SFCM: A Fuzzy Clustering Algorithm of Extracting the Shape Information of Data

Topological data analysis is a new theoretical trend using topological techniques to mine data. This approach helps determine topological data structures. It focuses on investigating the global shape of data rather than on local information of high-dimensional data. The Mapper algorithm is considered as a sound representative approach in this area. It is used to cluster and identify concise and meaningful global topological data structures that are out of reach for many other clustering methods. In this article, we propose a new method called the Shape Fuzzy C-Means (SFCM) algorithm, which is constructed based on the Fuzzy C-Means algorithm with particular features of the Mapper algorithm. The SFCM algorithm can not only exhibit the same clustering ability as the Fuzzy C-Means but also reveal some relationships through visualizing the global shape of data supplied by the Mapper. We present a formal proof and include experiments to confirm our claims. The performance of the enhanced algorithm is demonstrated through a comparative analysis involving the original algorithm, Mapper, and the other fuzzy set based improved algorithm, F-Mapper, for synthetic and real-world data. The comparison is conducted with respect to output visualization in the topological sense and clustering stability.