324 \\  & n-SVR & 0.0473 & 29.718 & 0.9353 \\ \hline \multirow{3}{*}{HPC flow capacity} & KELM & 0.0245 & 16.2676 & 1.0853 \\  & DKELMs & 0.0244 & 0.4203 & 0.1845 \\  & n-SVR & 0.0244 & 53.2591 & 1.1486 \\ \hline \multirow{3}{*}{HPT efficiency} & KELM & 0.0750 & 15.972 & 1.1660 \\  & DKELMs & 0.0752 & 0.4448 & 0.0823 \\  & n-SVR & 0.0755 & 57.5651 & 0.9393 \\ \hline \multirow{3}{*}{HPT flow capacity} & KELM & 0.0752 & 16.4246 & 1.1567 \\  & DKELMs & 0.0740 & 0.3281 & 0.0848 \\  & n-SVR & 0.0734 & 72.348 & 1.0358 \\ \hline \multirow{3}{*}{LPT efficiency} & KELM & 0.0951 & 15.8335 & 1.0149 \\  & DKELMs & 0.0939 & 0.4479 & 0.0933 \\ \cline{1-1}  & n-SVR & 0.0945 & 48.6863 & 0.9668 \\ \hline \multirow{3}{*}{LPT flow capacity} & KELM & 0.0471 & 15.8797 & 1.1166 \\  & DKELMs & 0.0466 & 0.3395 & 0.0855 \\ \cline{1-1}  & n-SVR & 0.0472 & 17.3455 & 0.9589 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Performance comparison for degeneration estimation.