Distance-based information granularity in neighborhood-based granular space

In granular computing, information granularity and hierarchical structures are the two main issues that are relevant in investigating the uncertainty measure and structure of all types of granular spaces. To represent and analyze granular structures for neighborhood-based granular space, a distance-based information granularity and the corresponding hierarchical structures of neighborhood-based granular space are discussed in this paper. First, we propose the representation and operations of neighborhood-based granular structures and examine four hierarchical structures of neighborhood-based granular space. Second, a distance between two neighborhood-based granular structures is introduced to differentiate them; this distance is used to establish the axiomatic approach of information granularity of neighborhood-based granular space. Third, the representation and operations of fuzzy neighborhood-based granular structures, a distance between two fuzzy neighborhood-based granular structures, a distance-based information granularity, and a distance-based hierarchical structure for fuzzy neighborhood-based granular space, are studied. Fourth, a novel distance is developed in multi-granulation neighborhood-based granular space. Using this distance, information granularity and a hierarchical structure for multi-granulation neighborhood-based granular space are provided. Finally, distance, information granularity, and hierarchical structure are examined in multi-granulation fuzzy neighborhood-based granular space. The presented distance between two neighborhood granular structures is an effective tool for representing information granularity and constructing hierarchical structures in neighborhood-based granular spaces.