## 5 Conclusions and Future Directions  
In this study, an improved hybrid ISOSK-means metaheuristic algorithm is presented. Several improvements were incorporated into each of the two classical algorithms combined in the hybridization. In the initialization phase of the SOS algorithm, the population size was constructed using \(40+2g\) for a well-distributed initial population of solutions that scaled well with the data size. This resulted in a convergence rate increase and lower computational time. For performance upgrade, a three-part mutualism phase, with a random weighted reflection coefficient, was also integrated into the SOS algorithm with a random probability for determining whether an organism would be engaged in the three-part mutualism. To improve the quality of the clustering solution, the benefit factor was modified, by incorporating consideration for fitness value relativity with respect to the maximum fitness value. The misleading effects of outliers in the dataset were addressed by the improvement incorporated into the K-means phase of the improved hybrid algorithm. A method for detecting and excluding putative outliers during the centroid update phase of the classical algorithm was added. The algorithm uses a point-to-centroid distance threshold for the centroid update, instead of using the means of data points. The point-to-centroid distance threshold uses the median absolute deviation, which is considered to be a robust measure of statistical dispersion and is known to be more resilient to outliers. This ensured that outliers were excluded from contributing to minimizing the mean square error in the K-means. This resulted in a more compact cluster output. The improved hybrid algorithm was evaluated on 42 datasets (18 synthetic and 24 real-life) with varying characteristics, such as being high dimensional datasets, low dimensional datasets, synthetically generated datasets, image datasets, shape datasets, and location datasets, with varied dataset sizes and clusters. The performance of the improved \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Cluster Validity** & \multicolumn{3}{c}{**DB Index**} & \multicolumn{3}{c}{**CS Index**} \\ \hline
**Datasets** & **ISOSK-Means** & **ISOSK-Means** & **ISOSK-Means** & **ISOSK-Means** & **ISOSK-Means** & **ISOSK-Means** \\  & **vs. SOS** & **vs. K-Means** & **vs. SOSK-Means** & **vs. SOS** & **vs. K-Means** & **vs. SOSK-Means** \\ \hline Breast & 0.0000 & 1.0000 & 1.0000 & 0.0000 & 0.0000 & 0.0010 \\ Compound & 0.0000 & 0.0000 & 0.0000 & 0.0010 & 0.0000 & 0.0430 \\ Flame & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.0000 \\ Glass & 0.0000 & 0.0000 & 0.0740 & 0.0000 & 0.0000 & 1.0000 \\ Iris & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ Jain & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 1.0000 \\ Pathbased & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ Spiral & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0040 \\ Thyroid & 0.0000 & 0.0000 & 0.0000 & 0.3170 & 0.0000 & 1.0000 \\ Two moons & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ Wine & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ Yeast & 0.0000 & 0.0000 & 0.0000 & 0.2760 & 0.0000 & 0.0000 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Wilcoxon signed-rank test for ISOSK-means, SOSK-means, SOS, and K-means. hybrid algorithm was compared with the standard hybrid and non-hybrid algorithms. The performance was also compared with the two standard algorithms SOS and K-means on 12 real life datasets. A Friedman means rank test was applied to analyze the significant difference between the ISOSK-means and the competing algorithms. A pairwise post hoc Wilcoxon signed-rank sum test was also performed to highlight the performance of the ISOSK-means in comparison with the competing algorithms. The ISOSK-means algorithm outperformed the three algorithms with lower computational time and higher convergence rate, as reflected in the convergence curve for the competing algorithms.