A New Look at Type-2 Fuzzy Sets and Type-2 Fuzzy Logic Systems

We propose the concept of a conditional fuzzy set and prove that a type-2 fuzzy set is equivalent to a conditional fuzzy set. We show that both the conditional fuzzy sets and the type-2 fuzzy sets are fuzzy relations on the product space of the primary and secondary variables, and the difference is...

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Bibliographic Details
Published in:IEEE transactions on fuzzy systems 2017-06, Vol.25 (3), p.693-706
Main Author: Wang, Li-Xin
Format: Article
Language:English
Subjects:
Compositional rule of inference
conditional fuzzy sets
Construction
Equivalence
Fuzzy logic
fuzzy relations
Fuzzy set theory
Fuzzy sets
Fuzzy systems
Inference
Probability density function
Random variables
type-2 fuzzy logic systems (T2 FLSs)
type-2 fuzzy sets (T2 FSs)
Uncertainty
Variables
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Summary:We propose the concept of a conditional fuzzy set and prove that a type-2 fuzzy set is equivalent to a conditional fuzzy set. We show that both the conditional fuzzy sets and the type-2 fuzzy sets are fuzzy relations on the product space of the primary and secondary variables, and the difference is that the primary and secondary variables in the conditional fuzzy set framework are usually independent to each other, whereas in the type-2 fuzzy set framework, the secondary variable depends on the primary variable by definition. It is this dependence between the primary and secondary variables that makes the type-2 fuzzy sets a complex subject, while the conditional fuzzy sets do not have this built-in dependence and, thus, are much easier to analyze. With the fuzzy relation formulation, powerful tools in fuzzy set theory such as Zadeh's compositional rule of inference can be used to obtain the marginal fuzzy sets of the type-2 and conditional fuzzy sets, transforming the type-2 problems back to the conventional type-1 domain. With the help of the marginal fuzzy set concept, we show that a type-2 fuzzy logic system can be designed in the same way as designing a type-1 fuzzy logic system.
ISSN:1063-6706
1941-0034
DOI:10.1109/TFUZZ.2016.2543746