Least-squares fuzzy regression with fuzzy random variables

Let (X) over tilde, (Y) over tilde be two convex fuzzy random variables on R-n. Using a suitable metric we prove that the conditional expectation E((Y) over tilde\(X) over tilde) is the best approximation of (Y) over tilde by measurable functions of (X) over tilde. This generalizes the analogous and well known property for real random variables. A further topic is the approximation of (Y) over tilde by a linear function of (X) over tilde. In special cases and by use of Hukuhara's difference between fuzzy sets, we obtain formulas which are analogous to the classical structure. Contrary to the classical fact, however, the conditional expectation of Gaussian fuzzy random variables in general does not coincide with the linear regression function. (C) 2002 Elsevier Science B.V. All rights reserved.