## 2 Methodology
### DBN Analysis Model of Reliability
In the Dynamic Bayesian Network (DBN) model established in this paper, in order to describe the running health state of the ship lubrication system, the key components of the system are modeled as root nodes, and each node can be in one of the following four typical health states: N (Normal)--the normal operation state, indicating that the lubrication system or its key components are running normally; without abnormal indicators, all performance parameters are within the design range, the lubricating oil quality is good, and the system is stable and reliable. I (Initial Abnormal)--the initial abnormal state, indicating that the system is showing mild signs of abnormality but is still functioning. This state often manifests as the temperature of the lubricating oil being high but not beyond the limit; a slight fluctuation in oil pressure; an increase in trace impurities or metal particles in the lubricating oil; equipment vibration; or noise being slightly abnormal. Although these phenomena do not directly cause failure, they indicate that the system has deviated from the normal operation track and may gradually degrade without intervention. D (Degradation)--the degraded state, indicating that the lubrication performance has decreased significantly, and some functions of the system have been affected, which may pose a threat to the operation safety of the equipment. The common manifestations include: oxidation deterioration in the lubricating oil; abnormal viscosity; sustained low oil pressure or insufficient oil flow; a continuous abnormal vibration or temperature rise; or the content of metal abrasive particles in the oil increases rapidly. At this time, the system is already in the stage of moderate failure, and if it is not maintained, it is very likely to further develop into a serious failure. C (Critical Failure)--a serious failure state, indicating that the lubrication system cannot continue to provide effective lubrication, or critical components may be severely worn or even stuck due to a lack of oil or dry friction, which may lead to equipment shutdown, damage, and other consequences. The lubrication system may randomly transition to the C state (severe failure state) from any state; the state transition rate is constant and follows an exponential distribution, which is used to model the system's degradation process; the current degradation state can be observed through lubrication-related monitoring parameters, and it is assumed that the time of state detection can be ignored; and after scheduled maintenance, the system is restored to a near-new state, meaning that the I and D states can be repaired to the N state through scheduled maintenance, restoring lubrication functionality. In Figure 4, we show the basic probability transition structure of the system. Other state transitions are not impossible but can be calculated by the known probabilities in the figure. Tables 3 and 4 list the complete transition probability matrix between each state, including the maintenance and recovery paths from D \(\rightarrow\) N and I \(\rightarrow\) N. The transition probabilities between all pairs of states can be consulted, and the model also fully considers these paths during inference. Therefore, in order to avoid redundant lines in the figure and affect the identification of the backbone structure, we only keep the state transition directions that need to collect probability data in Figure 4, and the other state transition probabilities can be deduced from the data in Tables 3 and 4. The state transition process of the root node in the DBN is shown in Figure 4. The corresponding state transition equations for the ship's voyage with maintenance are presented in Table 3, while those without maintenance are presented in Table 4. The relationship between the two tables is the calculation formula for the transition probabilities of various states at time t to the states at time \(t+\Delta t\). The structure of the DBN can be derived through fault tree transformation.