## 2 Methodology
### Bridge Deterioration Curve Fitting Based on Markov Chain  
State-based models use a transition probability matrix obtained from the difference between two condition states at a given discrete time interval to predict the long-term performance of bridges [25]. [PERSON] et al. [26] proposed a relationship between bridge condition and bridge age, concluding that the deterioration performance of bridges remains constant throughout the prediction period; however, this method does not accurately reflect the true deterioration process of bridges. Conversely, [PERSON] et al. [27] suggested using multiple transition probability matrices to predict bridge conditions by dividing bridge age into six-year age groups, assuming that the deterioration performance is consistent within  
Figure 2: Structure of Elman neural network.  
each age group, and the transition probability matrix for each group can be calculated by optimizing the objective function.  
\[Min{\sum_{d=1}^{N}}|A(t)\,-\,E(t)|\qquad 0\leq P(i)\leq 1\quad i\ =1,2,3 \ldots,U \tag{3}\]  
where \(N=\) the number of years in an age group; \(U=\) the number of unknown probabilities; \(A(t)=\) the conditional rating of the performance curve at time \(t\), estimated from the deterioration curve predicted by the neural network; and \(E(t)=\) the estimated value of the conditional rating at time \(t\), predicted by the Markov chain. The bridge conditional ratings to be predicted by the Markov chain are generated by the following equation:  
\[E(t,P)=P(0)P^{t}R^{\prime} \tag{4}\]  
where \(P(0)=\) the initial state vector, e.g., if the initial state of the bridge is Level 9, then \(P(0)\) is denoted by \([1\ 0\ 0\ 0\ 0\ 0\ 0]\); and \(P\) is the transition probability matrix, \(R=[9\ 8\ 7\ 6\ 5\ 4\ 3]\).  
The reliability of state-based Markov stochastic models depends on the accuracy of the transition probability matrix. This matrix is established in an \(n\) by \(n\) format, where \(n\) represents the number of possible condition ratings. It is used to calculate the distribution of bridge condition ratings over time. According to the National Bridge Inventory (NBI) bridge data, the lowest condition rating typically observed for a bridge component is 3. Thus, the minimum rating for bridge condition transitions is set at 3, below which no further deterioration is calculated.  
Transition probability matrices can be divided into two types: one-step and multi-step. In a one-step transition probability matrix, the condition state can only transition to the next lower state. In a multi-step transition probability matrix, the condition state can transition to multiple lower states. Although it is theoretically possible for a condition state to deteriorate by two or more levels within a year, such occurrences are rare in bridge condition data [28]. Therefore, the one-step transition probability matrix is described using the following equation:  
\[P=\begin{bmatrix}\mathrm{P}(1)&1-P(1)&0&0&0&0&0\\ 0&\mathrm{P}(2)&1-P(2)&0&0&0&0\\ 0&0&\mathrm{P}(3)&1-P(3)&0&0&0\\ 0&0&0&\mathrm{P}(4)&1-P(4)&0&0\\ 0&0&0&0&\mathrm{P}(5)&1-P(5)&0\\ 0&0&0&0&0&\mathrm{P}(6)&1-P(6)\\ 0&0&0&0&0&0&1\end{bmatrix} \tag{5}\]  
where \(P(i)=\) probability of staying at condition rating \(i\), \(i=1\), \(2\),..., \(6\); \(1\ -\ P(i)=\) probability of switching to the next condition rating. In the NBI condition rating scale, the lowest condition rating for maintenance is 3, and it is not possible to continue deteriorating to the next rating, so the corresponding probability \(P(7)=1\).