The uncertainty of probabilistic rough sets in multi-granulation spaces

Pawlak's rough sets model describes an uncertain target set (concept) with two crisp boundary lines (i.e. lower and upper approximation sets) and as an effective tool has successfully been used to deal with uncertain information systems. Based on the rough sets model, a probabilistic rough sets model with a pair of thresholds was proposed to improve the fault-tolerance ability of rough sets. The uncertainty of Pawlak's rough sets model is rooted in the objects contained in the boundary region of the target concept, while the uncertainty of probabilistic rough sets model comes from three regions, because the objects in the positive or negative regions are probably uncertain, and the membership degrees of these objects are not necessary equal to 1 or 0. In this paper, a method for measuring the uncertainty of probabilistic rough sets is proposed, and the change rules of uncertainty with changing knowledge spaces are presented and analyzed. Then, for an uncertain target concept, the uncertainties of the three regions are discussed, and the related change rules of uncertainty with changing knowledge spaces are revealed and successfully proved. Finally, a comparative analysis on the uncertainty of a target concept in rough sets and probabilistic rough sets model is presented. These results are important to further enrich and improve probabilistic rough sets theory, and effectively promote the development of uncertainty artificial intelligence.