A roughness measure of fuzzy sets from the perspective of distance

Abstract

Covering generalized rough set theory is an important extension of classical rough set theory. To characterize a fuzzy set in a given covering approximation space, a pair of fuzzy sets, called covering rough fuzzy lower and upper approximations, were introduced, but they do not describe well how much uncertainty is induced by the granularity of knowledge. In this paper, we first discuss the relationship between uncertainty and granularity of knowledge. Then we examine several commonly used distance measures, and indicate that some of them exhibit some limitations. Next we propose a roughness measure based on Minkowski distance, and examine some important properties of this measure. Finally, an illustrative example is provided to demonstrate the application of the roughness measure to incomplete information systems with fuzzy decision.

Keywords:

• Covering
• rough sets
• roughness
• granularity
• distance

Acknowledgements

The authors are thankful to Prof. Xiang Li for his valuable suggestions that significantly improved the paper.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This paper is supported by the Natural Science Foundation of China [grant numbers 61272060, 61472056, 61309014]; the Key Natural Science Foundation of Chongqing of China [grant number CSTC2013jjB40003]; the Natural Science Foundation of Chongqing of China [grant number cstc2012jjA40032]; and the Science & Technology Research Program of Chongqing Education Committee of China [grant number KJ130534].