Correlation Coefficient of Interval-Valued Fuzzy Sets with Interval-Valued Reference Functions and Applications in Medical Diagnosis and Selecting Appropriate Mediciness

Fuzzy set theory (FST) has become a dominant model to deal with imprecision, uncertainty, vagueness, and ambiguity arising from many real-world applications. There are a couple of extensions and generalizations of FST, and an interval-valued FST is one of the important generalizations where the membership values are expressed in terms of closed intervals in [0, 1]. Finding the correlation coefficient of different FSTs can be an interesting research problem, as it has many real applications, from social science to medical science. This article first proposes a new extension of fuzzy set in terms of an interval-valued membership function and an interval-valued reference function and then proposes a new method of finding the correlation coefficient using the statistical parameters, like covariance and variance of the proposed fuzzy sets. The proposed formulae can be applied to both discrete and continuous sets of the universe of discourse. It has also been found that the correlation coefficient computed by the proposed formula lies in the closed interval of −1 and 1, which establishes that the correlation coefficient of the proposed fuzzy sets not only gives the strength of the relationship but also tells whether they are positively or negatively correlated. Furthermore, two practical applications along with numerical examples are discussed in detail. The result convincingly establishes the efficacy of the proposed formula.