Least Squares for Fuzzy Vector Data Regression

摘要

We consider model fitting problems in which the mathematical model of an observed event is a system of equations involving model parameters and observables, and in which the data are fuzzy vectors. Such problems naturally arise in applications when data are scarce and information is vague about distributions of variances that are contained in the observations. Then it may only be possible to obtain estimates of membership functions of the data. A model can be fitted to such data by maximizing the membership values of the adjusted observations. We achieve this by minimizing the sum of squares of the the deviations of the membership values from one. The optimization problem simplifies significantly if the membership functions of the fuzzy vectors belong to a class of conical functions, which are defined in terms of an elliptic norm. Properties of these functions are discussed and a membership propagation formula derived which component-wise obeys the extension principle. (Author)