Hierarchical Clustering Problems and Analysis of Fuzzy Proximity Relation on Granular Space

In this paper, based on granular space, some hierarchical clustering problems and analysis for fuzzy proximity relation are developed by using rigorous mathematical descriptions, and four results are obtained. First, the granular representation of a fuzzy proximity relation is studied, an algorithm is obtained to compute the granular space (or hierarchical clustering structure) and the min-transitive closure derived from the same fuzzy proximity relation on a finite universe, and the consistent clustering properties are discussed. Second, a consistency index that is based on granular space is developed, and a global optimizing mathematical model is established to obtain the optimal clustering of a fuzzy proximity relation. Furthermore, for given two fuzzy proximity relations on different granulations of the same universe, a collaborative clustering method is studied by the intersection operation, and this method can provide a better hierarchical clustering structure. Finally, the theory for hierarchical clustering analysis of fuzzy proximity relations on granular space is established. For given two fuzzy proximity relations on the same universe, two sufficient conditions (Theorems 6.1 and 6.2) of isomorphism are given, and the necessary condition (Theorem 6.3) and the sufficient condition (Theorem 6.4) for ε-similarity are obtained. We also illustrate the effectiveness of these theories and methods with some practical examples. These results may help form a comprehensive theory and methodology for related potential application and will help us to obtain a deeper understanding of the essence of hierarchical clustering procedures.