Fuzzy-rough membership functions

This paper generalizes the concepts of rough membership functions in pattern classification tasks to fuzzy-rough membership functions. Unlike the rough membership value of a pattern, which is sensitive only towards the rough uncertainty associated with the pattern, the fuzzy-rough membership value of the pattern signifies the rough uncertainty as well as the fuzzy uncertainty associated with it. In absence of fuzziness, the fuzzy-rough membership functions reduce to the existing rough membership functions. Moreover, under certain conditions the fuzzy-rough membership functions are equivalent to fuzzy membership functions or characteristic functions. In this paper, various set theoretic properties of the fuzzy-rough membership functions are exploited to characterize the concept of fuzzy-rough sets. Some measures of the fuzzy-rough ambiguity associated with a given output class are also discussed.