Improvements on Correlation Coefficients of Hesitant Fuzzy Sets and Their Applications

Hesitant fuzzy set (HFS) can express the hesitancy and uncertainty according to human’s cognitions and knowledge. The decision making with HFSs can be regarded as a cognitive computation process. Decision making based on information measures is a hot topic, among which correlation coefficient is an important direction. Although many correlation coefficients of HFSs have been proposed in the previous papers, they suffer from different counter-intuitions to a certain extent. Therefore, we mainly focus on improving these counter-intuitions of the existing correlation coefficients of HFSs in this paper. We point out the counter-intuitions of the existing correlation coefficients of HFSs and analyze the reasons of them in the view of the rigorous mathematics and stochastic process rules. We improve these counter-intuitions and develop the correct versions. Moreover, we use two examples about medical diagnosis and cluster analysis to compare the improved correlation coefficients with the existing ones. The improved correlation coefficients can handle the examples well. Further, combining with the comparison analysis, the accuracy and discrimination property of the improved correlation coefficients are demonstrated in detail, which shows the advantages of them. The notion of the improved correlation coefficients can benefit other types of fuzzy sets too.