A note on the convexity assumption of possibilistic correlation

Abstract

In 2005, Carlsson, Fullér and Majlender introduced a measure of possibilistic correlation of fuzzy numbers A and B by their joint possibility distribution C as an average degree of interaction between the γ -level sets of A and B as compared to their individual dispersions. They proved that this possibilistic correlation coefficient can never exceed 1 in absolute value, if all γ -level sets of the joint possibility distribution C are convex. In this communication, we shall formulate a special class of joint possibility distributions with non-convex γ -level sets, for which the correlation coefficient can take values outside the interval [ − 1 , 1 ] . In particular, this result will show that the assumption about the convexity of the level sets of C is essential for the possibilistic correlation to be bounded by the interval [ − 1 , 1 ] .