# Ellipsoidal K-Means: An Automatic Clustering Approach for Non-Uniform Data Distributions
## 2 Adaptive Ellipsoidal Clustering Technique
### The Proposed Clustering Objective Function  
Measurement of the effectiveness of clustering problems is a complex task. Consequently, identifying an objective function that can serve as a universally applicable measure for all datasets undergoing cluster analysis is a challenge. For this reason, various research efforts have focused on developing new measures or improving the effectiveness of existing measures that have been shown to be effective with specific types of data. In addition, attempts have been made to integrate multi-objective functions to further improve results.  
This article presents a new measure of cluster validity as an additional decision support criterion in cluster analysis. Called the ellipse silhouette function \((ESF)\), this measure generalizes the average silhouette width criterion [26], which has proven effective in identifying optimal clusters in spherical data, but has shown limitations with datasets containing elliptical clusters. _ESF_ is specifically designed to improve the performance of the original silhouette criterion when applied to datasets containing elliptical clusters. Previous efforts to improve the performance of the silhouette function are reviewed in [27].  
Although _SF_ has been successful in determining the optimal number of clusters for circular and spherical data, it encounters difficulties with datasets containing elliptical clusters. This limitation is due to the fact that the function relies on the calculation of average distances per point within each cluster. In the case of elliptical clusters, these mean distances tend to be higher, which can lead to negative silhouette values (_cv_), as shown in Equation (3).  
\[cv(i)=\frac{\beta(i)-\alpha(i)}{max(\alpha(i),\beta(i))}, \tag{3}\]  
where \(\alpha(i)\) represents the average dissimilarity of point \(i\) with all other data points within the same cluster, while \(\beta(i)\) denotes the minimum average dissimilarity of point \(i\) with data points in another cluster. This lack of prioritization for elliptical clusters highlighted the need for an improved SF.  
For example, we generated a two-dimensional dataset with two clusters: one circular and the other ellipsoidal, as shown in Figure 1a. The application of the silhouette function produced three clusters (Figure 1b), while the actual number of clusters was two.  
To solve this problem, a concept was proposed for transforming non-spherical clusters, in particular elliptical clusters, into hyperspherical clusters. This specialized approach is designed to adapt to the characteristics of the dataset, while preserving the structure and mechanism of _SF_, thus improving its performance. The strategy involves converting elliptical clusters into two-dimensional circular clusters, three-dimensional spherical clusters, or even higher-dimensional hyperspherical clusters, while keeping the cluster centers unchanged. However, the challenge lies in efficiently transforming elliptical clusters into circular or spherical clusters, which requires an understanding of ellipse basic geometry.