## 2 Methodology
### DBN Analysis Model of Reliability  
To discuss the reliability correlation of various components in the system, the DBN analysis model is used for computation. A Dynamic Bayesian Network (DBN) is a probabilistic distribution model that combines the structure of Bayesian networks with time series principles [30]. The DBN can be defined as \((B_{1},B_{\rightarrow})\); \(B_{1}\) is a Bayesian network that defines the prior probability \(P(X_{1})\); and \(B_{\rightarrow}\) is a two-layer time slice Bayesian network that defines \(P(X_{t}\mid X_{t-1})\). The probability relationship between \((B_{1},B_{\rightarrow})\) is \(P(X_{t}\mid X_{t-1})\), which is expressed as:  
\[P(X_{t}\mid X_{t-1})=\prod_{i=1}^{N}P\left(X_{(t,i)}\mid Pa(X_{(t,i)})\right) \tag{7}\]  
\(X_{(t,i)}\) is the \(i\)-th node at time \(t\), \(Pa(X_{(t,i)})\) is the parent node of \(X_{(t,i)}\) at time \(t\), and \(P(X_{(t,i)}\mid Pa(X_{(t,i)}))\) is the conditional probability of the \(i\)-th node given its parent node. \(N\) represents the number of nodes in the Dynamic Bayesian Network (DBN). A DBN defines the joint probability distribution over a finite time segment T by expanding the 2-layer DBN, and the joint probability distribution over time segments 1 to \(T\) is:  
\[P\left(X_{(1:T)}\right)=\prod_{t=1}^{T}\prod_{i=1}^{N}P\left(X_{(t,i)}|Pa\left( X_{(t,i)}\right)\right) \tag{8}\]  
\(T\) is the number of time slices; \(P\left(X_{(1:T)}\right)\) is the joint probability distribution of the DBN from time slice 1 to T. The states of the ship's lubrication system can be divided into the normal operation state (N), initial abnormal state (I), degradation state (D), and severe failure state (C). When the lubrication system is in the initial abnormal state (I), although the system can still provide basic lubrication functions, it has already showed \"abnormal signs\" relative to the normal state (N). These signs are often early warning signals of failure, that if not handled in time, may lead to further deterioration in lubrication performance. According to the working principle and common fault characteristics of the ship lubrication system, these \"abnormal signs\" can be described more specifically as follows: Under the same working condition, the temperature of the lubricating oil is higher than usual, indicating that the heat of the friction pair increases or the cooling effect decreases. Lubricating oil will be gradually oxidized or polluted during normal use. If there is a rapid color change in a short period of time or the viscosity significantly deviates from the normal range, it indicates that there may be oil deterioration or abnormal conditions such as impurities and moisture. If the lubricating oil pressure or flow rate in the system shows significant instability or periodic fluctuations, it may indicate an internal pipeline blockage, seal wear, or abnormal pump operation. In general, the initial abnormal state (I) does not mean that there will be a serious failure or shutdown of the equipment immediately, but the above \"abnormal signs\" indicate that the lubrication performance has begun to degrade and need to be checked and handled in time. For example: replace or replenish lubricating oil, clean or replace the filter element, correct oil pressure and flow, check the wear of seals and key components, etc. Otherwise, the system may deteriorate further in a short period of time, enter a degraded state (D), and, eventually, a serious fault state (C), causing greater risks to the safety and normal operation of the equipment. In this paper, when using the Dynamic Bayesian Network (DBN) to analyze the multi-state health factors of the ship's lubrication system, the following assumptions are made: The key components in the system are treated as root nodes in the DBN, and these nodes can be in the N, I, D, or C states.