## 4 Performance Evaluation of Improved SOSK-Means
### Experiment 1
Under the DB validity index, the ISOSK-means recorded a shorter average execution time than the SOSK-means in nine datasets. This indicated that the ISOSK-means required less execution time to achieve convergence. Under the CS validity index, the ISOSK-means recorded lower execution time in six datasets. Out of the remaining six, only three datasets showed a considerable difference in execution time for the two algorithms in favor of the SOSK-means. Since the CS index usually requires higher computational time, this result showed that the ISOSK-means substantially outperformed SOSK-means in terms of computational cost. \begin{table}
\begin{tabular}{l l l l l l l l l} \hline \hline \multirow{2}{*}{**Datasets**} & \multirow{2}{*}{
\begin{tabular}{c} **Statistical** \\ **Measure** \\ \end{tabular} } & \multicolumn{2}{c}{**DB Index**} & \multicolumn{4}{c}{**CS Index**} \\  & & **SOS** & **K-Means** & **ISOSK-Means** & **SOS** & **K-Means** & **SOSK-Means** \\ \hline
**Breast** & Mean & 1.3520 & 0.8121 & 0.8121 & 0.8121 & 0.9946 & 1.1019 & 0.7606 & **0.6639** \\
**Compound** & Std Dev & 0.2858 & 0.0000 & 0.0000 & 0.0000 & 0.2667 & 0.0000 & 0.1217 & 0.0391 \\
**Compound** & Mean & 0.6924 & 0.9716 & 0.5046 & **0.4985** & 0.5670 & 1.2887 & 0.5072 & **0.5031** \\  & Std Dev & 0.1481 & 0.0748 & 0.0044 & 0.0021 & 0.1225 & 0.1486 & 0.0155 & 0.0006 \\
**Flame** & Mean & 0.8234 & 1.2306 & 0.7770 & **0.7760** & 1.2707 & 1.5806 & 0.3846 & 0.3846 \\  & Std Dev & 0.0180 & 0.0059 & 0.0008 & 0.0006 & 0.1006 & 0.0263 & 0.0000 & 0.0000 \\
**Glass** & Mean & 0.8164 & 1.2208 & 0.7113 & **0.6547** & 0.2200 & 1.4894 & 0.0608 & 0.0608 \\  & Std Dev & 0.1174 & 0.1570 & 0.1217 & 0.1727 & 0.2563 & 0.1904 & 0.0000 & 0.0000 \\
**Iris** & Mean & 0.8602 & 0.9167 & 0.6346 & **0.6165** & 0.8585 & 1.2404 & 0.5743 & **0.5510** \\  & Std Dev & 0.1809 & 0.0033 & 0.0188 & 0.0132 & 0.1922 & 0.0092 & 0.0237 & 0.0118 \\
**Jain** & Mean & 0.7007 & 0.8587 & 0.6518 & **0.6507** & 0.8196 & 1.0668 & 0.6546 & 0.6546 \\  & Std Dev & 0.0274 & 0.0001 & 0.0009 & 0.0008 & 0.0212 & 0.0003 & 0.0000 & 0.0000 \\
**Path-based** & Mean & 0.7578 & 0.7696 & 0.6708 & **0.6669** & 1.0021 & 0.9893 & 0.6511 & **0.6289** \\  & Std Dev & 0.0866 & 0.0066 & 0.0031 & 0.0050 & 0.1708 & 0.0086 & 0.0120 & 0.0299 \\
**Spiral** & Mean & 0.8013 & 0.9589 & 0.7437 & **0.7386** & 1.0818 & 1.1896 & 0.6812 & **0.6618** \\  & Std Dev & 0.0447 & 0.0109 & 0.0045 & 0.0033 & 0.2107 & 0.0053 & 0.0115 & 0.0329 \\
**Thyroid** & Mean & 1.0232 & 1.0298 & 0.6321 & **0.6039** & 0.6446 & 1.7863 & 0.6409 & 0.6409 \\  & Std Dev & 0.1479 & 0.2042 & 0.0316 & 0.0182 & 0.0238 & 0.3602 & 0.0000 & 0.0000 \\
**Two-moons** & Mean & 0.6128 & 0.7948 & 0.6032 & **0.6021** & 0.7701 & 0.9385 & 0.7498 & **0.7243** \\  & Std Dev & 0.0179 & 0.0000 & 0.0010 & 0.0006 & 0.0281 & 0.0000 & 0.0162 & 0.0113 \\
**Wine** & Mean & 1.1488 & 1.3053 & 1.0460 & **1.0053** & 1.1938 & 1.4425 & 0.8422 & **0.7614** \\  & Std Dev & 0.1394 & 0.0022 & 0.0207 & 0.0228 & 0.3318 & 0.0128 & 0.0527 & 0.0571 \\
**Yeast** & Mean & 1.2144 & 1.7176 & 0.8496 & **0.8019** & 0.5594 & 2.6417 & 0.5242 & **0.4777** \\  & Std Dev & 0.2911 & 0.1875 & 0.1588 & 0.1407 & 0.2847 & 0.5950 & 0.0437 & 0.0252 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Computation results for the four competing algorithms on 12 real-life datasets. Figure 4: Performance analysis of four competing algorithms, ISOSK-means, SOSK-means, SOS and K-means, under the DB index. Figure 5: Performance analysis of four competing algorithms, ISOSK-means, SOSK-means, SOS and K-means, under the CS index. Figure 6: Analysis of computational times of SOS, SOSK-means and ISOSK-means under the DB index. The convergence curve for the three algorithms, ISOSK-means, SOSK-means and SOS, are illustrated in Figures 8 and 9 for the DB index and CS index, respectively. A rapid and smooth declining curve reflected a superior performance. In all 12 datasets, the ISOSK-means recorded the best performance with rapidly declining smooth curves under the two validity indices. Figure 8: _Cont_.