# Cluster-Centered Visualization Techniques for Fuzzy Clustering Results to Judge Single Clusters
## 6 Conclusions  
The use of fuzzy clustering can lead to complex and overlapping cluster structures. For further interpretation, not all meaningful clusters within a data set should necessarily  
Figure 16: Sugar production data set with four clusters and visualized results with the CPC (**left**), A2C (**middle**), and AM (**right**) methods.  
Figure 15: Sugar production data set with three clusters and visualized results with the CPC (**left**), A2C (**middle**), and AM (**right**) methods.  
be found, but the focus is more on individual clusters. The novelty of the paper is an assessment of individual clusters using the proposed visualizations to better assess single clusters and identify their characteristics. The developed visualization techniques focus on a single cluster, which is placed in the center, and possible further clusters around it. Therefore, the data can be considered from the perspective of a selected cluster, making it possible to judge whether a specific cluster is meaningful or not. This is important because global representation is not always helpful. The developed CCV techniques are applied to different datasets to demonstrate their ability to identify patterns and relationships in the data. For each dataset, two to four clusters were visualized using the three CCV techniques: CPC, A2C, and AM. The datasets were an artificially good and ambiguous dataset with a clear separation of the clusters, but also a more difficult separation of the clusters due to mixed data points. The Iris dataset showed a clear separation of the different numbers of clusters, as did the HCV dataset, despite mixed data points. The sugar production dataset showed a clear separation of the clusters up to a number of three, before a larger proportion of the data points were mixed with four clusters. The visualization techniques allowed a characterization of individual clusters with varying degrees of uncertainty in the assignments. This is particularly relevant when traditional hard clustering might be too restrictive.  
The results show that the presented techniques can add value to data analysis and the interpretation of meaningful clusters. Focusing on cluster centers and membership degrees leads to the identification of patterns in the datasets that may have been missed by traditional clustering visualizations. The results contribute to the understanding and possible applications of fuzzy clustering and its visualization in data analysis. The integration of more advanced machine learning methods and adaptation to specific application areas can lead to further improvements in visualization techniques in the future.  
The following supporting information can be downloaded at: [[https://www.mdpi.com/article/10.3390/app14031102/s1](https://www.mdpi.com/article/10.3390/app14031102/s1)]([https://www.mdpi.com/article/10.3390/app14031102/s1](https://www.mdpi.com/article/10.3390/app14031102/s1)), R-Code: Function of the CCV techniques.  
Conceptualization, K.V. and F.K.; methodology, F.K.; software, K.V. and F.K.; validation, K.V. and F.K.; formal analysis, K.V. and F.K.; investigation, K.V. and F.K.; resources, F.K.; data curation, K.V. and F.K.; writing--original draft preparation, K.V.; writing--review and editing, F.K.; visualization, K.V.; supervision, F.K.; project administration, K.V.; funding acquisition, F.K. All authors have read and agreed to the published version of the manuscript.  
This research was funded partly by the German Federal Ministry of Education and Research, grant number 68518.  
Data Availability Statement: Data are contained within the article.  
[PERSON] is employed by the company Nordzucker AG. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.