# Cluster-Centered Visualization Techniques for Fuzzy Clustering Results to Judge Single Clusters
## 2 Clustering Methods  
Hard clustering is a common method in cluster analysis in which data points are clearly and exclusively assigned to a specific cluster. Each data point is assigned to exactly one cluster, and there are no overlaps or uncertainties regarding membership. One example is the \(k\)-means algorithm, which groups the data points into \(k\) predefined clusters [8]. This clear assignment makes hard clustering particularly easy to interpret and understand. While hard clustering is effective in many applications, it proves to be disadvantageous when the data has naturally fuzzy boundaries between the clusters. The same applies if it is necessary to consider uncertainties in the cluster assignment. If a data point belongs to a specific cluster, it cannot be included in another cluster [9]. In such cases, the fuzzy clustering approach proves to be more advantageous because data points can be assigned to different clusters via membership degrees. Clustering algorithms are based on different \"philosophies\" concerning the assignment of data objects to clusters and how this assignment is computed. Based on hard clustering, methods like DBSCAN, and hierarchical or \(k\)-means clustering assign each data object to a single cluster--including a possible noise cluster, as in the case of DBSCAN. Gaussian mixture models approximate the distribution of the data by (multivariate) normal distributions, yielding for each data object and each cluster a probability for the data object to be generated by the normal distribution that represents the cluster. Fuzzy clustering uses membership degrees between 0 and 1 to assign data objects to clusters, and for so-called probabilistic fuzzy clustering, the membership degrees of one data object to all clusters sum up to one. Given \(n\) data objects and a suitable distance function \(d\), probabilistic fuzzy clustering tries to minimize the objective function. \[f=\sum_{i=1}^{c}\sum_{j=1}^{n}u_{ij}^{m}d_{ij} \tag{1}\]  
under the constraints  
\[\sum_{i=1}^{c}u_{ij}=1\ for\ all\ j=1,\ldots,n \tag{2}\]The parameter \(c\) is the chosen number of clusters, and the parameter \(m>1\) is the so-called fuzzifier, which controls how \"fuzzy\" the clusters tend to be. The larger \(m\) is to chosen, the more data objects are assigned to multiple clusters. The value \(d_{ij}\) denotes the distance the of data object \(j\) to the cluster \(i\). In the simplest case of fuzzy \(k\)-means clustering, this is the distance of the data object \(j\) to the center of the cluster \(i\). It was shown by [10] that  
\[u_{ij}=\frac{1}{\sum_{k=1}^{c}\left(\frac{d_{ij}}{d_{ij}}\right)^{\frac{1}{m-1}}} \tag{3}\]  
must hold to minimize Equation (1). Although the membership degrees sum up to 1 for each data object, one should be careful with an interpretation of such membership degrees as probabilities. Of course, for clustering algorithms that do not yield a unique assignment of data objects to clusters, the probabilities computed in mixture models or membership degrees in fuzzy clustering can be converted to unique assignments by simply choosing for each data object the cluster with the highest probability or membership degree. In an ideal case, a data set can be partitioned into clusters that are perfectly separated in the data space; however, this assumption does not often apply to real data sets. Fuzzy clustering can take ambiguities in the assignment of data objects into account, resulting in slightly higher computational costs. Our visualization techniques allow the identification of single good clusters among sets of more or less overlapping clusters. Among many other possible applications, the identification of symptom clusters in medicine might be an important scenario when a disease has ambiguous manifestations, like, for example, in the case of Post-COVID-19 [11]. Apart from the different data to cluster assignment \"philosophies\", clustering algorithms also differ in the principal input that is used.