On tolerant fuzzy c-means clustering and tolerant possibilistic clustering

This paper presents two new types of clustering algorithms by using tolerance vector called tolerant fuzzy c-means clustering and tolerant possibilistic clustering. In the proposed algorithms, the new concept of tolerance vector plays very important role. The original concept is developed to handle data flexibly, that is, a tolerance vector attributes not only to each data but also each cluster. Using the new concept, we can consider the influence of clusters to each data by the tolerance. First, the new concept of tolerance is introduced into optimization problems. Second, the optimization problems with tolerance are solved by using Karush–Kuhn–Tucker conditions. Third, new clustering algorithms are constructed based on the optimal solutions for clustering. Finally, the effectiveness of the proposed algorithms is verified through numerical examples and its fuzzy classification function.