Measuring Uncertainty of Probabilistic Rough Set Model From Its Three Regions

The probabilistic rough set model is a generalized rough set model, which was proposed to improve the fault tolerance ability of the classical Pawlak's rough set model. Compared with the classical Pawlak's rough set model, the probabilistic rough set model and its several special forms (such as variable precision rough set model, Bayesian rough set model, and decision-theoretic rough set model) divide a domain into three regions by two parameters (α, β), and the objects belong to every region with a certain uncertainty. With the increase of information (attributes), the boundary region of the probabilistic rough set model may become smaller, bigger, or remain unchanged. This paper aims first to study some more complex uncertainty of the probabilistic rough set model. Then the uncertainty of the three regions is defined and analyzed. Besides, the changing regularities of uncertainty of a target concept in the probabilistic rough set model are discovered. Finally, three kinds of incremental information are defined, and their judging theorems are proposed. These results could enrich and improve rough set theory to deal with uncertain information systems.