Hierarchical structures and uncertainty measures for intuitionistic fuzzy approximation space.

The hierarchical structures and uncertainty measures in granular computing are the two main aspects for investigating the structure and uncertainty of all types of approximation spaces. Although several hierarchical structures and uncertainty measures have been proposed to represent and analyze different granular structures, these structures and uncertainty measures are studied under crisp or fuzzy conditions. Hierarchical structures and uncertainty measures for the intuitionistic fuzzy (IF) approximation space are addressed in this paper. First, we propose the representation and operations of IF granular structures, as well as examine four hierarchical structures and a lattice structure of IF approximation space. Second, the natural extensions of fuzzy information granularity, fuzzy information entropy, fuzzy rough entropy, and fuzzy information Shannon entropy, namely, IF granularity, IF information entropy, IF rough entropy, and IF information Shannon entropy, are developed and adopted to characterize the uncertainty of IF granular structures in the IF approximation space. Third, we provide the multi-granulation IF approximation space and study its four hierarchical structures. Furthermore, we discuss the relationships between the presented hierarchical structures and multi-granulation IF rough sets in the multi-granulation IF approximation space. Fourth, we propose four types of IF uncertainty measures to depict the uncertainty of the multi-granulation IF knowledge base in the multi-granulation IF approximation space with respect to optimistic and pessimistic multi-granulation IF rough sets, which is more reasonable compared with previous work conducted in a crisp context.