# Ellipsoidal K-Means: An Automatic Clustering Approach for Non-Uniform Data Distributions
## 4 Experimental Study
### Results and Technical Discussion
GCUK clustering was limited to datasets containing hyperspherical clusters when identifying the correct number of clusters. The KMC clustering technique was added to the comparison between the pure silhouette function measure and a newly modified silhouette width criterion ESF-NN based on the ellipsoidal clustering technique (ESF). KMC accurately identified the correct number of clusters in all eight artificial datasets used in the experiment, as these datasets contain hyperspherical clusters. Among the real-life datasets, only for the Cancer and LiverDisorder datasets was the method able to determine the appropriate number of clusters. However, it was unable to determine the correct number of clusters for the remaining four real-world datasets. This can be considered a real-life comparison between _SF_and ESF-NN to confirm the extent of the ability of ESF-NN to provide the appropriate number of clusters. Finally, the results show that the ability of ESF-NN is better than that of SF to achieve the appropriate clusters. Table 2 shows that the PESA-II-based clustering technique successfully identifies the correct partitioning for a single artificial dataset. SSym-PESAII is capable of detecting the appropriate number of clusters from the six real-life datasets. Figure 9: Automatically clustered Mixed_3_2 for \(K=3\) and Sym_3_2 for \(K=3\) after application of SAELLC clustering technique; (**a,c**) illustrate transformed spherical clusters and (**d,b**) show points belonging to each cluster. Overall, the results demonstrate that the proposed SAELLC clustering method outperforms the six compared automatic clustering techniques and several hybrid approaches in accurately identifying the optimal number of clusters across diverse datasets (see Figures 8-11). Notably, Figure 9 presents a specific case in which the identified clusters follow a distinct geometric pattern that does not align with the spatial clustering criterion employed in this study. Since our method currently focuses on a spatial or regional clustering basis, some points were not assigned to their expected clusters according to other potential clustering criteria. However, the proposed framework is inherently flexible and can be adapted to incorporate pattern-based or other clustering criteria beyond the spatial domain. This flexibility highlights more directions of future research, particularly in extending the method to different clustering paradigms. A solution's quality in light of the actual clustering is gauged by its Minkowski Score [19; 47]. Let \(S\) be the solution we want to measure and \(T\) be the \"true\" solution. The number of element pairs that are in the same cluster in both \(S\) and \(T\) is indicated by \(n_{11}\). The number of pairs in the same cluster in \(S\), but in different ones in \(T\), is shown by \(n_{01}\), while the number of pairs in the same cluster in \(T\), but in different ones in \(S\), is indicated by \(n_{10}\). The following equation is then used to define the Minkowski Score. \[MS(T,S)=\sqrt{\frac{n_{01}+n_{10}}{n_{11}+n_{10}}}. \tag{22}\]  
In this instance, a score of 0 is ideal, with lower numbers being \"better\". Figure 10: Automatically clustered Ellip_2_2 for \(K=2\) and Square1 for \(K=4\) after application of SAELLC clustering technique; (**a**,**c**) illustrate transformed spherical clusters and (**b**,**d**) show points belonging to each cluster. Table 3 proves the superiority of the proposed SAELLC in obtaining high-quality clusters for all the dataset used in the experimentation. This remarkable improvement is due to the fact that the input data clusters usually lie in two categories: spherical or non-spherical. By nature, ellipsoidal clustering captures both patterns in a better way than spherical-based clustering. Figure 11: Automatically clustered Square4 for \(K=4\) and Sizes5 for \(K=4\) after application of SAELLC clustering technique; (**a,c**) illustrate transformed spherical clusters and (**b,d**) show points belonging to each cluster.