## 2 Methodology
### Reliability Modeling of Marine Oil System  
Let \(d_{p}^{b}\) represent the between-class distance of the \(p\)-th feature parameter between samples of different operating states, and \(d_{p}^{w}\) represent the within-class distance of the \(p\)-th feature parameter between samples within the same operating state. The definition of the operating state feature set is \(\{q_{(m,t,p)},m=1,2,\ldots,M_{t};t=1,2,\ldots,T;p=1,2,\ldots,P\}\), where t represents different operating states, \(p\) represents the types of feature parameters, \(M_{t}\) represents the number of feature parameter samples under the \(t\)-th operating state, and \(q_{(m,t,p)}\) represents the \(m\)-th sample value of the \(p\)-th feature parameter under the \(t\)-th operating state. Calculate the average distance between different samples of each feature parameter within a given operating state:  
\[d_{(t,p)}=\frac{1}{M_{t}(M_{t}-1)}\sum_{m=1}^{M_{t}}\sum_{l=1,l\
eq m}^{M_{t}} |q_{(m,t,p)}-q_{(l,t,p)}| \tag{9}\]  
where \(l\
eq\) m.  
Compute the average within-class distance overall operating states:  
\[d_{p}^{w}=\frac{1}{T}\sum_{t=1}^{T}d_{t,p} \tag{10}\]  
The smaller the within-class distance, the more concentrated the values of the characteristic parameters of the samples within the same state, and the higher the stability of the features. \begin{table}
\begin{tabular}{c l} \hline \hline
**Component Name** & **Relevant Signal Parameters** \\ \hline Turbine lubricating oil pump & Turbine lubricating oil pump \\ Turbine lubricating oil pump & Turbine lubricating oil pump oil flow rate \\ Turbine lubricating oil pump & Turbine lubricating oil pump outlet oil temperature \\ Turbine lubricating oil pump temperature \\ \hline Power unit lubricating oil cooler & Total lubricating oil flow rate of the power unit \\ Power unit circulating cooling water flow rate \\ Main pipeline pressure of the power unit lubricating oil \\ Main pipeline temperature of the power unit lubricating oil \\ \hline Electric lubricating oil pump & Total lubricating oil flow rate of the generating unit \\ Electric lubricating oil pump & Generating unit pump outlet pressure \\ Main pipeline pressure of the generating unit lubricating oil \\ Main pipeline temperature of the generating unit lubricating oil \\ \hline Generator set lubricating oil cooler & Turbine generator unit lubricating oil flow rate \\ Turbine generator unit outlet oil temperature \\ Turbine generator unit temperature \\ \hline Return Oil Valve & Return oil pressure \\ Return oil flow rate \\ Lubricating oil temperature \\ \hline Overhead oil tank & Overhead oil tank oil level \\ Overhead oil tank oil temperature \\ \hline Safety Pressure Relief Valve & Relief valve switch status \\ Relief valve vibration \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Signal for each part. Calculate the average value of each feature parameter under different operating states:  
\[u_{(t,p)}=\frac{1}{M_{t}}\sum_{m=1}^{M_{t}}q_{(m,t,p)} \tag{11}\]  
Then, calculate the between-class distance between different operating states:  
\[d_{p}^{b}=\frac{1}{T(T-1)}\sum_{st=1}^{T}|u_{(t,p)}-u_{(s,p)}| \tag{12}\]  
where s \(\
eq\) t; the larger the distance between classes, the more significant the difference in the mean values of the feature parameters between different states, and the stronger the ability to distinguish between features. Calculate the distance evaluation factor \(\lambda=\left|\frac{d_{p}^{b}}{d_{p}^{b}}\right|\), and quantify the sensitivity of feature parameters to state classification. Ideal features should satisfy: small intra-class distances (stable in the same class) and large inter-class distances (significant differences between dissimilar classes). Sort \(\lambda_{p}\) in descending order. Use the contribution rate-based inflection point monitoring method to determine which \(\lambda_{p}\) to retain. Normalize all data: \(\vec{\lambda}_{p}=\lambda_{p}/\sum_{p=1}^{p}\lambda_{p}\), and calculate the cumulative contribution: \(C(k)=\sum_{p=1}^{k}\vec{\lambda}_{p},k=1,\dots,P\).