New fuzzy entropy via class-consistent technology with applications to attribute reduction for heterogeneous data

A fuzzy relation is a basic concept of fuzzy set theory. Fuzzy entropy as information entropy for a fuzzy relation is to measure the uncertainty of a fuzzy relation. Some measures on the uncertainty of a fuzzy relation have been presented in recent years by generalizing information entropy. However, a fuzzy relation can induce to a family of upper-fuzzy sets and a family of lower-fuzzy sets, respectively; the existing methods consider only the upper-fuzzy sets in a fuzzy relation and do not address the lower-fuzzy sets. Moreover, these methods sometimes need to use the equality between fuzzy sets, but the equality between fuzzy sets is actually very difficult to achieve. To solve the above problem, this paper proposes new fuzzy entropy based on class-consistent technology and considers its application in attribute reduction for heterogeneous data. Class-consistent technology is a technique for dealing with approximate equality of values. It replaces equality with approximate equality between two values in unit close interval. First of all, based on this technology, two equivalence relations are put forward by means of upper-fuzzy and the lower-fuzzy sets in a fuzzy relation. Then, upper-fuzzy entropy and lower-fuzzy entropy are presented by using these two equivalence relations. Next, new fuzzy entropy is constructed to measure the uncertainty of a fuzzy relation, and the constructed fuzzy entropy overcomes the weakness of the existing methods. Moreover, new fuzzy conditional entropy is defined. Finally, new fuzzy conditional entropy is applied to carry out attribute reduction for heterogeneous data. The experiment results confirm that the given reduction method is more effective than other methods.