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An Overview of Fuzzy Correlation Coefficients and Their Novel Proportional Fuzzy Extensions Using Transformative Functions
Irem Otay, Cengiz Kahraman and Vicenç Torra
Fuzzy correlation coefficients were introduced for fuzzy sets as a tool when data are imprecise. They permit to evaluate the correlation between two arbitrary fuzzy sets. For the same reason, fuzzy correlation coefficients have been introduced for many fuzzy set extensions such as type-2 fuzzy sets, intuitionistic fuzzy sets, neutrosophic sets, hesitant fuzzy sets, picture fuzzy sets, spherical fuzzy sets, Pythagorean fuzzy sets, Fermatean fuzzy sets, and T-spherical fuzzy sets. Recently, proportional fuzzy sets were introduced as a way to ease the determination of membership, nonmembership and hesitancy with sufficient sensitivity and accuracy. In this paper, we first review correlation coefficients for different types of fuzzy sets. Then, we review proportional fuzzy sets, discuss transformative functions, and introduce fuzzy correlation coefficients for proportional fuzzy sets through transformative functions. We provide examples for both existing fuzzy correlation coefficients and the new introduced ones.
Keywords: Transformative function, correlation coefficient, fuzzy set extensions, proportional fuzzy sets, vague proportions