# Cross-Correlation Wavelet-Domain-Based Particle Swarm Optimization for Lightning Mapping
## 3 Results and Discussions  
### Optimal Selection of PSO Parameters Using Simulated Data  
The proposed PSO was applied to four sets of iterations: 100, 200, 300, and 500. To ensure that the comparisons are fair, the population size for all optimization iterations was set to 10. The compared optimization iterations had several parameters settings, as listed in Table 1. To determine the reliability, strength, and efficiency of PSO in obtaining the global minimum value of the four iterations, the objective function was found by averaging the Euclidean distance of the simulated and estimated lightning map. Each iteration set was tested 20 times. The results that represent the best objective function are indicated in Table 2. In each optimization iteration, the window size for each simulated lightning signal was tuned, and the optimal value (fitness value) was calculated. The number of PSO iterations varied from one problem to another depending on the data being processed. In order to observe its influence on the performance of the PSO-based searching algorithm, this study selected a maximum iteration number of 500 since the performance was not improved beyond this number. Selecting a suitable iteration number can help to extend the search behavior to find the best global minimum and search for the best position. The best result for each iteration set is written in boldface in Table 2. Extending the search behavior by increasing the number of iterations up to 500 showed no significant change in minimizing the fitness value. From Table 2, the PSO reaches the best global minimum value at 100, 300, and 500 iterations with 0.6243, whereas the iteration at 200 reaches a near-global minimum value of 0.6328. The performance of PSO with 300 iterations is comparatively satisfactory in most of the tested optimization iterations. To illustrate this, Figure 4 depicts the PSO-convergence performance curves for lightning mapping estimation. The best optimal fitness value of the proposed algorithm is the one that exists at the lowest convergence time, which is achieved at optimization with 300 iterations. Hence, the optimization with 300 iterations was chosen for accurately estimating the lightning maps. \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameters** & **Description** \\ \hline \(W_{max}\) & 0.9 \\ \(W_{min}\) & 0.4 \\ Acceleration coefficients \(c_{1}\) and \(c_{2}\) & 2 \\ Initial window size (\(Ws\)) & \(W_{super}=512\), \(W_{linear}=100\), initial \(Ws=256\) \\ No of population & 2 \\ Number of iterations & 100, 200, 300, and 500 \\ Cross-correlation wavelet domain & CCWD \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the IPSO-WT based techniques. \begin{table}
\begin{tabular}{l l c c c c c c c c} \hline \hline \multirow{3}{*}{**Parameters**} & \multirow{3}{*}{
\begin{tabular}{c} **Sampling** \\ **Ratio** \\ \end{tabular} } & \multicolumn{6}{c}{**Iteration Sets**} \\ \cline{3-10}  & & \multicolumn{2}{c}{**100 Iteration**} & \multicolumn{2}{c}{**200 Iteration**} & \multicolumn{2}{c}{**300 Iteration**} & \multicolumn{2}{c}{**500 Iteration**} \\ \cline{3-10}  & & **Linear** & **Cubic** & **Linear** & **Cubic** & **Linear** & **Cubic** & **Linear** & **Cubic** \\ \hline Window size (Ws) & 4 & 282.71 & 446.33 & 290.41 & 446 & 282.55 & 425.56 & 282.46 & 419.72 \\  & 8 & 457.50 & 446.82 & 544.86 & 451.55 & 466.08 & 452.43 & 466.63 & 451.24 \\ \hline Objective function & 4 & 0.6243 & 0.7702 & 0.6328 & 0.7944 & **0.6243** & 0.7913 & 0.6243 & 0.7923 \\  & 8 & 0.7829 & 0.7271 & 0.7940 & 0.7273 & 0.7836 & 0.7325 & 0.7836 & 0.7273 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The performance of CCWD-PSO technique in terms of Euclidean distance. The comparison involves several cross-correlation-related parameters. The sampling parameter indicates the interpolation ratio of the cross-correlation output signal. Two interpolation methods were used, i.e., linear and cubic. With a given captured lightning signal of antenna B, the signals of antennas C and D were simulated.