Covariance of random variables with fuzzy states and their applications

Abstract

New definitions of scalar products, covariances, variations and correlation coefficients of random variables with fuzzy states are given in terms of their centers and radii. The following properties are established: symmetry, additivity and positive homogeneity of scalar products and covariances, the corresponding analogues of the Cauchy — Bunyakovsky inequality, and the characteristic properties of the correlation coefficient. The relationship between the introduced covariances and scalar products, as well as between various definitions of metrics on the set of fuzzy numbers and the introduced variations, is studied. The relationship with known results is shown. As an application, the solution to the problem of optimal linear regression of random variables with fuzzy states is considered. As another application, the problem of transforming a fuzzy random signal by a linear dynamic system is considered.