Abstract
In many contexts, type-2 fuzzy sets (T2 FS) are obtained from a type-1 fuzzy set to which we wish to add uncertainty. However, in the current type-2 representation, there is no restriction on the shape of the footprint of uncertainty and the embedded sets (ESs) that can be considered acceptable. This leads, usually, to the loss of the semantic relationship between the T2 FS and the concept it models. As a consequence, the interpretability of some of the ESs and the explainability of the uncertainty measures obtained from them can decrease. To overcome these issues, constrained type-2 (CT2) fuzzy sets have been proposed. However, no formal definitions for some of their key components [e.g., acceptable ESs (AESs)] and constrained operations have been given. In this article, we provide some theoretical underpinning for the definition of CT2 sets, their inferencing and defuzzification method. To conclude, the constrained inference framework is presented, applied to two real-world cases and briefly compared to the standard interval type-2 inference and defuzzification method.
Highlights
T YPE-2 (T2) fuzzy sets (T2 FS) were introduced by Zadeh [1] in 1975 as an extension of type-1 (T1) fuzzy sets (T1 FS) so that it would be possible to model the uncertainty of membership functions (MFs)
Proof: To do that, we show that it is possible to write the union of all the S ∈ CAESAas (14), by rewriting S as in (18)
Those embedded sets (ESs) have been obtained as the result of the defuzzification of the output of a constrained IT2 (CIT2) FLS generated through the learning framework described
Summary
T YPE-2 (T2) fuzzy sets (T2 FS) were introduced by Zadeh [1] in 1975 as an extension of type-1 (T1) fuzzy sets (T1 FS) so that it would be possible to model the uncertainty of membership functions (MFs). T2 fuzzy systems have required the creation of additional representations, definitions, and algorithms, including to allow the creation of complete rule-based inferencing systems One of these is the concept of the footprint of uncertainty (FOU), introduced by Mendel and John [5], which represents the existence of nonzero secondary membership values as a 2-D shaded area. Two properties, which we believe decrease the overall interpretability of T2 systems, are: 1) there is currently no agreed mechanism to derive the FOU, in the situation in which a concept being modeled by a T1 set has uncertainty added to form a T2 set representing the same concept; and 2) ESs may have any shape, including ones which bear no relationship to the concept being modeled To overcome these issues, constrained T2 (CT2) fuzzy sets (CT2 FS) have been proposed [9], [10]. Throughout, we stress that the proposed CIT2 approach, which may be used in contexts in which explainability and interpretability are considered important, is an alternative to other approaches including the conventional T2 approach
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