The covariance of uncertain variables: definition and calculation formulae

Uncertainty theory as a branch of axiomatic mathematics has been widely used to deal with human uncertainty. The two commonly used numerical characteristics of uncertain variables, the expected value and the variance together with their mathematical properties have been discussed and applied to real optimization problems in an uncertain environment. As a further study, in this paper, we focus on the covariance and correlation coefficient of uncertain variables. The definitions and calculation formulae of covariance and correlation coefficient of two uncertain variables are suggested by means of their inverse distributions. Then we show that the correlation coefficient of uncertain variables is essentially a measure of the relevance of distributions of uncertain variables. Finally, the relation between variance and covariance is analysed and represented with some equalities and inequalities.