# Simple K-Medoids Partitioning Algorithm for Mixed Variable Data
## 3 Demonstration on Artificial and Real Datasets
### Data Simulation
#### 3.1.4 Different Numbers of Objects  
The artificial data had variations in the number of objects. The datasets had 100, 500, 1000, and 2000 objects. Overall, both PAM and SKM had better results than SFKM (Table 7). They shared similar high Rand indices (\(\geq\)0.95) when the number of objects was 2000 in the Huang, Harikumar-PV, and Esimma GDFs. However, although SFKM had a high index in the Harikumar-PV GDF, it was significantly different from both the PAM and SKM algorithms. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(p\) & **Algorithm** & **[PERSON]** & **[PERSON]** & **[PERSON]** & **Huang** & **[PERSON]** & **Esimma** & **Marweco** \\ \hline \multirow{3}{*}{6} & PAM & 0.85 & 0.77 & 0.83 & 0.69 & 0.69 & 0.69 & 0.79 \\  & SFKM & 0.79 \({}^{a}\) & 0.75 \({}^{b}\) & 0.81 \({}^{a}\) & 0.70 \({}^{b}\) & 0.70 \({}^{c}\) & 0.70 & 0.77 \({}^{c}\) \\  & SKM & 0.84 & 0.76 & 0.84 & 0.69 & 0.69 & 0.69 & 0.79 \\ \hline \multirow{3}{*}{8} & PAM & 0.80 & 0.93 & 0.79 & 0.90 & 0.90 & 0.92 & 0.86 \\  & SFKM & 0.74 \({}^{a}\) & 0.86 \({}^{a}\) & 0.75 \({}^{a}\) & 0.88 \({}^{a}\) & 0.84 \({}^{a}\) & 0.88 \({}^{a}\) & 0.79 \({}^{a}\) \\  & SKM & 0.80 & 0.93 & 0.79 & 0.91 & 0.90 & 0.92 & 0.86 \\ \hline \multirow{3}{*}{10} & PAM & 0.74 & 0.77 & 0.73 & 0.76 & 0.76 & 0.76 & 0.76 \\  & SFKM & 0.71 \({}^{a}\) & 0.73 \({}^{a}\) & 0.71 \({}^{a}\) & 0.74 \({}^{a}\) & 0.73 \({}^{a}\) & 0.74 \({}^{a}\) & 0.70 \({}^{a}\) \\  & SKM & 0.74 & 0.76 & 0.73 & 0.75 & 0.76 & 0.76 & 0.75 \\ \hline \multirow{3}{*}{14} & PAM & 0.69 & 0.85 & 0.68 & 0.95 & 0.95 & 0.95 & 0.89 \\  & SFKM & 0.66 \({}^{a}\) & 0.78 \({}^{a}\) & 0.66 \({}^{a}\) & 0.85 \({}^{a}\) & 0.81 \({}^{a}\) & 0.86 \({}^{a}\) & 0.79 \({}^{a}\) \\ \cline{1-1}  & SKM & 0.69 & 0.85 & 0.67 & 0.95 & 0.95 & 0.95 & 0.89 \\ \hline \end{tabular}  
Significantly different (\(\alpha=0.05\)) from: \({}^{a}\) PAM and SKM, \({}^{b}\) PAM, \({}^{c}\) SKM.  
\end{table} 
Table 6: Rand indices of the different numbers of variables. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(k\) & **Algorithm** & **[PERSON]** & **[PERSON]** & **[PERSON]** & **[PERSON]** & **[PERSON]** & **[PERSON]** & **[PERSON]** \\ \hline \multirow{3}{*}{3} & PAM & 0.83 & 0.87 & 0.81 & 0.81 & 0.83 & 0.84 & 0.88 \\  & SFKM & 0.79 \({}^{a}\) & 0.83 \({}^{a}\) & 0.76 \({}^{a}\) & 0.79 \({}^{a}\) & 0.81 & 0.81 \({}^{a}\) & 0.84 \({}^{a}\) \\  & SKM & 0.83 & 0.87 & 0.81 & 0.79 & 0.81 & 0.84 & 0.88 \\ \hline \multirow{3}{*}{4} & PAM & 0.76 & 0.88 & 0.74 & 0.89 & 0.90 & 0.90 & 0.84 \\  & SFKM & 0.72 \({}^{a}\) & 0.84 \({}^{a}\) & 0.73 & 0.86 \({}^{a}\) & 0.86 \({}^{a}\) & 0.88 \({}^{a}\) & 0.80 \({}^{a}\) \\  & SKM & 0.77 & 0.88 & 0.74 & 0.89 & 0.90 & 0.90 & 0.84 \\ \hline \multirow{3}{*}{8} & PAM & 0.82 & 0.91 & 0.81 & 0.91 & 0.90 & 0.91 & 0.88 \\  & SFKM & 0.80 \({}^{a}\) & 0.87 \({}^{a}\) & 0.80 \({}^{a}\) & 0.89 \({}^{a}\) & 0.87 \({}^{a}\) & 0.89 \({}^{a}\) & 0.84 \({}^{a}\) \\  & SKM & 0.81 & 0.90 & 0.81 & 0.91 & 0.90 & 0.90 & 0.87 \\ \hline \multirow{3}{*}{10} & PAM & 0.86 & 0.92 & 0.85 & 0.91 & 0.91 & 0.91 & 0.89 \\  & SFKM & 0.84 \({}^{a}\) & 0.88 \({}^{a}\) & 0.85 \({}^{a}\) & 0.90 \({}^{a}\) & 0.88 \({}^{a}\) & 0.90 \({}^{a}\) & 0.85 \({}^{a}\) \\ \cline{1-1}  & SKM & 0.85 & 0.90 & 0.85 & 0.91 & 0.90 & 0.91 & 0.87 \\ \hline \end{tabular}  
Significantly different (\(\alpha=0.05\)) from: \({}^{a}\) PAM and SKM.  
\end{table} 
Table 5: Rand indices of the different numbers of clusters. Tables 4-7 show that the proposed k-medoids (SKM) had better performance than the evaluated algorithm (SFKM). In all settings, the Rand indices between the \"true cluster\" and SKM algorithm were higher than those between the \"true cluster\" and SFKM algorithm. On the other hand, from the running time perspective, theoretically, the SKM algorithm required more time than the SFKM algorithm due to repetition sets vs. a single set of initial medoids. By applying \(s\) times the multiple random seeding, the SKM finished a partitioning algorithm in a maximum of \(s\) multiplied by the required time of the SFKM.