# Cluster-Centered Visualization Techniques for Fuzzy Clustering Results to Judge Single Clusters
## 1 Introduction  
Cluster analysis is often defined as the task of partitioning a data set into subsets--called clusters--such that data objects within the same cluster are similar and data objects from different clusters are dissimilar. The notion of clustering as a partitioning task is, for instance, explicitly stated in [1; 2] and implicitly in [3] where data objects are assumed to \"fall into relatively distinct groups\". In real applications, data sets might not satisfy the idealized assumption of a partition in the form of almost perfect clusters. In many cases, cluster analysis is \"misused\" to identify some interesting clusters without the intention of partitioning the whole data set into more or less well-separated clusters.  
Sometimes it is not necessary to identify all existing patterns or groups with a clustering algorithm within a data set. Instead, the focus is on certain specific clusters that are important for analyzing or understanding a particular problem. Therefore, individual clusters are the focus of the analyzed data sets in order to understand specific patterns or trends within them. This is particularly relevant when different clusters have different characteristics. In addition, too many clusters can also make interpretation more difficult. If there are only a few well-defined clusters, it is often easier to gain insights and interpret them. The examples in this paper comprise artificial data sets, the Iris data set, a medical data set (Hepatitis-C-Virus (HCV)), and a production data set from a sugar factory. A single cluster within the HCV data set could be more interesting if, for instance, only the severely diseased patients were analyzed in more detail.  
In order to judge whether single clusters are \"meaningful\", special methods are required. Cluster validity indices are designed to evaluate a clustering result as a whole [4], but are usually not meant for judging single clusters. We also believe that abstract indicesand measures are an important issue in cluster analysis, and they add a certain objective perspective to the judgment of clustering results. Nevertheless, they usually summarize a complex clustering result into a single number, leading to a very strong loss of information.  
Visualization techniques are less objective but can provide much more information on clustering results than a single number provided by a cluster validity measure. Dimension reduction techniques like principal component analysis, multidimensional scaling, or more advanced techniques like t-SNE or UMAP [5; 6] can help to identify clusters in multidimensional data visually or to check whether a clustering result might make sense. Like cluster validity measures, dimension reduction techniques focus on lower-dimensional representations of the data set as a whole and do not focus on specific parts or clusters within the data set. In addition, when cluster analysis techniques are used that are based on modified or individual distance measures for each cluster, like the Gustafson-Kessel algorithm [7] or Gaussian mixture models, dimension reduction techniques do not take the individual local scaling for each cluster into account and might lead to misleading representations.  
In this paper, we focus on visualization techniques that focus on single clusters, i.e., the visualizations do not aim to provide a global representation of all clusters but show the data from the perspective of a selected cluster, making it possible to judge whether a specific cluster is meaningful or not.  
The paper is divided into six sections. After the introduction, Section 2 presents the basic principles of clustering methods. In Section 3, cluster validity and visualization techniques are briefly reviewed. The cluster-centered visualization techniques are described in detail and compared to other approaches in Section 4. Then, the visualization techniques are validated by different examples in Section 5, before Section 6 briefly concludes.