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Interval Type-2 Fuzzy Systems

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Uncertain Rule-Based Fuzzy Systems

Abstract

This chapter explores many aspects of the interval type-2 fuzzy system that was introduced in Chap. 1. As was done for type-1 fuzzy systems, it provides a very comprehensive and unified description of the two major kinds of interval type-2 fuzzy systems that are widely used in real-world applications—IT2 Mamdani and TSK fuzzy systems. Importantly, it also distinguishes between IT2 fuzzy systems that include type-reduction followed by defuzzification and those that bypass type-reduction and use direct defuzzification. The coverage of this chapter includes IT2 rules, three kinds of fuzzifiers (singleton , type-1 non-singleton , and IT2 non-singleton ), input–output formulas for the fuzzy inference engine (also valid for GT2 fuzzy systems), the effects of the three kind of fuzzifiers on the input–output formulas (valid for IT2 fuzzy systems), IT2 first -and second-order rule partitions , combining or not combining fired-rule output sets on the way to defuzzification , type-reduction (centroid, height, and center-of-sets) + defuzzification for an IT2 Mamdani fuzzy system, type-reduction + defuzzification for four kinds of IT2 TSK fuzzy systems, novelty partitions , approximate type-reduction and defuzzification (the Wu–Mendel Uncertainty Bounds ), direct defuzzification (Nie–Tan and Biglarbegian–Melek–Mendel), IT2 fuzzy basis functions which provide a mathematical description of an IT2 fuzzy system from its input to its output, remarks, and insights about an IT2 fuzzy system (including layered architecture interpretations for it, fundamental differences between type-1 and IT2 fuzzy systems, universal approximation by it, continuity of it, rule explosion and some ways to control it , and rule interpretability for it), and historical notes . Seventeen examples are used to illustrate the important concepts and there is also a comprehensive numerical example in Sects. 9.7 and 9.11.

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Notes

  1. 1.

    Separable MFs are assumed, as is discussed in Sect. 6.10.

  2. 2.

    Equation (9.12) is the MF of a vertical slice, which is a T1 FS, whose name is \( \tilde{B}^{l} (y|{\mathbf{x}}^{\prime } ) \), hence, the notation \( \mu_{{\tilde{B}^{l} (y|{\mathbf{x}}^{\prime } )}} \).

  3. 3.

    This uses the Associative and Commutative Laws for the meet, which is okay to do for both the minimum and product t-norms and any kind of secondary MFs (see Tables 7.1 and 7.2).

  4. 4.

    This example can be skipped without any loss of continuity, because it only provides a different derivation of the already-derived results that are in Corollaries 9.1 and 9.2.

  5. 5.

    Although it is unnecessary to use the subscript 1 on x for a single-antecedent rule, by doing so the multiple-antecedent case in Mendel et al. (2006) will be easier to understand because of the presence of the subscript 1 in all of the notation and formulas.

  6. 6.

    In (3.20), the superscript l denotes rule number. Since the focus here is on a single rule, this superscript is not used here. Instead, superscripts are associated with specific embedded type-1 FSs.

  7. 7.

    Readers who are not interested in non-singleton fuzzification can immediately go to Sect. 9.4.2.4.

  8. 8.

    For a type-1 fuzzy system there were three possible ways to combined fired rule output sets, one of which was combining by using a weighted combination. To-date, this approach has not been used for T2 fuzzy systems, due arguably to the additional complexity of having to add T2 FSs.

  9. 9.

    Much of this example was provided by Dongrui Wu, and is used with his permission.

  10. 10.

    Note, e.g., that in (9.142) {LMFs, left} refers to the fact that this centroid only uses lower MFs of the firing interval and left-endpoint values of the consequent set centroid. In addition, in (9.142)–(9.145), (9.149) and (9.150), \( \underline{f}^{s} ({\mathbf{x}}) \) and \( \overline{f}^{s} ({\mathbf{x}}) \) have been shortened to \( \underline{f}^{s} \) and \( \overline{f}^{s} \), respectively.

  11. 11.

    Li et al. (2011) have extended (9.158) to the situation where the same \( y_{i} \) does not appear in both terms of (9.158).

  12. 12.

    The IT2 FBFs are shown as a function of x rather than of \( {\mathbf{x}}^{\prime } \) since they are valid for \( {\mathbf{x}} \in {\mathbf{X}} \).

  13. 13.

    An equivalent way of saying this is that in (9.111) and (9.112) L and R are not known ahead of time.

  14. 14.

    Proofs of these findings are provided in Wu and Mendel (2011).

  15. 15.

    A function \( f(x) \) has a jump discontinuity at c if: \( f(c) \) is defined but \( { \lim }_{{x \rightarrow c^{ + } }} f(x) \ne { \lim }_{{x \rightarrow c^{ - } }} f(x) \), i.e. both \( f(c) \) and \( f(c + \delta ) \) are defined, but \( f(c + \delta ) \) does not approach \( f(c) \) as \( \delta \) approaches 0.

  16. 16.

    The colorized Figs. 9.17 and 9.18 were provided by Dongrui Wu.

  17. 17.

    This kind of hierarchical approach can also be used in a T1 fuzzy system.

  18. 18.

    This reference incorrectly shows Gorzalczany as the sole author of this paper.

  19. 19.

    Their function is explained here using this book’s notation, which is quite different from their notation.

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  1. University of Southern California, Los Angeles, CA, USA

    Jerry M. Mendel

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  1. Jerry M. Mendel
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Mendel, J.M. (2017). Interval Type-2 Fuzzy Systems. In: Uncertain Rule-Based Fuzzy Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-51370-6_9

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