# Ellipsoidal K-Means: An Automatic Clustering Approach for Non-Uniform Data Distributions
## 4 Experimental Study
### Results and Technical Discussion  
To evaluate the effectiveness of the proposed approach in determining optimal partition numbers and suitable partitioning from datasets, we applied it to eight artificial datasets and six datasets sourced from the UCI Machine Learning Repository (UCI) [43]. In [19], in addition to the SAELLC algorithm, other clustering techniques were included that fell into groups of single-objective and multiobjective optimization techniques. For each of those groups, the selected techniques were as follows: a recently developed multi-objective optimization method based on simulated annealing combined with a stability-induced symmetry-based clustering approach (SSym-AMOSA) [19]; another multi-objective clustering method known as MOCK [21]; a single-objective automatic clustering approach, VGAPS clustering [22]; an optimization method for a newly introduced symmetry-based cluster validity index, GCUK clustering [23]; and a proposed clustering algorithm using the PESA-II optimization technique, SSym-PESAII [19]; as well as the pure silhouette width criterion used in the simulated annealing based on the \(K\)-means algorithm [17; 46]. The results on the eight artificial and six real-life datasets showed that SAELLC performed the best in detecting the appropriate partitioning in all cases except one. The number of clusters automatically determined by these clustering techniques (VGAPS, MOCK, GCUK, KMC, SSym-PESAII, SSym-AMOSA) for the fourteen datasets are reported in Table 2.  
Figure 7: Other four artificial datasets used in comparison. It can also be seen from the results in Table 2 that the proposed SAELLC algorithm was capable of finding the appropriate clustering. Figures 8-11 show that SSym-AMOSA and SSym-PESAII succeeded in most cases, while VGAPS, MOCK, and KMC failed. The results on the real-life Newthyroid, Lungcancer, and Wine data showed that SAELLC was able to detect symmetric clusters irrespective of their densities, while the densities (nested overlapping) of Data_5_2 stood in the way of determining the actual clusters. Figure 8a,b show that the partitions identified by the best solution of SAELLC had four clusters. MOCK [21] was capable of detecting the appropriate number of clusters for four of these 14 datasets. MOCK usually failed for datasets that had symmetric overlapping clusters. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Datasets** & **AC** & **MOCK** & **VGAPS** & **GCUK** & **KMC** & \(\textsf{SSym}_{PESAII}\) & \(\textsf{SSym}_{AMOSA}\) & **SAELLC** \\ \hline Data\_5\_2 & 5 & 6 & 5 & 5 & 5 & 5 & 4 \\ Data\_10\_2 & 10 & 6 & 7 & 10 & 10 & 13 & 10 & 10 \\ Mixed\_3\_2 & 3 & 2 & 3 & 6 & 2 & 6 & 3 & 3 \\ Sym\_3\_2 & 3 & 2 & 2 & 7 & 3 & 3 & 3 & 3 \\ Ellip\_2\_2 & 2 & 4 & 2 & 3 & 3 & 7 & 2 & 2 \\ Square1 & 4 & 4 & 4 & 5 & 4 & 9 & 4 & 4 \\ Square4 & 4 & 4 & 5 & 4 & 4 & 7 & 4 & 4 \\ Sizes5 & 4 & 2 & 5 & 2 & 4 & 6 & 4 & 4 \\ Iris & 3 & 2 & 3 & 3 & 2 & 3 & 3 & 3 \\ Cancer & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\ Newthyroid & 3 & 2 & 3 & 7 & 2 & 3 & 2 & 3 \\ Lungcancer & 3 & 7 & 2 & 2 & 2 & 3 & 5 & 3 \\ Wine & 3 & 3 & 6 & 9 & 3 & 3 & 6 & 3 \\ LiverDisorder & 2 & 3 & 2 & 3 & 2 & 2 & 2 \\ \hline \end{tabular}
\end{table} 
Table 2: Comparison of SAELLC with seven clustering technique indices for determining number of clusters for different artificial and real-world datasets  
Figure 8: Automatically clustered Data_5_2 for \(K=4\) and Data_10_2 for \(K=10\) after application of SAELLC clustering technique; (**a,c**) illustrate transformed spherical clusters and (**b,d**) show points belonging to each cluster. For real-world datasets, VGAPS clustering successfully identified the correct number of clusters in four out of six cases, with the exception of the Newthyroid dataset. However, it did not perform well on three artificial datasets: Data_10_2, Square1, and Sizes5. On the other hand, the GCUK clustering optimization technique accurately determined the number of clusters in only three of the eight artificial datasets tested.