Uncertainty measures of roughness of knowledge and rough sets in incomplete information systems

In this paper we address uncertainty measures of roughness of knowledge and rough sets by introducing rough entropy in incomplete information systems. We make only one assumption about unknown values: the real value of a missing attribute is one from the attribute domain. However, we do not assume which one. We prove that the rough entropy of knowledge and the rough entropy of rough sets decrease monotonously as the granularity of information grows smaller through finer partitionings. These conclusions are helpful to understand the essence of rough set theory and essential to seek new efficient algorithm of knowledge reduction in incomplete information systems.