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General Chebyshev Type Inequality for Seminormed Fuzzy Integral

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Modeling Decisions for Artificial Intelligence (MDAI 2019)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 11676))

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Abstract

We give a necessary and sufficient condition for Chebyshev type inequality in the class of seminormed fuzzy integrals with respect to m-positively dependent functions. Reviewing the literature many known results are generalized. We also present the Chebyshev type inequality for Shilkret integral for independent random variables.

This work was supported by the Slovak Research and Development Agency under the contract No. APVV-16-0337 and also cofinanced by bilateral call Slovak-Poland grant scheme No. SK-PL-18-0032 with the Polish National Agency for Academic Exchange PPN/BIL/2018/1/00049/U/00001.

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Notes

  1. 1.

    Functions \(f,g:X\rightarrow {\mathbb R}\) are called comonotone on \(D\subset X\) if \((f(x)-f(y))(g(x)-g(y))\geqslant 0\) for all \(x,y\in D.\) In the case \(D=X\) we will simply say “f, g are comonotone”.

References

  1. Agahi, H., Eslami, E.: A general inequality of Chebyshev type for semi(co)normed fuzzy integrals. Soft Comput. 15(4), 771–780 (2011)

    Article  Google Scholar 

  2. Agahi, H., Eslami, E., Mohammadpour, A., Vaezpour, S.M., Yaghoobi, M.A.: On Non-additive Probabilistic Inequalities of Hölder-type. Results Math. 61, 179–194 (2012)

    Article  MathSciNet  Google Scholar 

  3. Agahi, H., Mesiar, R., Ouyang, Y.: New general extensions of Chebyshev type inequalities for Sugeno integrals. Int. J. Approx. Reason. 51, 135–140 (2009)

    Article  MathSciNet  Google Scholar 

  4. Agahi, H., Mesiar, R., Ouyang, Y.: Further development of Chebyshev type inequalities for Sugeno integral and T-(S-)evaluators. Kybernetika 46, 83–95 (2010)

    MathSciNet  MATH  Google Scholar 

  5. Agahi, H., Mesiar, R., Ouyang, Y.: On some advanced type inequalities for Sugeno integral and T-(S-)evaluators. Inform. Sci. 190, 64–75 (2012)

    Article  MathSciNet  Google Scholar 

  6. Agahi, H., Mesiar, R., Ouyang, Y., Pap, E., Štrboja, M.: General Chebyshev type inequalities for universal integral. Inform. Sci. 187, 171–178 (2012)

    Article  MathSciNet  Google Scholar 

  7. Agahi, H., Mohammadpour, A., Vaezpour, S.M.: Predictive tools in data mining and k-means clustering: universal inequalities. Results Math. 63, 779–803 (2013)

    Article  MathSciNet  Google Scholar 

  8. Agahi, H., Yaghoobi, M.A.: On an extended Chebyshev type inequality for semi(co)normed fuzzy integrals. Int. J. Uncertain Fuzziness Knowl. Based Syst. 19, 781–798 (2011)

    Article  MathSciNet  Google Scholar 

  9. Armstrong, T.E.: Chebyshev inequalities and comonotonicity. Real Anal. Exchange 19, 266–268 (1993/94)

    Google Scholar 

  10. Boczek, M., Kaluszka, M.: On comonotone commuting and weak subadditivity properties of seminormed fuzzy integrals. Fuzzy Set Syst. 304, 35–44 (2016)

    Article  MathSciNet  Google Scholar 

  11. Borzová-Molnárová, J., Halčinová, L., Hutník, O.: The smallest semicopula-based universal integrals I: properties and characterizations. Fuzzy Set Syst. 271, 1–17 (2015)

    Article  MathSciNet  Google Scholar 

  12. Flores-Franulič, A., Román-Flores, H.: A Chebyshev type inequality for fuzzy integrals. Appl. Math. Comput. 190, 1178–1184 (2007)

    MathSciNet  MATH  Google Scholar 

  13. Girotto, B., Holzer, S.: A Chebyshev type inequality for Sugeno inegral and comonotonicity. Int. J. Approx. Reason. 52, 444–448 (2011)

    Article  Google Scholar 

  14. Kaluszka, M., Okolewski, A., Boczek, M.: On Chebyshev type inequalities for generalized Sugeno integrals. Fuzzy Set Syst. 244, 51–62 (2014)

    Article  MathSciNet  Google Scholar 

  15. Mao, Q.-S.: A note on the Chebyshev-type inequality of Sugeno integrals. Appl. Math. Comput. 212, 275–279 (2009)

    MathSciNet  MATH  Google Scholar 

  16. Mesiar, R., Ouyang, Y.: General Chebyshev type inequalities for Sugeno integrals. Fuzzy Set Syst. 160, 58–64 (2009)

    Article  MathSciNet  Google Scholar 

  17. Narukawa, Y., Murofushi, T.: Choquet integral and Sugeno integral as aggregation functions. In: Torra, V. (ed.) Information Fusion in Data Mining, Studies in Fuzziness and Soft Computing, vol. 123, pp. 27–39. Springer, Heidelberg (2003). https://doi.org/10.1007/978-3-540-36519-8_3

    Chapter  MATH  Google Scholar 

  18. Ouyang, Y., Fang, J., Wang, L.: Fuzzy Chebyshev type inequality. Int. J. Approx. Reason. 48, 829–835 (2008)

    Article  MathSciNet  Google Scholar 

  19. Ouyang, Y., Mesiar, R.: On the Chebyshev type inequality for seminormed fuzzy integral. Appl. Math. Lett. 22, 1810–1815 (2009)

    Article  MathSciNet  Google Scholar 

  20. Ouyang, Y., Mesiar, R., Agahi, H.: An inequality related to Minkowski type for Sugeno integrals. Inform. Sci. 180, 2793–2801 (2010)

    Article  MathSciNet  Google Scholar 

  21. Suárez-García, F., Álvarez-Gil, P.: Two families of fuzzy integrals. Fuzzy Set Syst. 18, 67–81 (1986)

    Article  MathSciNet  Google Scholar 

  22. Wang, Z., Klir, G.: Generalized Measure Theory. Springer, New York (2009). https://doi.org/10.1007/978-0-387-76852-6

    Book  MATH  Google Scholar 

  23. Wu, L., Sun, J., Ye, X., Zhu, L.: Hölder type inequality for Sugeno integral. Fuzzy Set Syst. 161, 2337–2347 (2010)

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Institute of Mathematics, Lodz University of Technology, ul. Wólczańska 215, 90-924, Lodz, Poland

    Michał Boczek

  2. Institute of Mathematics, Faculty of Science, Pavol Jozef Šafárik University in Košice, Jesenná 5, 040 01, Košice, Slovakia

    Anton Hovana & Ondrej Hutník

Authors
  1. Michał Boczek
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  2. Anton Hovana
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  3. Ondrej Hutník
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Corresponding author

Correspondence to Ondrej Hutník .

Editor information

Editors and Affiliations

  1. Maynooth University, Maynooth, Ireland

    Vicenç Torra

  2. Tamagawa University, Machida, Tokyo, Japan

    Yasuo Narukawa

  3. University of Milano-Bicocca, Milan, Italy

    Gabriella Pasi

  4. University of Milano-Bicocca, Milan, Italy

    Marco Viviani

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Boczek, M., Hovana, A., Hutník, O. (2019). General Chebyshev Type Inequality for Seminormed Fuzzy Integral. In: Torra, V., Narukawa, Y., Pasi, G., Viviani, M. (eds) Modeling Decisions for Artificial Intelligence. MDAI 2019. Lecture Notes in Computer Science(), vol 11676. Springer, Cham. https://doi.org/10.1007/978-3-030-26773-5_1

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  • DOI: https://doi.org/10.1007/978-3-030-26773-5_1

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-26772-8

  • Online ISBN: 978-3-030-26773-5

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