Super-Exponential Convergence of the Karnik–Mendel Algorithms for Computing the Centroid of an Interval Type-2 Fuzzy Set

Author

Mendel, Jerry M. ; Liu, Feilong

Author_Institution

Dept. of Electr. Eng., Univ. of Southern California, Los Angeles, CA

Volume

15

Issue

2

fYear

2007

fDate

4/1/2007 12:00:00 AM

Firstpage

309

Lastpage

320

Abstract

Computing the centroid of an interval T2 FS is an important operation in a type-2 fuzzy logic system (where it is called type-reduction), but it is also a potentially time-consuming operation. The Karnik-Mendel (KM) iterative algorithms are widely used for doing this. In this paper, we prove that these algorithms converge monotonically and super-exponentially fast. Both properties are highly desirable for iterative algorithms and explain why in practice the KM algorithms have been observed to converge very fast, thereby making them very practical to use

Keywords

convergence; fuzzy set theory; iterative methods; Karnik-Mendel iterative algorithms; interval type-2 fuzzy set centroid; super-exponential convergence; type-2 fuzzy logic system; Convergence; Extraterrestrial measurements; Frequency selective surfaces; Fuzzy logic; Fuzzy sets; Fuzzy systems; Image processing; Iterative algorithms; Measurement uncertainty; Signal processing; Centroid; Karnik–Mendel (KM) algorithms; interval type-2 fuzzy sets; type-2 fuzzy sets;

fLanguage

English

Journal_Title

Fuzzy Systems, IEEE Transactions on

Publisher

ieee

ISSN

1063-6706

Type

jour

DOI

10.1109/TFUZZ.2006.882463

Filename

4142760