Probabilistic rough sets: Approximations, decision-makings, and applications

The main objective of this special issue is to present readers with the significantly extended and improved versions of the articles presented at the International Conference on Rough Sets, Fuzzy Sets, and Granular Computing (RSFDGrC’05) held in Regina, Canada in September 2005. In the standard rough set model, the lower and upper approximations are defined based on the two extreme cases (full inclusion or non-empty overlap) regarding the relationships between an equivalence class and a target set. This requirement unnecessarily limits the applications of rough sets for some practical problems. In Pawlak’s formulation of rough sets the degree of set overlap was not considered. Many researchers were motivated to investigate probabilistic generalizations of the theory. Probabilistic approaches to rough sets have appeared in many forms, such as the decision-theoretic rough set model, the variable precision rough set model, the Bayesian rough set model, and others. There is a fast growing interest in these probabilistic extensions. We are delighted to have six papers in this special issue discussing developments and recent trends of research on probabilistic rough set approaches and applications. The first two articles are general and review papers with new results. ‘‘Probabilistic Rough Set Approximations’’, authored by Yao, provides a comprehensive and well balanced view of the state of the probabilistic approaches in rough set theory. A general framework for comparing and synthesizing probabilistic rough set approximations is provided. Within the framework, the existing approaches are described and compared in depth. ‘‘Probabilistic Approach to Rough Sets’’, authored by Ziarko, is a review of existing notions and approaches. It summarizes the recent development of probabilistic rough sets, as well as a variant of rough set theory that makes use of probabilities and dependency in probabilistic sense. The paper also discusses several measures for the evaluation of the proposed models. The complementary views of the two papers not only provide insights into existing research but also give directions for further research. The incorporation of Bayesian inference and decision theory into probabilistic rough sets is a very promising new research topic. The next three papers present specific extensions and alternatives. The paper entitled ‘‘Parameterized Rough Set Model using Rough Membership and Bayesian Confirmation Measures’’, authored by Greco, Matarazzo and Slowinski, proposes a parameterized rough set model as a generalization of the rough set model and the variable precision rough set model. Two kinds of membership, relative and absolute membership are introduced to model different degrees to which the condition attribute values confirm the decision attribute values. ‘‘Fuzzy Rough Approximations of Process Data’’, authored by Mieszkowicz-Rolka and Rolka, presents variable precision fuzzy rough sets model with asymmetric bounds. A unified form of expressing the lower and upper crisp approximations is considered. The model is used for describing and analyzing the control actions that are accomplished by a human operator. The decision model is expressed by means of a decision table with fuzzy attributes and a T-similarity relation is used for comparing elements of the universe. In Eberbach’s paper, ‘‘Approximate Reasoning in the Algebra of Bounded Rational Agents’’, $-calculus proposed and developed by others over the past twenty years is used to model approximate reasoning. The