## 4 Results and Discussion
### Performance Analysis of Parameter Estimation with HPSO Algorithm  
The parameter settings for the simulation data of the generalized Pareto distribution are as follows: \(a=3.5\) and \(b=0.5\), with a total of 5 \(\times\) 10\({}^{4}\) samples of noise. We use this simulated data to test the effectiveness of the HPSO algorithm and compare it with the performance of the PSO algorithm. By calculating the fitness values of the simulated data, we can compare the performance differences between these two algorithms in optimizing the objective function. Figure 13 illustrates the iteration process of the HPSO algorithm and the PSO algorithm. From Figure 13, it can be observed that the HPSO algorithm rapidly converges to the global optimum, while the PSO algorithm exhibits slower convergence and may suffer from local convergence. The output parameter estimation results for the optimal positions of both algorithms in the figure are \(a=3.2748\), \(b=0.5729\) (HPSO algorithm) and \(a=3.2747\), \(b=0.5729\) (PSO algorithm), indicating that the final parameter estimation results are similar for both algorithms. However, the PSO algorithm requires more computation time, with the HPSO algorithm only requiring 18 iterations (while the PSO algorithm requires 35 iterations) to effectively search for the estimated parameters. To further investigate the performance of the HPSO algorithm in optimizing the objective function for parameter estimation of the generalized Pareto distribution, we fixed either the shape parameter or the scale parameter and varied the other parameter. The PSO algorithm and the HPSO algorithm were employed to search for the global optimal value of the fitness function for parameter estimation of the generalized Pareto distribution. In this way, the corresponding parameter estimation values were obtained. Firstly, we fixed the scale parameter as \(b=0.3\) and varied the shape parameter of each set of simulated clutter samples from 0.1 to 10 with equal intervals. The number of simulated samples for the generalized Pareto distribution was set as 5 \(\times\) 10\({}^{4}\). The experimental results were obtained by averaging the results of 10 repeated experiments. Figure 14a shows the variation curve of the absolute distance between the parameter estimation values obtained by the PSO algorithm and the HPSO algorithm and the actual values. Figure 12: The fitting effect chart of the SMSD test. From Figure 14a, it can be observed that the estimation error of the shape parameter \(a\) increases with the increase in the simulated data parameter \(a\), while the estimation error of the scale parameter undergoes a trend of decreasing and then increasing. In addition, the parameter estimation error of the PSO algorithm shows significant variations when the shape parameter \(a\) is small, indicating the poor robustness of the parameter estimation using the PSO algorithm. To fix the shape parameter at \(a=2.5\), we varied the scale parameter of the clutter samples from 0.1 to 10 with equal intervals, while keeping the length of the simulated data constant. Similarly, the variation curves of the parameter estimation values for both algorithms were plotted (Figure 14b). From Figure 14b, it can be observed that as the scale parameter increases, there is no significant change in the parameter estimation error of the shape parameter \(a\), while the parameter estimation error of the scale parameter \(b\) shows an increasing trend. The PSO algorithm still exhibits pronounced oscillations, indicating that the particles in the algorithm are prone to premature convergence, resulting in the population being trapped in local optimal positions. Based on the experimental results mentioned above, it can be concluded that the HPSO algorithm not only achieves higher parameter estimation accuracy compared to the PSO algorithm but also exhibits better stability in parameter estimation. Therefore, in this study, the HPSO algorithm  
Figure 14: Parameter estimation error curves of two different algorithms.