Hierarchical FCM in a stepwise discovery of structure in data

This paper is concerned with a stepwise mode of objective function-based fuzzy clustering. A revealed structure in data becomes refined in a successive manner by starting with the most dominant relationships and proceeding with its more detailed characterization. Technically, the proposed process develops a so-called hierarchy of clusters. Given the underlying clustering mechanism of the fuzzy C means (FCM), the produced architecture is referred to as a hierarchical FCM or hierarchical FCM tree (HFCM tree). We discuss the design of the tree demonstrating how its growth is guided by a certain mapping criterion. It is also shown how a structure at the higher level is effectively used to build clusters at the consecutive level by making use of the conditional FCM. Detailed investigations of computational complexity contrast a stepwise development of clusters with a single-step clustering completed for the equivalent number of clusters occurring in total at all final nodes of the HFCM tree. The analysis quantifies a significant reduction of the stepwise refinement of the clusters. Experimental studies include synthetic data as well as those coming from the machine learning repository.