## 3 Method  
### Particle Swarm Optimization Algorithm  
The particle swarm optimization (PSO) algorithm is a swarm intelligence class optimization algorithm that originated from the study of bird predation behavior, which was proposed by [PERSON] and [PERSON] in 1995 [33] and has been widely used in many fields after more than a decade of continuous research and development [34; 35]. The PSO algorithm randomly initializes particles in the feasible solution space of the problem, and each particle is given a certain velocity of motion at its initial position, and the trajectory of the particles is influenced by its factors and the overall behavior of the population during the whole iteration cycle. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter Estimation Methods** & **Shape Parameter Estimated Range** & **Estimated Expressions** \\ \hline Positive 2 nd/4 th-order moment estimation (MoM) & (2, +\(\infty\)) & Closure \\ Positive 0.5 th/1 st-order moment estimation (MfoM) & (0.5, +\(\infty\)) & Non-closed \\ Positive 1 st-order logarithmic moment estimation (0) & (1, +\(\infty\)) & Closure \\ Positive 0.25 th-order logarithmic moment estimation (Zlog2) & (0.25, +\(\infty\)) & Non-closed \\ Maximum Likelihood Estimation (MLE) & (0, +\(\infty\)) & Non-closed \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Feature comparison of parameter estimation methods. The fitness function is the target object of the particle population activity because it determines the activity space and search path of the particles. The particle population has the ability of memory, and all particles calculate the corresponding fitness value in each iteration, then compare it with the historical value and select the optimal value of the historical record, and then guide the whole population toward the direction of the global optimal solution. The basic process of the particle swarm algorithm is: 1. The positions \(\mathbf{X}\) and velocities \(\mathbf{V}\) of the particles are randomly initialized in the D-dimensional space of feasible solutions, where the \(i\)-th particle position and velocity can be expressed as \[ \begin{cases}\mathbf{X}_{i}=(x_{i1},x_{i2},\cdots,x_{iD})\\ \mathbf{V}_{i}=(v_{i1},v_{i2},\cdots,v_{iD})\end{cases} ,i=1,2,\cdots,N\] (16) 2. Based on the determined fitness function (particle population search object) and the corresponding position of each particle, the corresponding fitness value is calculated, and then the global optimum is evaluated, where the historical optimum of the particle and the global optimum of the population is assumed to be \(P_{best}\) and \(G_{best}\), respectively, that is, we have \[ \begin{cases}\mathbf{P}_{best}=(p_{1i},p_{2i},\cdots,p_{iD})\\ G_{best}=(g_{1i},g_{2i},\cdots,g_{iD})\end{cases} ,i=1,2,\cdots,N\] (17) 3. The particle population is continuously updated iteratively to search for the extreme value solution of the fitness function, and the velocity and position of each particle in the next iteration are updated by the individual historical optimal value \(P_{best}\) and the current velocity \(\mathbf{V}_{i}\), and the \(k+1\)th update of the particle is given by \[ \begin{split} V_{id}^{k+1}=\omega V_{id}^{k}+c_{1}r_{1}\Big{(}P_{ id}^{k}-X_{id}^{k}\Big{)}+c_{2}r_{2}\Big{(}P_{gd}^{k}-X_{id}^{k}\Big{)}\\ \hskip 142.26378 ptX_{id}^{k+1}=X_{id}^{k}+V_{id}^{k+1}0\end{split} \] (18) where \(w\) is the inertia weight, which gives the particle the inertia of motion trend; \(k\) is the number of current iterations; \(c_{1}\) and \(c_{2}\) are learning factors; and \(r_{1}\) and \(r_{2}\) are random numbers distributed in the interval of [0, 1]. To prevent the particles from searching blindly and falling into the risk of local optimum, the velocity and position of the particles are usually limited to a certain interval.