Abstract
This article provides a high-level retrospective of type-2 fuzzy sets and fuzzy logic systems. It explains how type-2 fuzzy sets can be used to model membership function uncertainties, and how by doing this smoother performance can be obtained than by using type-1 fuzzy sets. It also summarizes the notation that should be used for type-2 fuzzy sets, describes four important mathematical representations for these fuzzy sets, explains the differences between type-1 and type-2 fuzzy logic systems and which of the four representations is most useful when designing an optimal type-2 fuzzy logic system, provides a very useful strategy for optimal designs of fuzzy logic systems – one that guarantees performance improvement as one goes from a type-1 fuzzy logic system to a type-2 fuzzy logic system design – , and describes four methods for simplifying the designs of type-2 fuzzy logic systems. Finally, it explains why type-2 fuzzy sets can capture two kinds of linguistic uncertainties simultaneously (the uncertainty of an individual and the uncertainties of a group about a word), whereas type-1 fuzzy sets cannot, and that such type-2 fuzzy set word models are what should be used to implement Zadeh’s Computing With Words paradigm.
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References
Aisbett J, Rickard JT, Morgenthaler DG (2010) Type-2 fuzzy sets as functions on spaces. IEEE T Fuzzy Syst 18:841–844
Biglarbegian M, Melek WW, Mendel JM (2010) On the stability of interval type-2 TSK fuzzy logic control systems. IEEE Trans Syst Man Cybern – Part B: Cybern 40:798–818
Biglarbegian M, Melek WW, Mendel JM (2011) On the robustness of type-1 and interval type-2 fuzzy logic systems in modeling. Inform Sci 181:1325–1347
Bustince H, Fernandez J, Hagras H, Herrera F, Pagola M, Barrenechea E (2015) Interval type-2 fuzzy sets are generalization of interval-valued fuzzy sets: towards a wide view on their relationship. IEEE T Fuzzy Syst, early access, doi:10.1109/TFUZZ.2014.2362149
Castillo O, Martinez-Marroquin R, Melin P, Valdez F, Soria J (2012) Comparative study of bio-inspired algorithms applied to the optimization of type-1 and type-2 fuzzy controllers for an autonomous mobile robot. Inform Sci 192:19–38
Castillo O, Melin P, Alanis A, Montiel O, Sepulveda R (2011) Optimization of interval type-2 fuzzy logic controllers using evolutionary algorithms. J Soft Comp 15:1145–1160
Derrac J, Garcia S, Hui S, Suganthan PN, Herrera F (2014) Analyzing convergence performance of evolutionary algorithms: a statistical approach. Inform Sci 289:41–58
Karnik NN, Mendel JM (2001) Centroid of a type-2 fuzzy set. Inform Sci 132:195–220
Karnik NN, Mendel JM, Liang Q (1999) Type-2 fuzzy logic systems. IEEE T Fuzzy Syst 7:643–658
Kayacan E, Ahmadieh M (2015) Fuzzy Neural Networks for Real Time Applications. Elsevier, Amsterdam
Kennedy J, Eberhart R (1995) Particle swarm optimization. In: Proc IEEE Int’l Conf on Neural Networks, pp 1942–1948
Liang Q, Mendel JM (2000) Interval type-2 fuzzy logic systems: theory and design. IEEE T Fuzzy Syst 8:535–550
Liu F (2008) An efficient centroid type reduction strategy for general type-2 fuzzy logic system. Inform Sci 178:2224–2236
Lynch C, Hagras H, Callaghan V (2006) Using uncertainty bounds in the design of embedded real-time type-2 neuro-fuzzy speed controller for marine diesel engines. In: Proc IEEE FUZZ Conf, Vancouver, BC, Canada, pp 7217–7224
Mendel JM (2001) Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions. Prentice-Hall, Upper Saddle River, NJ
Mendel JM (2004) Computing derivatives in interval type-2 fuzzy logic systems. IEEE T Fuzzy Syst 12:84–98
Mendel JM (2007a) Advances in type-2 fuzzy sets and systems. Inform Sci 177:84–110
Mendel JM (2007b) Type-2 fuzzy sets and systems: an overview. IEEE Comput Intell Mag 2:20–29
Mendel JM (2009) On answering the question ‘Where do I start in order to solve a new problem involving type-2 fuzzy sets?’. Inform Sci 179:3418–3431
Mendel JM (2013a) On KM algorithms for solving type-2 fuzzy set problems. IEEE T Fuzzy Syst 21:426–446
Mendel JM (2013b) Type-2 fuzzy sets and beyond. In: Seising R, Trillas E, Moraga C, Termini S (eds) On Fuzziness: a Homage to Lotfi A. Zadeh, vol. 2, Ch 34. Springer, New York
Mendel JM (2014) General type-2 fuzzy logic systems made simple: a tutorial. IEEE T Fuzzy Syst 22:1162–1182
Mendel JM, Hagras H, Bustince H, Herrera F (2015) Comments on ‘Interval type-2 fuzzy sets are generalization of interval-valued fuzzy sets: towards a wide view on their relationship’ accepted for publication in IEEE T Fuzzy Syst
Mendel JM, Hagras H, Wan-Tan W, Melek W, Ying H (2014) Introduction to type-2 fuzzy logic control: theory and applications. Wiley and IEEE Press, Hoboken, NJ
Mendel JM, John RI (2002a) Type-2 fuzzy sets made simple. IEEE T Fuzzy Syst 10:117–127
Mendel JM, John RI (2002b) Footprint of uncertainty and its importance to type-2 fuzzy sets. In: Proc 6th IASTED Int’l Conf on Artificial Intelligence and Soft Computing, Banff, Canada
Mendel JM, John RI, Hagras H (2006) Standard background material about interval type-2 fuzzy logic systems that can be used by all authors, IEEE Computational Intelligence Society standard: can be accessed at http://cis.ieee.org/standards-committee.html, last access: 18.10.2015
Mendel JM, John RI, Liu F (2006) Interval type-2 fuzzy logic systems made simple. IEEE T Fuzzy Syst 14:808–821
Mendel JM, Liu F, Zhai D (2009) Alpha-plane representation for type-2 fuzzy sets: theory and applications. IEEE T Fuzzy Syst 17:1189–1207
Mendel JM, Liu X (2013) Simplified interval type-2 fuzzy logic systems. IEEE T Fuzzy Syst 21:1056–1069
Mendel JM, Rajati MR (2015a) Advanced computing with words: status and challenges. In: Seising R, Trillas E, Kacprzyk J (eds) Fuzzy Logic: Towards the Future. Springer, New York
Mendel JM, Rajati MR (2015b) “On clarifying some notations used for type-2 fuzzy sets as well as some recommended notational changes,” revised submission sent to Information Sciences
Mendel JM, Wu D (2010) Perceptual Computing: Aiding People in Making Subjective Judgments. Wiley and IEEE Press, Hoboken, NJ
Mizumoto M, Tanaka K (1976) Some properties of fuzzy sets of type-2. Inform Control 31:312–340
Mizumoto M, Tanaka K (1981) Fuzzy sets of type-2 under algebraic product and algebraic sum. Fuzzy Sets Syst 5:277–290
Nie M, Tan WW (2008) Towards an efficient type-reduction method for interval type-2 fuzzy logic systems. In: Proc IEEE FUZZ Conf: Paper # FS0339, Hong Kong, China
Trawinski B, Smetek M, Telec Z, Lasota T (2012) Nonparametric statistical analysis for multiple comparison of machine learning regression algorithms. Int J Appl Math Comput Sci 22:867–881
Wagner C, Hagras H (2008) z slices – towards bridging the gap between interval and general type-2 fuzzy logic. In: Proc IEEE FUZZ Conf, Paper # FS0126, Hong Kong, China, pp 489–497
Wagner C, Hagras H (2010) Towards general type-2 fuzzy logic systems based on zslices. IEEE T Fuzzy Syst 18:637–660
Wang L-X, Mendel JM (1992) Fuzzy basis function, universal approximation, and orthogonal least-squares learning. IEEE T Neural Netw 3:807–814
Wei F, Jun S, ** XZ, Xu WB (2010) Convergence analysis of quantum-behaved particle swarm optimization algorithm and study on its control parameter. Acta Phys Sin 59:3686–3694
Wu D, Mendel JM (2011) On the continuity of type-1 and interval type-2 fuzzy logic systems. IEEE T Fuzzy Syst 19:179–192
Wu H, Mendel JM (2002) Uncertainty bounds and their use in the design of interval type-2 fuzzy logic systems. IEEE T Fuzzy Syst 10:622–639
Zadeh LA (1965) Fuzzy sets. Inform Control 8:338–353
Zadeh LA (1996) Fuzzy logic = computing with words. IEEE T Fuzzy Syst 4:103–111
Zadeh LA (1999) From computing with numbers to computing with words – from manipulation of measurements to manipulation of perceptions. IEEE T Circuits-1 4:105–119
Zadeh LA (2012) Computing with Words: Principal Concepts and Ideas. Springer, New York
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Mendel, J. Type-2 Fuzzy Sets and Systems: a Retrospective. Informatik Spektrum 38, 523–532 (2015). https://doi.org/10.1007/s00287-015-0927-4
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DOI: https://doi.org/10.1007/s00287-015-0927-4