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Fuzzy Rule Interpolation in Multidimensional Input Spaces: Techniques and Case Study Applications

This paper explores fuzzy rule interpolation methods for addressing gaps in sparse fuzzy rule bases within multidimensional input spaces. It discusses techniques including the original KH fuzzy interpolation, MACI (modified -cut interpolation), and a new improved method (IMUL). The findings demonstrate the applicability of these methods in engineering problems, particularly through a well log analysis case study. The proposed techniques allow for effective interpolation of fuzzy sets, eliminating the need for defuzzification and enhancing system control without increasing rule complexity.

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Fuzzy Rule Interpolation in Multidimensional Input Spaces: Techniques and Case Study Applications

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  1. Fuzzy Rule Interpolation for Multidimensional Input Spaces With Applications: A Case Study IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL 13, NO. 6, DECEMBER Kok Wai Wong, Domonkos Tikk, Tamas D. Gedeon and Laszlo T. Koczy Speaker: Yuan Kai Ko

  2. Outline • 1.Introduction • 2.Overview of fuzzy rule interpolation techniques • 3.Fuzzy rule interpolation for multidimensional input spaces • 4.Case study • 5.Conclusion

  3. 1.Introduction • When a fuzzy rule base contains gaps, which is called sparse rule base, classical fuzzy reasoning methods can no longer be used. This fact is due to the lack of traditional inference mechanism in the case when observations find no fuzzy rule to fire. • Fuzzy rule interpolation techniques provide a tool for specifying an output fuzzy set even when one or all of the input spaces are sparse. Kóczy and Hirota (KH) introduced the first interpolation approach known as (linear) KH interpolation.

  4. Introduction ?

  5. 2.Overview of fuzzy rule interpolation techniques

  6. The KH Rule Interpolation Technique • Every fuzzy sets can be approximated with the use of the family of its -cuts. • In the trapezoidal or triangular cases forα=0 and α=1. • A partial ordering can be introduced among CNF sets of the input by means of their -cuts

  7. 3.Fuzzy rule interpolation for multidimensional input spaces • In this paper, we will limit ourselves to the analysis of only three techniques that can be extended for use in multidimensional input spaces: the original KH fuzzy interpolation technique, the modified –cut fuzzy interpolation (MACI) technique and finally the new improved fuzzy interpolation technique for multidimensional input spaces (IMUL) proposed here.

  8. A. KH Fuzzy Rule Interpolation for Multidimensional Input Spaces

  9. B. MACI Fuzzy Rule Interpolation for Multidimensional Input Spaces

  10. C. IMUL Fuzzy Rule Interpolation for Multidimensional Input Spaces

  11. 4.Case study

  12. Application to Well Log Analysis

  13. 5.Conclusion • This technique can be used to interpolate the gaps between the rules for engineering problems with multidimensional input spaces • It does not require the application of any defuzzification methods when the observations are crisp. • This is significant as this will allow the use of a fuzzy system as an alternative for most engineering problems, at the same time without increasing the number of fuzzy rules that allows more human control.

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