Belief and plausibility functions of type-2 fuzzy rough sets

The rough set theory is closely related with evidence theory. It has been demonstrated that certain rough approximation spaces are associated with certain belief structures such that the lower and upper approximation operations induced by the approximation spaces may be used to interpret the corresponding belief and plausibility functions induced by the belief structures. This paper studies the belief and plausibility functions of type-2 fuzzy rough sets. First, measurable interval type-2 fuzzy sets are defined and a new probability measure of interval type-2 fuzzy sets is proposed using the integral operation. Then the probability measure of type-2 fuzzy sets is given based on α-plane representation of type-2 fuzzy sets. Finally, belief and plausibility functions of type-2 fuzzy rough sets are defined and an example is employed to illustrate their applications.