# Fatigue Crack and Residual Life Prediction Based on an Adaptive Dynamic Bayesian Network
## 2 Principle and Methods
### Brief Introduction to Dynamic Bayesian Network  
Figure 2 illustrates a simple example of a DBN. Nodes \(A_{i}\) and \(C_{i}\) represent hidden variables, indicating states that cannot be directly observed. In some cases, the variable \(C_{i}\) corresponds to a physical model, and \(A_{i}\) represents parameters of \(C_{i}\). The node \(D_{i}\) denotes observed variables. The subscripts correspond to different moments. The arrows between nodes describe the conditional dependencies between variables, pointing from parent nodes to child nodes. This directionality from \"cause\" to \"effect\" signifies a causal relationship or non-conditional independence between parent and child nodes, which is mathematically characterized by conditional probability distributions (CPDs). Variables with the same subscript form an independent BN, and BNs across different time slices are also connected by arrows, depicting the influence of variables from one moment to the next.  
According to Figure 2, a DBN consists of two parts [29]: prior network B and transition network B\({}_{\rightarrow}\). The prior network defines the joint probability distribution \(C_{1}\) of the initial state B\({}_{1}\) at \(t\) = 1. According to Bayesian inference, the probability formula for the hidden variable \(C\) can be written on the prior network as follows:  
\[\mathrm{P}(C|A,D)=\frac{\mathrm{P}(D|C,A)\mathrm{P}(A|C)}{\mathrm{P}(D|A)} \tag{5}\]  
The transition network defines the transition probability between \(C_{t}\) and \(C_{t+1}\) at \(t\) and \(t+1\), respectively. That is to say, the DBN composed of (B \(|\)B\({}_{\rightarrow}\)) is a semi-infinite network structure on \(B_{1}\), \(B_{2}\), \(\ldots\), \(B_{t}\)(\(t\) = 1, 2, \(\ldots\)\(\infty\)). Based on the first-order Markov assumption, the probability of variable C at the next moment only depends on the current moment and is independent of the past moments, that is, \(\mathrm{P}(C_{t+1}|C_{1:t})=\mathrm{P}(C_{t+1}|C_{t})\). The probability formula for variable a on the transition network can be articulated as follows:  
\[\mathrm{P}(C_{1:t})=\prod_{t=2}^{T}\mathrm{P}(C_{t}|C_{t-1}) \tag{6}\]  
The predictive probability distribution of the hidden variable \(C\) at cycles \(T\) can be obtained by applying the total probability theorem as follows:  
\[P(\hat{C}_{T}|D_{1:T})=\int P(\hat{C}_{T}|A_{1:T},D_{1:T})P(A_{1:T}|D_{1:T})dA \tag{7}\]  
After obtaining the posterior probability distribution of the variable \(C\), posterior updating and prediction of the parameter variable \(A\) can be carried out. Within a DBN, the EM algorithm, Monte Carlo method, or the Metropolis-Hastings algorithm is commonly  
Figure 2: A simple example of a DBN.  
used for sample parameter selection. However, for models involving long-term observation and updates, multiple iterations might lead to sample impoverishment, resulting in a decrease in accuracy. Therefore, this paper employs a particle filtering algorithm, replacing the backward propagation inference and the update and prediction of parameter \(A\), as shown in Equation (7).  
In DBNs, the joint probability density function across the time dimension \(T\) can be constructed by combining the prior network probability distribution Equation (6) at the beginning of the time series and the state transition network probability distribution Equation (7) between subsequent time points.  
\[\mathrm{P}(A|C_{1:T},D)=\mathrm{P}(A)\times\mathrm{P}(C_{1}|A)\times\prod_{t =2}^{T}\mathrm{P}(C_{t}|C_{t-1})\times\mathrm{P}(D|C_{T}) \tag{8}\]  
As shown, it is possible to compute the joint distribution probability of any node in the DBN.