# \(K\)-Means Cloning: Adaptive Spherical \(K\)-Means Clustering
## 3 KMC Algorithm
Then, we provide the results obtained over four synthetic datasets Separate 3, Overlap 3, Overlap 5 and Separate 6, shown in Figure 7, and iris dataset. The first two datasets are produced by sampling from normal distributions with a cluster generator, and the other two datasets used in previous performance shown in Figure 7c,d. We used ten instances of each of the dataset Overlap 3 in our evaluation and multiple runs were distributed evenly across the instances. We compare the performance of KMC with a cooperative co-evolutionary algorithm (CCEA) and a variable string-length GA (VGA) [17; 80; 81]. ## 5 Results and Discussions

The clustering results of the KMC algorithm are shown in Tables 3 and 4. Two major indicators of performance can be found in these tables: the average computed number of clusters and the average classification error, accompanied by the corresponding standard deviations. In most real datasets the number of clusters is not priori known. So, to illustrate the potential of KMC in accurate estimation of the number of clusters in non-obvious datasets, KMC was applied to ten well-known real datasets [75]. In order to test the performance of the proposed method in higher dimensional datasets, two datasets called \"Hill-Valley\" and \"LSVT\" are used [75]. The details of these datasets are given in Table 6. The KMC method could obtained the exact number of cluster for both datasets as stated in Table 6. Figure 8 shows the performance of the KMC method in improving the misclassification errors for these datasets. The average misclassification errors for Hill-Valley and LSVT datasets are 14.95% and 27.98%, respectively. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Data sets** & **Algorithm** & **Clusters** & **Misclass** \\ \hline Separate 3 & KMC & 3.00 \(\pm\) 0.00 & 0.00 \(\pm\) 0.00 \\  & CCEA & 3.00 \(\pm\) 0.00 & 0.00 \(\pm\) 0.00 \\  & VGA & 3.06 \(\pm\) 0.07 & 0.34 \(\pm\) 0.40 \\ \hline Overlap 3 & KMC & 3.00 \(\pm\) 0.00 & 8.624 \(\pm\) 0.41 \\  & CCEA & 3.48 \(\pm\) 0.21 & 16.22 \(\pm\) 5.21 \\  & VGA & 4.22 \(\pm\) 0.28 & 26.56 \(\pm\) 5.11 \\ \hline Overlap 5 & KMC & 5.00 \(\pm\) 0.00 & 12.68 \(\pm\) 2.55 \\  & CCEA & 5.00 \(\pm\) 0.00 & 9.90 \(\pm\) 0.62 \\  & VGA & 5.16 \(\pm\) 0.13 & 16.82 \(\pm\) 3.78 \\ \hline Separate 6 & KMC & 6.00 \(\pm\) 0.00 & 0.00 \(\pm\) 0.00 \\  & CCEA & 6.00 \(\pm\) 0.00 & 0.30 \(\pm\) 0.13 \\  & VGA & 6.12 \(\pm\) 0.11 & 3.08 \(\pm\) 2.52 \\ \hline Iris & KMC & 2.00 \(\pm\) 0.00 & 50.00 \(\pm\) 0.00 \\  & CCEA & 3.00 \(\pm\) 0.00 & 27.96 \(\pm\) 2.39 \\  & VGA & 3.00 \(\pm\) 0.00 & 16.76 \(\pm\) 0.12 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Performance of FCM, VGA and CCEA. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Name** & **Points** & **Dimensions** & **Actual Clusters** & **Computed Clusters** \\ \hline LiverDisorder & 345 & 6 & 2 & 2 \\ Heart & 297 & 13 & 2 & 2 \\ Sonar & 208 & 60 & 2 & 2 \\ Wine & 178 & 13 & 3 & 3 \\ Diabetes & 768 & 8 & 2 & 2 \\ Voting & 435 & 16 & 2 & 2 \\ Ionosphere & 351 & 34 & 2 & 2 \\ Bavaria1 & 89 & 3 & 2 & 2 \\ Bavaria2 & 89 & 4 & 2 & 2 \\ German\_towns & 59 & 2 & 2 & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Results of clustering in real world datasets with known classes. ## 6 Conclusions

In this paper, we have introduced \(K\)-Means Cloning (KMC) as an adaptive clustering approach that is based on \(K\)-means clustering. It is a reliable evolving clustering technique that comprises two novel merging and splitting procedures. KMC uses meta-heuristics global search that is based on the silhouette function. Experiments on synthetic and real datasets proved the potential of KMC in accurate estimation of the number of clusters and as an adaptive clustering method. Conceptualization, A.-R.H., A.-M.I., A.E.A.-H., A.A.S. ; Methodology, A.-R.H., A.-M.I., A.E.A.-H., A.A.S. ; Programming & Coding, A.-R.H., A.-M.I. ; Writing-original Draft Preparation, A.-R.H., A.-M.I. ; Writing-review & Editing, A.-R.H., A.E.A.-H., A.A.S. ; Visualization, A.-R.H., A.E.A.-H.; Funding Acquisition, A.E.A.-H.