Multi-granulation rough sets in multi-scale information systems

The key to granular computing (GrC) is to make use of granules in problem solving. With the view point of GrC, the notion of a granule may be interpreted as one of the numerous small particles forming a larger unit. In many situations, there are different granules at different levels of scale in data sets having hierarchical scale structures. The multi-scale information system is a new and interesting topic in the theory of GrC. In this paper, the multi-granulation rough set approach is introduced into the multi-scale information systems. Firstly, the notion of multi-scale information systems is introduced. In multi-scale information systems, a hierarchical structure of granules can be obtained. At every level, the optimistic and the pessimistic multi-granulation rough sets can be defined based on a family of the equivalence relations. Analogously, the boundary region can be defined as usual at every level. Furthermore, the properties of the lower approximations, the upper approximations, the boundary region, the approximation accuracy, and the roughness between different levels are discussed respectively.