Estimation of a Fuzzy Regression Model Using Fuzzy Distances

Regression analysis is a powerful statistical tool that has many applications in different areas. The problem of regression analysis under a fuzzy environment has been treated in the literature from different points of view and considering a variety of input/output data (crisp or fuzzy). However, we realize that, in general, most research papers have a conflict between the solution of the fuzzy regression problem using crisp distances (minimizing a real error function) and the interpretation of fuzzy data as possibility distributions. The main aim of this paper is to develop a methodology to solve this problem introducing a fuzzy partial order and a family of fuzzy distance measures on the whole set of fuzzy numbers. The new approach allows us to obtain linear and nonlinear models that reach the lowest fuzzy error; the estimation process, in general, can be considered easier to apply in practice, and it is not limited to triangular fuzzy numbers. Numerical examples are provided to illustrate the usefulness and applicability of these results, and comparisons with existing methodologies show that the performance of the proposed solution is very satisfactory.