# \(K\)-Means Cloning: Adaptive Spherical \(K\)-Means Clustering
## 3 KMC Algorithm
The clusters presented in this dataset are highly overlapping and each of them consists of 50 data points. This dataset is shown in Figure 7c. 4. Separate 6: This dataset, used in [74], consists of 300 data points that are distributed over six different clusters with 50 points in each cluster. This dataset is shown in Figure 7d. 5. Separate 4: This dataset consists of 400 data points in three-dimensional space distributed over four hyper-spherical disjoint clusters where each cluster contains 100 data points, as shown in Figure 7e. It was used in [76]. 6. Iris: This dataset consists of 150 data points distributed over three clusters [77]. Each cluster has 50 points. This dataset represents different categories of irises characterized by four feature values. It has three classes Setosa, Versicolor and Virginica. It is known that two classes (Versicolor and Virginica) have a large amount of overlap, while the class Setosa is linearly separable from the other two. 7. Breast cancer: This Wisconsin Breast Cancer dataset consists of 683 sample points. Each pattern has nine features corresponding to clump thickness, cell size uniformity, cell shape uniformity, marginal adhesion, single epithelial cell size, bare nuclei, bland chromatin, normal nucleoli and mitoses. There are two categories in the data: malignant and benign. The two classes are known to be linearly separable. ### Parameter Setting and Tuning

The KMC algorithm was programmed in MATLAB and its parameters are set according to their standard setting or our preliminary experiments. The details of setting these parameter are given as follows. The cooling schedule consists of the initial temperature \(T_{max}\), the cooling function, the epoch length \(M\) and the stopping condition. We choose the value of \(T_{max}\) large enough to make the initial probability of accepting transition close to 1. We set the initial probability to 0.9. Then, \(T_{max}\) is calculated from the equation \(T_{max}=-\frac{f(x_{p+1})-f(x_{p})}{\ln(0.9)}\). The temperature is reduced from \(T=\lambda T\), where \(\lambda=0.9\) is a parameter called cooling ratio. Epoch length \(M\) is the number of trials allowed at each temperature and we set it to 8. Initial number of clusters is chosen between \(K_{max}=min(20,\sqrt{n})\), where \(n=\text{number of points}\), and \(K_{min}=2\). We have observed that the diversification and intensification procedure is more effective when setting \(\delta=(K_{max}-K_{min})/3\). The stopping condition is satisfied when a minimum allowed temperature \(T_{min}=min(0.001,0.001*T_{max})\) is hit. The parameter setting values are summarized in Table 2. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Definition** & **Value** \\ \hline \(T_{max}\) & Initial temperature & \(-\frac{\Delta f}{\ln(0.9)}\) \\ \(T_{min}\) & Final temperature & \(\min(0.001,0.001*T_{max})\) \\ \(\lambda\) & Cooling ratio & 0.9 \\ \(M\) & Annealing epoch length & 8 \\ \(K_{max}\) & Max. number of clusters & \(\min(20,\sqrt{n})\) \\ \(K_{min}\) & Min. number of clusters & 2 \\ \(\delta\) & Diversification parameter & \((K_{max}-K_{min})/3\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameter Setting. ### Numerical Comparisons

We first provide the results obtained over the aforementioned three well-chosen synthetic datasets Overlap 5, Separate 6 and Separate 4, shown in Figure 7 and the real dataset breast cancer used on [17]. The performance of KMC has been compared with: the MEPSO [78] algorithm based on a modified the Particle Swarm Optimization (PSO) algorithm with improved convergence properties, the fuzzy c-means (FCM) [79] algorithm, which is the most popular algorithm in the field of fuzzy clustering. The later algorithm is referred in literature as Fuzzy clustering with Variable length Genetic Algorithm (FVGA) [17]. Figure 7: Synthetic datasets used in comparison, (**a**) Separate 3, (**b**) Overlap 3, (**c**) Overlap 5, (**d**) Separate 6 and (**e**) Separate 4.