Introduction

A rough set, first described by a Polish computer scientist Zdzisław I. Pawlak, is a formal approximation of a vague set in terms of a pair of sets which give the lower and the upper approximation of the original set. In the standard version of rough set theory, the lower- and upper-approximation sets are crisp sets, but in other variations, the approximating sets may be fuzzy sets. Rough sets can be used to represent ambiguity, vagueness and general uncertainty. Fuzzy-rough sets further extend the rough set concept through the use of fuzzy equivalence classes. Rough set methods can be applied as a component of hybrid solutions in machine learning and data mining. They have been found to be particularly useful for rule induction and feature selection (semantics-preserving dimensionality reduction). Rough set-based data analysis methods have been successfully applied in bio-informatics, economics and finance, medicine, multimedia, web and text mining, signal and image processing, software engineering, robotics, and engineering (e.g. powersystems and control engineering).