Properties of interval type-2 defuzzification operators

IEEE Trans. Fuzzy Syst.

Coupland

et al. 2018

Self Cite

Abstract-Type-reduction of type-2 fuzzy sets is considered to be a defuzzification bottleneck because of the computational complexity involved in the process of type-reduction. In this research, we prove that the closed-form Nie-Tan operator, which outputs the average of the upper and lower bounds of the footprint of uncertainty, is actually an accurate method for defuzzifing interval type-2 fuzzy sets.

Section: Discussionmentioning

confidence: 99%

On Nie-Tan Operator and Type-Reduction of Interval Type-2 Fuzzy Sets

IEEE Trans. Fuzzy Syst.

Coupland

et al. 2018

Self Cite

“…This section follows a similar approach as early work on properties of type-1 defuzzification operators [24] and their extension to interval type-2 defuzzification operators [22]. Based on intuitive requirements we define five mathematical properties of type reduction methods for interval type-2 defuzzification and examine relations between these properties.…”

Section: Properties Of Type Reductionmentioning

confidence: 99%

“…One of the most popular methods for interval type-2 defuzzification is the Karnik-Mendel algorithm [11]. The mathematical properties of interval type-2 defuzzification have been studied in [22].…”

Section: Introductionmentioning

confidence: 99%

Type reduction operators for interval type–2 defuzzification

Runkler

Chen

2018

Information Sciences

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Fuzzy sets are an important approach to model uncertainty. Defuzzification maps fuzzy sets to non-fuzzy (crisp) values. Type-2 fuzzy sets model uncertainty in the degree of membership in a fuzzy set. Type-2 defuzzification maps type-2 fuzzy sets to non-fuzzy values. Type reduction maps type-2 fuzzy sets to type-1 fuzzy sets, in order to make type-2 defuzzification easier and to implement more efficient type-2 defuzzification algorithms. This paper is a first step towards a theoretical foundation of the emerging field of type reduction. Five mathematical properties of type reduction are defined, and two existing type reduction methods (Nie-Tan and uncertainty weight) are examined with respect to our five properties. Furthermore, two new type reduction methods are proposed: consistent linear type reduction and consistent quadratic type reduction. All our five properties are satisfied by consistent quadratic type reduction.

“…Then NT computes the conventional type-1 centroid (4) of this type-1 membership function. Type-1 conversion (11) and computation of the type-1 centroid using (4) is computationally much cheaper than iteratively minimizingc l (8) and maximizingc r (9). Therefore, the NT method is a popular low effort approximation of the KM method.…”

Section: Karnik-mendel Interval Type-defuzzificationmentioning

confidence: 99%

“…A set of desirable properties of interval type-2 defuzzification operators has been proposed in [11]. A popular method for interval type-2 defuzzification is the Karnik-Mendel (KM) method [3], which will be described in more detail in section 2.…”

Section: Introductionmentioning

confidence: 99%

Interval Type–2 Defuzzification Using Uncertainty Weights

Runkler

Coupland

Frontiers in Computational Intelligence

et al. 2017

Self Cite

One of the most popular interval type-2 defuzzification methods is the Karnik-Mendel (KM) algorithm. Nie and Tan (NT) have proposed an approximation of the KM method that converts the interval type-2 membership functions to a single type-1 membership function by averaging the upper and lower memberships, and then applies a type-1 centroid defuzzification. In this paper we propose a modification of the NT algorithm which takes into account the uncertainty of the (interval type-2) memberships. We call this method the uncertainty weight (UW) method. Extensive numerical experiments motivated by typical fuzzy controller scenarios compare the KM, NT, and UW methods. The experiments show that (i) in many cases NT can be considered a good approximation of KM with much lower computational complexity, but not for highly unbalanced uncertainties, and (ii) UW yields more reasonable results than KM and NT if more certain decision alternatives should obtain a larger weight than more uncertain alternatives.