Numerical construction of membership functions and aggregation operators from empirical data

A good choice of membership functions and aggregation operators is crucial for the behaviour of fuzzy systems. Goodness of fit to empirical data and flexibility in modelling various situations are the main criteria used by developers. This paper provides a general method for a non-parametric representation of membership functions and aggregation operators using constrained spline functions. Tensor-product monotone splines are used to approximate aggregation operators directly, while univariate splines are used to approximate their additive generators. Examples based on published empirical data are provided.