Inference based on α-cut and generalized mean in representing fuzzy-valued functions

It is mathematically proved that inference based on alpha;-cuts and generalized mean (alpha;-GEMII) deduces consequences converging to fuzzy sets mapped by linear fuzzy-valued functions, to be represented with alpha;-GEMII, as the number of fuzzy rules increases. The proof indicates that alpha;-GEMII satisfies axiomatic properties and can contribute to presenting interpretability in designing fuzzy systems in the rule base. Such properties do not hold in conventional methods based on the compositional rule of inference. Simulation results show that the difference between deduced consequences and fuzzy sets mapped by linear fuzzyvalued functions is smaller as the number of fuzzy rules increases, in which the difference is evaluated by mean square errors. The discussions may lead to improvements of the interpretability in representing nonlinear fuzzy-valued functions by using alpha;-GEMII.