## 3 Method
### Improved Parameter Estimation for Particle Swarm Hybridization
Set the position \(X\) and velocity \(V\) of each particle using uniform distribution random numbers within a certain range. **Step 2. ** Compute the fitness value for each particle based on the objective function and store it in the variable \(P_{id}\). Select the best fitness value among the particles and store it in the variable \(P_{gd}\). The variables \(P_{id}\) and \(P_{gd}\), respectively, represent the fitness value of the particles and the fitness value of the best position in the population for the current iteration. **Step 3. ** Update the inertia weight factor \(\omega\) using the nonlinearly decreasing Formula (20). **Step 4. ** Update the velocity and position of each particle using the current iteration's inertia weight, learning factors, and other algorithm parameters, following Formula (18). **Step 5. ** Compute the fitness value for each particle in the current iteration and compare it with the particle's personal best position. Update the variable \(P_{id}\) based on the fitness value. Then, compare all updated \(P_{id}\) values with the global best position, updating the swarm's variable \(P_{gd}\). **Step 6. ** Update the crossover probability \(P_{c}\) using the nonlinearly decreasing Formula (22). **Step 7. ** Select a specified number of particles specified by \(P_{s}\) and place them in the crossover pool. Randomly select two particles from the pool to participate in a crossover. Update the position and velocity of the resulting offspring using Formula (21). Crossover operations generate the same number of particles as the parent generation. **Step 8. ** Check whether the termination condition of the algorithm is met. If satisfied, stop the swarm search and save the global best position of the particles. Otherwise, return to Steps 3 to 7 to continue the search. The HPSO algorithm provides two ways to terminate the particle swarm search. The first approach is to set a maximum number of iterations for the particles, while the second approach involves specifying the number of generations to maintain the global best particle. Since determining the maximum number of iterations depends on the nature of the objective function and particle activities exhibit inherent randomness during each iteration, this section adopts the method of setting the number of generations to retain the global best particle to ensure an effective search for the optimal position of the objective function. Different parameters of the generalized Pareto distribution can describe the amplitude distribution characteristics of ocean clutter under different background environments. The shape and scale parameters are the main factors influencing the statistical characteristics curve of the generalized Pareto distribution. To achieve an accurate fit of the generalized Pareto distribution to clutter data, the statistical histogram of clutter data is compared with the cumulative error on the theoretical distribution curve, which serves as the objective function for optimizing the parameter estimation of the generalized Pareto distribution. Furthermore, to balance the fitting discrepancies between the probability density function (PDF) curve and the cumulative distribution function (CDF) curve of clutter data, an adapted fitness function is constructed by appropriately weighting and combining two discrepancy functions. The HPSO algorithm is then employed to perform the corresponding optimal parameter search task. The fitness function for the parameter estimation of the generalized Pareto distribution is defined as follows:  
\[f_{fitness}(\mathbf{\theta})=\frac{1}{1+\lambda_{1}\sum\limits_{n=1}^{N}\left(f(n ;\mathbf{\theta})-h(n)\right)^{2}+\lambda_{2}\sum\limits_{n=1}^{N}\left(F(n;\mathbf{ \theta})-H(n)\right)^{2}} \tag{23}\]  
where \(f(n;\theta)\) and \(F(n;\theta)\), respectively, denote the PDF and CDF of the generalized Pareto distribution. \(h(n)\) and \(H(n)\) represent the values of the clutter statistical histogram at the PDF and CDF curve sampling intervals.