A fuzzy inference method is proposed on the basis of α-cuts, which can mathematically prove to deduce consequences in both convex and symmetric forms under the required conditions, studied here, when fuzzy sets in the consequent parts of fuzzy rules are all convex and symmetric. The inference method can reflect the distribution forms of fuzzy sets in consequent parts of fuzzy rules, guaranteeing the convexity in deduced consequences. It also has a control scheme for the fuzziness and specificity in deduced consequences. The controllability provides a way to suppress excessive fuzziness increase and specificity decrease in deduced consequences. Simulation studies show that the proposed method can deduce consequences in convex and symmetric forms under the required conditions. It is also demonstrated that the distribution forms of consequent parts are reflected to deduced consequences. Moreover, it is found that the fuzziness and specificity of deduced consequences can be effectively controlled in the simulations.
A governing scheme is proposed for fuzzy constraint propagation from given facts to consequences in convex forms, which is applied to the inference method based on α-cuts and generalized mean. The governing is performed by self-tuning which can reflect the distribution forms of fuzzy sets in consequent parts to the forms of deduced consequences. Thereby, the proposed scheme can solve the problems in the conventional inference based on the compositional rule of inference that deduces fuzzy sets with excessive fuzziness increase and specificity decrease. In simulations, it is confirmed that the proposed scheme can effectively perform the constraint propagation from given facts to consequences in convex forms while reflecting fuzzy-set distributions in consequent parts. It is also demonstrated that consequences are deduced without excessively large fuzziness and small specificity.
This paper clarifies that inference based on α-cut and generalized mean (α-GEMII) is effective in suppressing consequence deviations. The suppression effect of α-GEMII is numerically evaluated in comparison to conventional inference based on the Compositional Rule of Inference (CRI). CRI-based parallel inference causes discontinuous deviations in the least upper and greatest lower bounds of deduced fuzzy sets even when it models the continuous input-output relation of a system and given facts change continuously. In contrast, α-GEMII can suppress the deviations because of its schemes originally developed for constraint propagation control. In simulations, indices are defined for numerically evaluating the degree to which deduced consequences follow the change in fuzzy outputs of given systems. Simulation results show that α-GEMII is effective in suppressing the deviations, compared to CRI-based parallel inference. In effective use of the schemes for suppressing the consequence deviations, α-GEMII can be applied to nonlinear prediction filters for complex time series, especially with fluctuations that do not always originate from a correlation between time series data.
Theoretical aspects are provided for inference based on α-cuts and generalized mean (α-GEMII). In order to clarify the basic properties of the inference, fuzzy tautological rules (FTRs) are focused on, which are composed by setting fuzzy sets in consequent parts identical to those in antecedent parts of initially given fuzzy rules. It is mathematically proved that the consequences deduced with FTRs are closer to given facts as the number of FTRs increases. The aspects provided in this paper are appropriate from axiomatic viewpoints and can contribute to interpretability in fuzzy systems constructed with α-GEMII. They are not obtained in conventional methods based on the compositional rule of inference. Simulations are performed by evaluating difference (mean square errors) between given facts and deduced consequences under the condition that convex and symmetric fuzzy sets are given as facts. Their results show that the difference becomes smaller as the number of FTRs increases. Thereby, it is confirmed that α-GEMII has an advantage in the interpretability with respect to FTRs over the conventional methods.
It is mathematically proved that inference based on α-cuts and generalized mean (α-GEMII) deduces consequences converging to fuzzy sets mapped by linear fuzzy-valued functions, to be represented with α-GEMII, as the number of fuzzy rules increases. The proof indicates that α-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 α-GEMII.
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