## 2 Methodology
### Multivariate Signal Processing of Components  
Covariate Calculation Based on Feature Distance Evaluation  
The lubricating oil system contains numerous sensor signals, and traditional Bayesian networks (BNs) rely on predefined static conditional probability tables, making it difficult to dynamically integrate multi-source heterogeneous data (such as sensor time-series data, maintenance records, and environmental parameters); this is shown in Table 5 below. Directly using all dynamic sensor data can lead to data explosion, resulting in an excessive computational load beyond system capacity. Time-domain statistical feature analysis of data focuses on analyzing the characteristics manifested concerning time as the independent variable. Time-domain statistical features can be used to calculate the magnitude, amplitude variation, and energy distribution of the data, which can effectively reflect the operational status of the lubrication system. Frequency-domain features, on the other hand, can reflect the dynamic characteristics and changing patterns of the lubrication system [31]. Based on the above characteristics, to analyze the equipment's behavior under different operating states, the within-class and between-class distances of feature parameters are calculated. The ratio of these distances is used as a distance evaluation factor. The greater the value of the distance evaluation factor, the more sensitive the feature parameter is in recognizing different operating states of the equipment, and it can represent equipment degradation or failure information [32]. \begin{table}
\begin{tabular}{c c c c c} \hline \(\mathbf{t}\) & \(\mathbf{t+\Delta t}\) & **Formula** & \(\mathbf{t}\) & \(\mathbf{t+\Delta t}\) & **Formula** \\ \hline N & N & \(e^{-(\lambda_{1}+\lambda_{2}+\lambda_{3})\Delta t}\) & D & N & \(\lambda_{4}(\lambda_{4}+\lambda_{5})^{-1}\Big{(}1-e^{-(\lambda_{4}+\lambda_{5 })\Delta t}\Big{)}\) \\ N & I & \(\lambda_{1}(\lambda_{1}+\lambda_{2}+\lambda_{3})^{-1}\Big{(}1-e^{-(\lambda_{1 }+\lambda_{2}+\lambda_{3})\Delta t}\Big{)}\) & D & I & 0 \\ N & D & \(\lambda_{2}(\lambda_{1}+\lambda_{2}+\lambda_{3})^{-1}\Big{(}1-e^{-(\lambda_{1 }+\lambda_{2}+\lambda_{3})\Delta t}\Big{)}\) & D & D & \(e^{-\lambda_{6}\Delta t}\) \\ N & C & \(\lambda_{3}(\lambda_{1}+\lambda_{2}+\lambda_{3})^{-1}\Big{(}1-e^{-(\lambda_{1 }+\lambda_{2}+\lambda_{3})\Delta t}\Big{)}\) & D & C & \(1-e^{-\lambda_{6}\Delta t}\) \\ I & N & 0 & C & N & 0 \\ I & I & \(e^{-(\lambda_{5}+\lambda_{4})\Delta t}\) & C & I & 0 \\ I & D & \(\lambda_{5}(\lambda_{4}+\lambda_{5})^{-1}\Big{(}1-e^{-(\lambda_{4}+\lambda_{5 })\Delta t}\Big{)}\) & C & D & 0 \\ I & C & \(\lambda_{4}(\lambda_{4}+\lambda_{5})^{-1}\Big{(}1-e^{-(\lambda_{4}+\lambda_{5 })\Delta t}\Big{)}\) & C & C & 1 \\ \hline \end{tabular}
\end{table} 
Table 4: Relationship of root node state transition in DBN without maintenance. \begin{table}
\begin{tabular}{c c c c c c} \hline \(\mathbf{t}\) & \(\mathbf{t+\Delta t}\) & **Formula** & \(\mathbf{t}\) & \(\mathbf{t+\Delta t}\) & **Formula** \\ \hline N & N & \(e^{-(\lambda_{1}+\lambda_{2}+\lambda_{3})\Delta t}\) & I & C & \(\lambda_{4}(\lambda_{4}+\lambda_{5})^{-1}\Big{(}1-e^{-(\lambda_{4}+\lambda_{5 })\Delta t}\Big{)}\) \\ N & I & \(\lambda_{1}(\lambda_{1}+\lambda_{2}+\lambda_{3})^{-1}\Big{(}1-e^{-(\lambda_{1 }+\lambda_{2}+\lambda_{3})\Delta t}\Big{)}\) & D & N & 0 \\ N & D & \(\lambda_{2}(\lambda_{1}+\lambda_{2}+\lambda_{3})^{-1}\Big{(}1-e^{-(\lambda_{1 }+\lambda_{2}+\lambda_{3})\Delta t}\Big{)}\) & D & I & 0 \\ N & C & \(\lambda_{3}(\lambda_{1}+\lambda_{2}+\lambda_{3})^{-1}\Big{(}1-e^{-(\lambda_{1 }+\lambda_{2}+\lambda_{3})\Delta t}\Big{)}\) & D & D & \(e^{-\lambda_{6}\Delta t}\) \\ I & N & 0 & D & C & \(1-e^{-\lambda_{6}\Delta t}\) \\ I & I & \(e^{-(\lambda_{5}+\lambda_{4})\Delta t}\) & C & N & \(1-e^{-\mu_{1}\Delta t}\) \\ I & D & \(\lambda_{5}(\lambda_{4}+\lambda_{5})^{-1}\Big{(}1-e^{-(\lambda_{4}+\lambda_{5 })\Delta t}\Big{)}\) & C & I & 0 \\ C & C & \(e^{-\mu_{1}\Delta t}\) & C & D & 0 \\ \hline \end{tabular}
\end{table}