Type-2 Fuzzy Sets

This chapter formally introduces type-2 fuzzy sets and is the backbone for the rest of this book. It includes a lot of new terminologies. Coverage includes: the concept of a type-2 fuzzy set, definitions of general type-2 fuzzy sets and associated concepts, definitions of interval type-2 fuzzy sets and associated concepts, examples of two popular footprints of uncertainty, interval type-2 fuzzy numbers, a hierarchy of different kinds of type-2 fuzzy sets, mathematical representations of type-2 fuzzy sets including the vertical slice, wavy slice, and horizontal slice representations, which mathematical representations are most useful for optimal design applications, how to represent non-type-2 fuzzy sets as type-2 fuzzy sets, returning to linguistic labels for type-2 fuzzy sets, and multivariable MFs. 24 examples are used to illustrate the important concepts.

[1]  Jerry M. Mendel,et al.  Comments on "alpha -Plane Representation for Type-2 Fuzzy Sets: Theory and Applications" , 2010, IEEE Trans. Fuzzy Syst..

[2]  Jerry M. Mendel,et al.  General Type-2 Fuzzy Logic Systems Made Simple: A Tutorial , 2014, IEEE Transactions on Fuzzy Systems.

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

[4]  Jerry M. Mendel,et al.  Encoding Words Into Interval Type-2 Fuzzy Sets Using an Interval Approach , 2008, IEEE Transactions on Fuzzy Systems.

[5]  Jerry M. Mendel,et al.  On answering the question "Where do I start in order to solve a new problem involving interval type-2 fuzzy sets?" , 2009, Inf. Sci..

[6]  Jerry M. Mendel,et al.  Type-2 fuzzy sets and systems: an overview , 2007, IEEE Computational Intelligence Magazine.

[7]  Robert Ivor John,et al.  Learning of interval and general type-2 fuzzy logic systems using simulated annealing: Theory and practice , 2016, Inf. Sci..

[8]  Jerry M. Mendel,et al.  Comments on “Interval Type-2 Fuzzy Sets are Generalization of Interval-Valued Fuzzy Sets: Towards a Wide View on Their Relationship” , 2015, IEEE Transactions on Fuzzy Systems.

[9]  John T. Rickard,et al.  Type-2 Fuzzy Sets as Functions on Spaces , 2010, IEEE Transactions on Fuzzy Systems.

[10]  Robert Ivor John,et al.  Geometric Type-1 and Type-2 Fuzzy Logic Systems , 2007, IEEE Transactions on Fuzzy Systems.

[11]  Hani Hagras,et al.  Towards a linear general type-2 fuzzy logic based approach for computing with words , 2013, Soft Comput..

[12]  Jerry M. Mendel,et al.  $\alpha$-Plane Representation for Type-2 Fuzzy Sets: Theory and Applications , 2009, IEEE Transactions on Fuzzy Systems.

[13]  Jerry M. Mendel,et al.  Uncertainty measures for interval type-2 fuzzy sets , 2007, Inf. Sci..

[14]  Feilong Liu,et al.  An efficient centroid type-reduction strategy for general type-2 fuzzy logic system , 2008, Inf. Sci..

[15]  Hooman Tahayori,et al.  Modeling Complex Concepts with Type-2 Fuzzy Sets: The Case of User Satisfaction of Online Services , 2013, Advances in Type-2 Fuzzy Sets and Systems.

[16]  Hani Hagras,et al.  zSlices Based General Type-2 Fuzzy Sets and Systems , 2013, Advances in Type-2 Fuzzy Sets and Systems.

[17]  Humberto Bustince,et al.  Indicator of inclusion grade for interval-valued fuzzy sets. Application to approximate reasoning based on interval-valued fuzzy sets , 2000, Int. J. Approx. Reason..

[18]  M. Gorzałczany.  A method for inference in approximate reasoning based on interval-valued fuzzy sets , 1987 .

[19]  Robert Ivor John,et al.  Type-2 Fuzzy Logic: Challenges and Misconceptions [Discussion Forum] , 2012, IEEE Computational Intelligence Magazine.

[20]  Hani Hagras,et al.  Towards a general type-2 fuzzy logic approach for Computing With Words using linear adjectives , 2012, 2012 IEEE International Conference on Fuzzy Systems.

[21]  M. Mizumoto,et al.  Fuzzy sets and type 2 under algebraic product and algebraic sum , 1981 .

[22]  Jerry M. Mendel,et al.  On clarifying some definitions and notations used for type-2 fuzzy sets as well as some recommended changes , 2016, Inf. Sci..

[23]  Jerry M. Mendel,et al.  Perceptual Computing: Aiding People in Making Subjective Judgments , 2010 .

[24]  Masaharu Mizumoto,et al.  Some Properties of Fuzzy Sets of Type 2 , 1976, Inf. Control..

[25]  Hani Hagras,et al.  A Self-Tuning zSlices-Based General Type-2 Fuzzy PI Controller , 2015, IEEE Transactions on Fuzzy Systems.

[26]  Jerry M. Mendel,et al.  Type-2 fuzzy logic systems , 1999, IEEE Trans. Fuzzy Syst..

[27]  Witold Pedrycz,et al.  Concepts and Design Aspects of Granular Models of Type-1 and Type-2 , 2015, Int. J. Fuzzy Log. Intell. Syst..

[28]  Janusz T. Starczewski.  Efficient triangular type-2 fuzzy logic systems , 2009, Int. J. Approx. Reason..

[29]  Li Lushu.  Random fuzzy sets and fuzzy martingales , 1995 .

[30]  Müjde Güzelkaya,et al.  Granular type-2 membership functions: A new approach to formation of footprint of uncertainty in type-2 fuzzy sets , 2013, Appl. Soft Comput..

[31]  Hooman Tahayori,et al.  Approximated Type-2 Fuzzy Set Operations , 2006, 2006 IEEE International Conference on Fuzzy Systems.

[32]  Jerry M. Mendel.  Plotting 2-1/2 D figures for general type-2 fuzzy sets by hand or by PowerPoint , 2012, 2012 IEEE International Conference on Fuzzy Systems.

[33]  Lotfi A. Zadeh,et al.  The concept of a linguistic variable and its application to approximate reasoning-III , 1975, Inf. Sci..

[34]  Pratyusha Rakshit,et al.  Secondary membership evaluation in Generalized Type-2 Fuzzy Sets by evolutionary optimization algorithm , 2013, 2013 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE).

[35]  Robert Ivor John,et al.  Optimised Generalised Type-2 Join and Meet Operations , 2007, 2007 IEEE International Fuzzy Systems Conference.

[36]  Christian Wagner,et al.  Measuring the similarity between zSlices general type-2 fuzzy sets with non-normal secondary membership functions , 2016, 2016 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE).

[37]  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).

[38]  Hani Hagras,et al.  zSlices — towards bridging the gap between interval and general type-2 fuzzy logic , 2008, 2008 IEEE International Conference on Fuzzy Systems (IEEE World Congress on Computational Intelligence).

[39]  Jonathan M. Garibaldi,et al.  The Association between Non-Stationary and Interval Type-2 Fuzzy Sets: A Case Study , 2005, The 14th IEEE International Conference on Fuzzy Systems, 2005. FUZZ '05..

[40]  Hani Hagras,et al.  A general type-2 fuzzy logic approach for adaptive modeling of perceptions for Computing With Words , 2012, 2012 12th UK Workshop on Computational Intelligence (UKCI).

[41]  Hooman Tahayori,et al.  Concave type-2 fuzzy sets: properties and operations , 2010, Soft Comput..

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

[43]  Qihao Chen,et al.  On fuzzy-valued fuzzy reasoning , 2000, Fuzzy Sets Syst..

[44]  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).

[45]  Janusz T. Starczewski.  Extended triangular norms , 2009, Inf. Sci..

[46]  Simon Coupland,et al.  Type-2 Fuzzy Arithmetic using Alpha-planes , 2009, IFSA/EUSFLAT Conf..

[47]  Jerry Mendel,et al.  Type-2 Fuzzy Sets and Systems: An Overview [corrected reprint] , 2007, IEEE Computational Intelligence Magazine.

[48]  M. Beer,et al.  Fuzzy Randomness: Uncertainty in Civil Engineering and Computational Mechanics , 2004 .

[49]  Jerry M. Mendel,et al.  Operations on type-2 fuzzy sets , 2001, Fuzzy Sets Syst..

[50]  Hani Hagras,et al.  An experience based linear general type-2 fuzzy logic approach for Computing With Words , 2013, 2013 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE).

[51]  Jerry M. Mendel,et al.  Interval Type-2 Fuzzy Logic Systems Made Simple , 2006, IEEE Transactions on Fuzzy Systems.

[52]  A. Konar,et al.  A type-2 fuzzy classifier for gesture induced pathological disorder recognition , 2016, Fuzzy Sets Syst..

[53]  Hani Hagras,et al.  Dynamic Profile-Selection for zSlices based type-2 fuzzy agents controlling multi-user Ambient Intelligent Environments , 2012, 2012 IEEE International Conference on Fuzzy Systems.

[54]  Hani Hagras,et al.  Toward General Type-2 Fuzzy Logic Systems Based on zSlices , 2010, IEEE Transactions on Fuzzy Systems.

[55]  Lennart Ljung,et al.  System Identification: Theory for the User , 1987 .

[56]  Robert Ivor John,et al.  A novel alpha-cut representation for type-2 fuzzy sets , 2010, International Conference on Fuzzy Systems.

[57]  Humberto Bustince,et al.  The Notion of Weak-Contradiction: Definition and Measures , 2015, IEEE Transactions on Fuzzy Systems.