# Turbofan Engine Health Prediction Model Based on ESO-BP Neural Network
## 3 Model Construction
### Feature Selection  
In the context of predicting the health of turbofan engines, there are several redundant features. Appropriate feature selection methods can enhance model performance, reduce computational costs, and provide an improved model interpretability. This study employs a feature selection method based on a random forest. In the random forest model, each decision tree is built by conducting bootstrap sampling from the original dataset [39]. The corresponding out-of-bag data are used to evaluate the prediction error of the \(j\)-th tree in the random forest. In order to assess the importance of the \(i\)-th feature, the values of \(i\)-th feature are randomly permuted and the out-of-bag data prediction error is reevaluated. The importance of the \(i\)-th feature is determined by calculating the ratio of the average and standard deviation of out-of-bag data errors for all trees in the entire random forest, both before and after disrupting the \(i\)-th feature. Equation (6) is used to calculate the standard deviation \(\sigma_{i}\), and Equation (7) is used to calculate the feature importance \(C_{i}\). \[\sigma_{i}=\sqrt{\frac{\sum_{j=1}^{n_{t}}\left(err_{i,j}-err_{0,j}\right)^{2}} {n_{t}}} \tag{6}\]  
\[C_{i}=\frac{\sum_{j=1}^{n_{t}}\left(err_{i,j}-err_{0,j}\right)}{n_{t}\cdot \sigma_{i}} \tag{7}\]  
Here, \(\sigma_{i}\) represents the standard deviation of the errors, \(err_{i,j}\) represents the prediction error after disrupting the \(i\)-th feature, \(err_{0,j}\) represents the prediction error before disrupting the \(i\)-th feature, \(n_{t}\) is the number of trees, and \(C_{i}\) is the importance of the \(i\)-th feature. In this section, we present the feature importance analysis for FD001, FD002, FD003, and FD004. The results are summarized in Table 2. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Feature Description**} & \multirow{2}{*}{**Units**} & \multirow{2}{*}{**Symbol**} & \multicolumn{6}{c}{**Feature Importance \(C_{i}\)**} \\ \cline{3-6}  & & & **FD001** & **FD002** & **FD003** & **FD004** \\ \hline Number of operating cycles & - & CYCLE & 5.574 & 10.141 & 5.386 & 9.868 \\ Height & km & H & 0.090 & 0.867 & 0.047 & 0.231 \\ Mach & - & MA & 0.086 & 0.343 & 0.002 & 0.493 \\ Throtte resolver angle & \({}^{\circ}\) & TRA & 0.000 & 0.141 & 0.000 & 0.146 \\ Total temperature at fan inlet & \({}^{\circ}\)R & T2 & 0.000 & 0.193 & 0.000 & 0.224 \\ Total temperature at LPC outlet & \({}^{\circ}\)R & T24 & 1.625 & 0.710 & 0.800 & 1.855 \\ Total temperature at HPC outlet & \({}^{\circ}\)R & T30 & 1.134 & 1.525 & 1.191 & 0.301 \\ Total temperature at LPT outlet & \({}^{\circ}\)R & T50 & 1.285 & 0.866 & 0.891 & 1.297 \\ Pressure at fan inlet & psia & P2 & 0.000 & 0.146 & 0.000 & 0.118 \\ Total pressure in bypass-duct & psia & P15 & 0.017 & 0.291 & 1.973 & 0.655 \\ Total pressure at HPC outlet & psia & P30 & 1.334 & 0.698 & 0.966 & 0.812 \\ Physical fan speed & rpm & NF & 1.649 & 0.766 & 0.825 & 1.085 \\ Physical core speed & rpm & NC & 1.137 & 0.645 & 1.011 & 0.928 \\ Engine pressure ratio (P50/P2) & - & EPR & 0.000 & 0.149 & 0.264 & 0.518 \\ Static pressure at HPC outlet & psia & PS30 & 0.971 & 1.569 & 1.004 & 1.026 \\ Ratio of fuel flow to Ps30 & pps/psi & PHI & 1.357 & 0.497 & 1.586 & 0.886 \\ Corrected fan speed & rpm & NRF & 1.261 & 1.378 & 0.813 & 2.271 \\ Corrected core speed & rpm & NRC & 0.799 & 0.959 & 1.290 & 1.124 \\ Bypass ratio & - & BPR & 1.371 & 1.084 & 1.158 & 1.835 \\ Burmer fuel-air ratio & - & FARB & 0.000 & 0.677 & 0.000 & 0.355 \\ Bleed enthalpy & - & HT\_BLEED & 0.829 & 0.925 & 0.317 & 0.594 \\ Demanded fan speed & rpm & NF\_DMD & 0.000 & 0.259 & 0.000 & 0.180 \\ Demanded corrected fan speed & rpm & PCNFR\_DMD & 0.000 & 0.025 & 0.000 & 0.101 \\ HPT coolant bleed & lbm/s & W31 & 0.752 & 0.554 & 1.652 & 0.382 \\ LPT coolant bleed & lbm/s & W32 & 1.636 & 0.816 & 1.548 & 1.009 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Feature importance of 4 sub-datasets. The underlined numbers represent the redundant features in the corresponding sub-datasets.