# Gaslike Social Motility: Optimization Algorithm with Application in Image Thresholding Segmentation
## 3 GSM Algorithm Performance Results
#### Wilcoxon Sing Rank Test  
The statistical results presented in Tables 5 and 6 do not provide sufficient evidence to determine whether there are significant differences between the algorithms. To address this, a pairwise statistical test is often employed for more robust comparisons. In this study, the Wilcoxon Signed-Rank Test is applied to the results obtained from 100 runs of each algorithm to assess their relative efficiency across the set of benchmark functions. This statistical technique is widely used for comparing paired samples in optimization studies [1]. The test assumes \(n\) experimental data sets, each with two observations, \(a_{i}\) and \(b_{i}\), corresponding to the performance of two algorithms on function \(i\). This results in two paired samples: \(a_{1},\ldots,a_{n}\) and \(b_{1},\ldots,b_{n}\). The \(T\)-statistic is calculated as the sum of the negative ranks derived from the differences \(z_{i}=b_{i}-a_{i}\) for all \(i=1,\ldots,n\). The null hypothesis for this test is defined as follows:  
\(H_{0}\): There is no significant difference between the mean solutions produced by algorithm \(A\) and algorithm \(B\) for the set of benchmark functions. The ranks provided by the Wilcoxon Signed-Rank Test, denoted as \(T+\) and \(T-\), are then examined to evaluate this hypothesis, as described in [1]. The critical values for the \(T\)-statistic are evaluated using a normal distribution \(Z\). A 95% significance level (\(\alpha=0.05\)) is applied, with \(n=22\) (the number of benchmark functions). Table 8 presents the pairwise statistical results comparing the GSM algorithm to the other algorithms. The GSM algorithm demonstrates superior performance compared to the PSO, DE, BA, ABC, AHA, AHA-AO, CBO, ECBO, and SNS algorithms. In all pairwise comparisons, the null hypothesis is rejected, indicating significant differences in performance. Specifically, the PSO algorithm underperforms relative to the GSM algorithm. While the DE algorithm is highly competitive, the statistical test results confirm that GSM outperforms it. The BA algorithm, a powerful swarm-based approach, has been successfully applied to various problems. However, its performance falls short when compared to the GSM algorithm. As  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**No** & \begin{tabular}{c} **GSM vs.** \\ **PSO** \\ \end{tabular}  & \begin{tabular}{c} **GSM vs.** \\ **DE** \\ \end{tabular}  & \begin{tabular}{c} **GSM vs.** \\ **BA** \\ \end{tabular}  & \begin{tabular}{c} **GSM vs.** \\ **ABC** \\ \end{tabular}  & \begin{tabular}{c} **GSM vs.** \\ **AHA** \\ \end{tabular}  & \begin{tabular}{c} **GSM vs.** \\ **CBO** \\ \end{tabular}  & \begin{tabular}{c} **GSM vs.** \\ **ECBO** \\ \end{tabular}  & \ \begin{tabular}{c} **GSM vs.** \\ **AHA-AO** \\ \end{tabular}  & \begin{tabular}{c} **GSM vs.** \\ **SNS** \\ \end{tabular}  \\ \hline \(F_{1}\) & + & + & + & + & + & + & + & + & + \\ \hline \(F_{2}\) & + & + & + & + & + & + & + & + \\ \hline \(F_{3}\) & + & + & + & + & \(-\) & + & + & +  
shown in Table 8, the key difference between the GSM and ABC algorithms lies in their exploration capabilities. The ABC algorithm struggles with certain functions, resulting in poorer performance. The AHA algorithm, another swarm-based approach, delivers excellent results and excels in optimizing smooth functions, often finding the optimum quickly. However, it faces challenges with multimodal functions, where the GSM algorithm achieves better results. Swarm-based algorithms CBO and ECBO exhibit lower performance in most problems, even underperforming compared to older algorithms. The AHA-AO algorithm, an improved version of AHA, delivers results very similar to its predecessor. It excels in optimizing smooth functions and performs slightly better in multimodal functions. The SNS algorithm, a recent swarm-based approach, demonstrates strong performance across various test functions.