Super-Exponential Convergence of the Karnik-Mendel Algorithms Used for Type-reduction in Interval Type-2 Fuzzy Logic Systems

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.

[1]  Jerry M. Mendel,et al.  Interval type-2 fuzzy logic systems , 2000, Ninth IEEE International Conference on Fuzzy Systems. FUZZ- IEEE 2000 (Cat. No.00CH37063).

[2]  Jerry M. Mendel,et al.  Equalization of nonlinear time-varying channels using type-2 fuzzy adaptive filters , 2000, IEEE Trans. Fuzzy Syst..

[3]  Dong Hoon Lee,et al.  An efficient algorithm for fuzzy weighted average , 1997, Fuzzy Sets Syst..

[4]  Jerry M. Mendel,et al.  Type-2 fuzzy sets made simple , 2002, IEEE Trans. Fuzzy Syst..

[5]  F. S. Wong,et al.  Fuzzy weighted averages and implementation of the extension principle , 1987 .

[6]  K. Wu.  Fuzzy interval control of mobile robots , 1996 .

[7]  Jerry M. Mendel,et al.  Type-2 Fuzzistics for Symmetric Interval Type-2 Fuzzy Sets: Part 2, Inverse Problems , 2007, IEEE Transactions on Fuzzy Systems.

[8]  Mao-Jiun J. Wang,et al.  Fuzzy weighted average: an improved algorithm , 1992 .

[9]  Jerry M. Mendel,et al.  Modeling MPEG VBR video traffic using type-2 fuzzy logic systems , 2001 .

[10]  Hani Hagras,et al.  A hierarchical type-2 fuzzy logic control architecture for autonomous mobile robots , 2004, IEEE Transactions on Fuzzy Systems.

[11]  Frank Chung-Hoon Rhee,et al.  An Interval Type-2 Fuzzy C Spherical Shells , 2004 .

[12]  E. Stanley Lee,et al.  Fuzzy weighted average: The linear programming approach via Charnes and Cooper's rule , 2001, Fuzzy Sets Syst..

[13]  Jerry M. Mendel,et al.  Connection admission control in ATM networks using survey-based type-2 fuzzy logic systems , 2000, IEEE Trans. Syst. Man Cybern. Part C.

[14]  N. N. Karnik,et al.  Introduction to type-2 fuzzy logic systems , 1998, 1998 IEEE International Conference on Fuzzy Systems Proceedings. IEEE World Congress on Computational Intelligence (Cat. No.98CH36228).

[15]  Jerry M. Mendel,et al.  New results about the centroid of an interval type-2 fuzzy set, including the centroid of a fuzzy granule , 2007, Inf. Sci..

[16]  Jerry M. Mendel,et al.  Designing interval type-2 fuzzy logic systems using an SVD-QR method: Rule reduction , 2000, Int. J. Intell. Syst..

[17]  J. Mendel,et al.  Overcoming time-varying co-channel interference using type-2 fuzzy adaptive filters , 2000 .

[18]  E. Lee,et al.  Fuzzy weighted average: A max-min paired elimination method , 1996 .

[19]  Jerry M. Mendel,et al.  Centroid uncertainty bounds for interval type-2 fuzzy sets: forward and inverse problems , 2004, 2004 IEEE International Conference on Fuzzy Systems (IEEE Cat. No.04CH37542).

[20]  Jerry M. Mendel,et al.  Type-2 Fuzzistics for Symmetric Interval Type-2 Fuzzy Sets: Part 1, Forward Problems , 2006, IEEE Transactions on Fuzzy Systems.

[21]  Jerry M. Mendel,et al.  Centroid of a type-2 fuzzy set , 2001, Inf. Sci..

[22]  J. Mendel.  Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions , 2001 .

[23]  Janusz T. Starczewski,et al.  Interval Type 2 Neuro-Fuzzy Systems Based on Interval Consequents , 2003 .

[24]  Jerry M. Mendel,et al.  Uncertainty bounds and their use in the design of interval type-2 fuzzy logic systems , 2002, IEEE Trans. Fuzzy Syst..