Generalizations of Fuzzy C-Means Algorithm to Granular Feature Spaces, based on Underlying Metrics: Issues and Related Works

This paper considers dissimilarity measures and clustering techniques for two special cases of set-defined objects: fuzzy granules and subsequence time series. To deal with clustering of such kind of objects, we propose two implementations that generalize the Fuzzy C-Means algorithm to granular feature spaces. Granular computing is a paradigm oriented towards capturing and processing meaningful pieces of information, the socalled information granules. In a granular feature space, such as a space populated with p-dimensional fuzzy granules, we are concerning with both granular data samples and granular centroids (center of clusters). In order to accommodate clustering algorithms to work in a granular environment we have to choose and/or define appropriate metrics and descriptors. Either a crisp distance between granules or the defuzzified value of a fuzzy distance has to be chosen. On the other hand, subsequence time series clustering requires a generalization of Fuzzy C-Means algorithm in a similar way. It involves a set-defined centroid and appropriate dissimilarity measures to determine the degree to which time sequences are different from their centroid. Furthermore, we discuss related work in granular clustering and subsequence time series clustering. Keywords—Granular clustering, Metrics in granular feature spaces, Fuzzy C-Means clustering, Agglomerative granular clustering algorithms, Subsequence time series clustering. 1 P-dimensional fuzzy granules: representation and cardinality measure In what follows, the attention will be restricted to the class p of normal fuzzy convex granules on p , whose level sets are nonempty compact convex sets for all 0 . Each fuzzy granule p M  is uniquely characterized by its support function ) , ( u sM }, | , { sup M x x u ] 1 , 0 ( , 1 p S u , where 1 p S is the (p-1)-dimensional unit sphere of p (i.e. 1 u ) and , is the inner product in p . The support function is a mapping from the class of fuzzy sets p into the space of functions ] 1 , 0 [ 1 p S L , which preserves addition and multiplication with non-negative scalars. A p-dimensional fuzzy granule p A A A 1 is defined on the product space p X X X 1 . Representation and cardinality measure for A can be given either in terms of membership functions: ], 1 , 0 [ :  p A X x x x x p A A A p , ) ) ( , ), ( ( min ) ( 1 1