Fundamentals of Fuzzy Clustering

Clustering is an unsupervised learning task that aims at decomposing a given set of objects into subgroups or clusters based on similarity. The goal is to divide the data-set in such a way that objects (or example cases) belonging to the same cluster are as similar as possible, whereas objects belonging to different clusters are as dissimilar as possible. The motivation for finding and building classes in this way can be manifold (Bock, 1974). Cluster analysis is primarily a tool for discovering previously hidden structure in a set of unordered objects. In this case one assumes that a ‘true’ or natural grouping exists in the data. However, the assignment of objects to the classes and the description of these classes are unknown. By arranging similar objects into clusters one tries to reconstruct the unknown structure in the hope that every cluster found represents an actual type or category of objects. Clustering methods can also be used for data reduction purposes. Then it is merely aiming at a simplified representation of the set of objects which allows for dealing with a manageable number of homogeneous groups instead of with a vast number of single objects. Only some mathematical criteria can decide on the composition of clusters when classifying data-sets automatically. Therefore clustering methods are endowed with distance functions that measure the dissimilarity of presented example cases, which is equivalent to measuring their similarity. As a result one yields a partition of the data-set into clusters regarding the chosen dissimilarity relation. All clustering methods that we consider in this chapter are partitioning algorithms. Given a positive integer c, they aim at finding the best partition of the data into c groups based on the given dissimilarity measure and they regard the space of possible partitions into c subsets only. Therein partitioning clustering methods are different from hierarchical techniques. The latter organize data in a nested sequence of groups, which can be visualized in the form of a dendrogram or tree. Based on a dendrogram one can decide on the number of clusters at which the data are best represented for a given purpose. Usually the number of (true) clusters in the given data is unknown in advance. However, using the partitioning methods one is usually required to specify the number of clusters c as an input parameter. Estimating the actual number of clusters is thus an important issue that we do not leave untouched in this chapter.