Possibilistic Correlation: Illustration, Ex- Planation and Computation of Some Im- Portant Cases

In 1987 Dubois and Prade defined an interval-valued expectation of fuzzy numbers, viewing them as consonant random sets. In 2001 Carlsson and Fullér defined an interval-valued mean value of fuzzy numbers, viewing them as possibility distributions, and they introduced the notation of crisp possibilistic mean value and crisp possibilistic variance of continuous possibility distributions, which are consistent with the extension principle. In 2003 Fullér and Majlender introduced the notation of weighted interval-valued possibilistic mean value of fuzzy numbers and investigate its relationship to the interval-valued probabilistic mean. They also introduced the notations of crisp weighted possibilistic mean value, variance and covariance of fuzzy numbers, which are consistent with the extension principle. In 2004 Fullér and Majlender introduced the notion of covariance between marginal distributions of a joint possibility distribution to measure the degree to which they interact. In 2005 Carlsson, Fullér and Majlender presented the concept of possibilistic correlation representing an average degree of interaction between marginal distributions of a joint possibility distribution as compared to their respective dispersions. Moreover, they formulated the classical Cauchy-Schwarz inequality in this possibilistic environment and showed that the measure of possibilistic correlation satisfies the same property as its probabilistic counterpart. In particular, applying the idea of transforming level sets of possibility distributions into uniform probability distributions, they identified a fundamental relationship between our proposed possibilistic approach and the classical probabilistic approach to measuring correlation. In this work we give a geometrical interpretation and explanation of the possibilistic correlation by some important cases.