## 4 Performance Evaluation of Improved SOSK-Means
### Statistical Analysis  
The statistical analysis experiment involved using a nonparametric statistical analysis technique to further validate the algorithms' computational results. The [PERSON]'s nonparametric test was conducted to draw a statistically verified meaningful conclusion from the reported performances. The reports for the computed [PERSON]'s mean rank on the four competing algorithms (the ISOSK-means, SOSK-means, SOS, and K-means) are presented in Table 11. The results showed that the ISOSK-means had the best performance, recording 1.13 as its minimum rank value and 2.00 as its maximum rank value, under the DB index, with a 1.15 minimum rank value and 1.99 maximum rank value under the CS index. \begin{table}
\begin{tabular}{l c c c c c c c c c c} \hline \hline
**High-** & \multicolumn{3}{c}{**ISOSK-Means**} & \multicolumn{1}{c}{**DE**} & \multicolumn{3}{c}{**PSO**} & \multicolumn{3}{c}{**FA**} & \multicolumn{3}{c}{**IWO**} \\
**Dimensional** & **Mean** & **Std Dev** & **Mean** & **Std Dev** & **Mean** & **Std Dev** & **Mean** & **Std Dev** & **Mean** & **Std Dev** \\ \hline
**A1** & **0.5911** & 0.0003 & 0.6016 & 0.0116 & 0.6662 & 0.0042 & 0.6089 & 0.0341 & 0.6308 & 0.0265 \\
**A2** & **0.6781** & 0.0002 & 0.7081 & 0.0348 & 0.7134 & 0.0284 & 0.6842 & 0.0143 & 0.7483 & 0.0468 \\
**A3** & 0.7945 & 0.0011 & 0.7349 & 0.0184 & 0.7279 & 0.0287 & **0.6844** & 0.0241 & 0.7545 & 0.0339 \\
**Birch 1** & 0.8030 & 0.0006 & 0.748 & 0.0171 & 0.7528 & 0.0169 & **0.7091** & 0.0203 & 0.7644 & 0.0211 \\
**Birch 2** & **0.5071** & 0.0001 & 0.5086 & 0.0016 & 0.5876 & 0.0358 & 0.5087 & 0.0078 & 0.5168 & 0.0052 \\
**Birch 3** & 0.7168 & 0.0004 & 0.7488 & 0.0297 & 0.7213 & 0.0306 & **0.6974** & 0.0127 & 0.7568 & 0.0248 \\
**D31** & 0.8125 & 0.0070 & 0.8757 & 0.0302 & **0.763** & 0.0514 & 0.7808 & 0.0436 & 0.7896 & 0.0312 \\
**Housec5** & **0.5377** & 0.0123 & 0.619 & 0.0412 & 0.7456 & 0.0679 & 0.5642 & 0.029 & 0.7034 & 0.043 \\
**Housec8** & 0.4919 & 0.5305 & 0.5245 & 0.0139 & 0.6418 & 0.0858 & **0.4584** & 0.0111 & 0.6206 & 0.0546 \\
**Leaves** & 0.7618 & 0.0745 & 1.1345 & 0.0798 & 1.0335 & 0.2854 & **0.6036** & 0.0587 & 1.5132 & 0.049 \\
**Letter** & 0.9683 & 1.0545 & 0.9852 & 0.0303 & 1.0354 & 0.1084 & **0.8071** & 0.0246 & 1.2245 & 0.0416 \\
**Average** & 0.6966 & 0.1529 & 0.7444 & 0.0281 & 0.7626 & 0.0676 & **0.6461** & 0.0255 & 0.8203 & 0.0343 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Performance comparison of ISOSK-means with other non-hybrid metaheuristic algorithms on high-dimensional datasets. \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline
**Datasets** & **SOS** & **K-Means** & **DSDSK-Means** & **ISOSK-Means** & **SOS** & **K-Means** & **SOSK-Means** & **SOSK-Means** & **ISOSK-Means** \\ \hline
**Breast** & 4 & 2 & 2 & 2 & 3.13 & 3.45 & 2.03 & 1.4 \\
**Compound** & 2.9 & 3.95 & 1.98 & 1.18 & 2.34 & 4 & 1.91 & 1.75 \\
**Flame** & 3 & 4 & 1.83 & 1.18 & 3 & 4 & 1.5 & 1.5 \\
**Glass** & 2.42 & 4 & 1.94 & 1.64 & 2.68 & 4 & 1.6 & 1.6 \\
**Iris** & 3.3 & 3.48 & 1.98 & 1.25 & 2.85 & 4 & 1.98 & 1.18 \\
**Jain** & 3 & 4 & 1.88 & 1.13 & 3 & 4 & 1.5 & 1.5 \\
**Path-based** & 3.18 & 3.45 & 2.08 & 1.3 & 3.63 & 3.23 & 1.94 & 1.21 \\
**Spiral** & 2.8 & 4 & 1.9 & 1.3 & 3.18 & 3.58 & 1.79 & 1.46 \\
**Thyroid** & 3.4 & 3.53 & 1.9 & 1.18 & 2.03 & 4 & 1.99 & 1.99 \\
**Two-moons** & 2.61 & 4 & 2.14 & 1.25 & 2.63 & 4 & 2.16 & 1.21 \\
**Wine** & 3 & 3.7 & 2.17 & 1.13 & 3.09 & 3.63 & 2.14 & 1.15 \\
**Yeast** & 3 & 3.88 & 1.73 & 1.4 & 1.63 & 4 & 2.69 & 1.69 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Friedman mean rank for ISOSK-means, SOSK-means, SOS, and K-means. For further verification of the specific significant differences among the competing algorithms, a post hoc test on the Friedman mean-rank test results, using the Wilcoxon signed-rank test, was conducted. The Wilcoxon signed-rank test presents a set of \(p\) values which statistically measures whether there is a significant difference between the competing algorithms at a significance level of 0.05.