A Comparative Investigation of Type-2 Fuzzy Sets, Nonstationary Fuzzy Sets and Cloud Models

Abstract

Since Zadeh introduced fuzzy sets, a lot of extensions of this concept have been proposed, such as type-2 fuzzy sets, nonstationary fuzzy sets, and cloud models, to represent higher levels of uncertainty. This paper provides a comparative investigation of type-2 fuzzy sets, nonstationary fuzzy sets, and cloud models. Type-2 fuzzy sets study the fuzziness of the membership function (MF) using primary MF and secondary MF based on analytic mathematical methods; nonstationary fuzzy sets study the randomness of the MF using primary MF and variation function based on type-1 fuzzy sets theory; cloud models study the randomness of the distribution of samples in the universe and generate random membership grades (MGs) using two random variables based on probability and statistic mathematical methods. They all concentrate on dealing with the uncertainty of the MF or the MG which type-1 fuzzy sets do not consider, and thus have many similarities. Moreover, we find out that, the same qualitative concept “moderate amount” can be represented by an interval type-2 fuzzy set, a nonstationary fuzzy set or a normal cloud model, respectively. Then, we propose a unified mathematical expression for the interval type-2 fuzzy set, nonstationary fuzzy set and normal cloud model. On the other hand, we also find out that, the theory fundament and underlying motivations of these models are quite different. Therefore, We summarize detailed comparisons of distinctive properties of type-2 fuzzy sets, nonstationary fuzzy sets, and cloud models. Further, we study their diverse characteristics of distributions of MGs across vertical slices. The comparative investigation shows that these models are complementary to describe the uncertainty from different points of view. Thus, this paper provides a fundamental contribution and makes a basic reference for knowledge representation and other applications with uncertainty.