Novel correlation coefficients between hesitant fuzzy sets and their application in decision making

Hesitant fuzzy set (HFS) is now attracting more and more scholars' attention due to its efficiency in representing comprehensively uncertain and vague information. Considering that correlation coefficient is one of the most widely used indices in data analysis, in this paper, after pointing out the weakness of the existing correlation coefficients between HFSs, we propose a novel correlation coefficient formulation to measure the relationship between two HFSs. As a departure, some new concepts, such as the mean of a hesitant fuzzy element (HFE), the hesitant degree of a HFE, the mean of a HFS, the variance of a HFS and the correlation between two HFSs are defined. Based on these concepts, a novel correlation coefficient formulation between two HFSs is developed. Afterwards, the upper and lower bounds of the correlation coefficient are defined. A theorem is given to determine these two bounds. It is stated that the correlation coefficient between two HFSs should also be hesitant and thus the upper and lower bounds can further help to identify the correlation coefficient between HFSs. The significant characteristic of the introduced correlation coefficient is that it lies in the interval [-1, 1], which is in accordance with the classical correlation coefficient in statistics, whereas all the old correlation coefficients between HFSs in the literature are within unit interval [0,1]. The weighted correlation coefficient is also proposed to make it more applicable. In order to show the efficiency of the proposed correlation coefficients, they are implemented in medical diagnosis and cluster analysis. Some numerical examples are given to support the findings and also illustrate the applicability and efficiency of the proposed correlation coefficient between HFSs. (C) 2015 Elsevier B.V. All rights reserved.