## 3 Estimation of Model Parameters
### MLE with Marine Predators Algorithm  
Let \(\Delta x_{k,m}=x_{k,m}-x_{k,m-1}\) represent the deterioration amount of the SDLL for track segment \(k\) between two consecutive inspection times \(t_{k,m-1}\) and \(t_{k,m}\). In accordance with the properties of the PTT-WP, \(\Delta x_{k,m}\) follows a normal distribution with the expectation \(E(\Delta x_{k,m})\) and variance \(\mathrm{Var}(\Delta x_{k,m})\) given by Equations (6) and (7):  
\[E(\Delta x_{k,m})=\beta_{k}\cdot\Delta\Lambda(t;\theta_{k})_{m}, \tag{6}\]  
\[\mathrm{Var}(\Delta x_{k,m})=\sigma_{k}^{2}\cdot\Delta\Lambda(t;\theta_{k})_{ m}\,, \tag{7}\]  
where \(\Delta\Lambda(t;\theta_{k})_{m}=t_{k,m}^{\theta_{k}}-t_{k,m-1}^{\theta_{k}}\). Based on the principles of the MLE method, the historical inspection data at the longitudinal level for track segment \(k\) can be used to construct the log-likelihood function, given by Equation (8):  
\[\ln[L(\mathbf{\gamma}_{k}|\mathbf{Data}_{k})]=\ln[\prod_{m=2}^{M_{k}}\frac{1}{\sqrt{2 \pi\sigma_{k}^{2}\Delta\Lambda(t;\theta_{k})_{m}}}\mathrm{exp}[\ -\ \frac{\left(\Delta x_{k,m}-\beta_{k}\Delta\Lambda(t;\theta_{k})_{m} \right)^{2}}{2\sigma_{k}^{2}\Delta\Lambda(t;\theta_{k})_{m}}]], \tag{8}\]  
where \(\mathbf{\gamma}_{k}=\left\{\gamma_{k1},\gamma_{k2},\gamma_{k3}\right\}=\left\{ \beta_{k},\theta_{k},\sigma_{k}\right\}\), is the vector composed of the model parameters to be estimated, \(\mathbf{Data}_{k}=\left\{\left(t_{k,m},x_{k,m}\right)|\ m=1,2,\ldots,M_{k}\right\}\) is the collected dataset of the total of \(M_{k}\) inspections of track segment \(k\), \(t_{k,m}\) is the inspection date of inspection sample \(m\), and \(x_{k,m}\) is the SDLL value obtained for inspection sample \(m\). According to the theory of MLE, the values of the parameters \(\beta_{k}\), \(\sigma_{k}\), and \(\theta_{k}\) that maximize the log-likelihood function, as shown in Equation (8), are the estimated results, denoted as \(\hat{\beta}_{k}\), \(\hat{\sigma}_{k}\), and \(\hat{\theta}_{k}\), respectively. Therefore, this parameter estimation problem is an optimization challenge that involves determining the optimal solution in Equations (9) and (10):  
\[\max\ln[L(\mathbf{\gamma}_{k}|\mathbf{Data}_{k})], \tag{9}\]s.t. \(\beta_{k}>0,\sigma_{k}>0,\theta_{k}\geq 1\), (10)  
Considering the intricate and multimodal characteristics of the objective function as well as the nonlinear and multidimensional characteristics of the solution space, gradient-based optimization methods are susceptible to getting trapped in local optima. Consequently, in this study, we employed a high-performance heuristic global optimization solver known as the marine predators algorithm (MPA). This algorithm was proposed by [PERSON] et al. [43]. The performance of the MPA in 29 test functions and several practical engineering cases was evaluated in their study. The proposed MPA performed better than the genetic algorithm, particle swarm optimization, gravity search algorithm, and covariance matrix adaptive evolutionary strategy algorithm [43]. Algorithm 1 presents the pseudocode of MPA used to solve Equations (9) and (10). ```
Initialization:\(n\)--population size, \(max\_iter\)--maximum number of iterations, \(lb\)--lower bound of the parameter value range, \(ub\)--upper bound of the parameter value range, set the number of iterations \(Iter=0\), initial population \(Prey_{k}=\left\{\mathbf{\gamma}_{k}^{i},i=1,\ldots,n\right\}\) Iterative process: While\(Iter<max\_iter\), 1. Calculate the fitness of the population based on Equation (8), i.e., \(\ln[L\left(\mathbf{\gamma}_{k}^{i}|\mathbf{Data}_{k}\right),\) construct the Elite matrix \(\mathbf{Elite}_{k}\), and accomplish memory saving. 2. Update the population \(Prey_{k}\): If\(Iter<max\_iter/3\) \(\mathbf{s}_{i}=R_{\mathbb{B}}\otimes\left(\mathbf{Elite}_{k}^{i}-R_{\mathbb{B}}\otimes \mathbf{\gamma}_{k}^{i}\right)\), \(\mathbf{\gamma}_{k}^{i}=\mathbf{\gamma}_{k}^{i}+P*\mathbf{R}\otimes\mathbf{s}_{i},i=1,\ldots,n\) Or elseif\(max\_iter/3<Iter<2*max\_iter/3\)  Divide the population into two parts: prey and predators.