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Results in Mathematics

, Volume 63, Issue 3–4, pp 779–803 | Cite as

Predictive tools in data mining and k-means clustering: Universal Inequalities

  • Hamzeh Agahi
  • A. Mohammadpour
  • S. Mansour Vaezpour
Article

Abstract

Grouping data into meaningful clusters is very important in data mining. K-means clustering is a fast method for finding clusters in data. The integral inequalities are a predictive tool in data mining and k-means clustering. Many papers have been published on speeding up k-means or nearest neighbor search using inequalities that are specific for Euclidean distance. An extended inequality related to Hölder type for universal integral is obtained in a rather general form. Previous results of Agahi et al. (Results Math, 61:179–194, 2012) are generalized by relaxing some of their requirements, thus closing the series of papers on this topic. Chebyshev’s, Hölder’s, Minkowski’s, Stolarsky’s, Jensen’s and Lyapunov’s type inequalities for the universal integral are obtained.

Mathematics Subject Classification (2000)

Primary 60E15 Secondary 39B62 

Keywords

Monotone measure Universal integral Data mining Chebyshev’s inequality Stolarsky’s inequality Minkowski’s inequality Jensen’s inequality Lyapunov’s inequality Hölder’s inequality 

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References

  1. 1.
    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)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Agahi H., Mesiar R., Ouyang Y., Pap E., Štrboja M.: General Chebyshev type inequalities for universal integral. Inf. Sci 187, 171–178 (2012)MATHCrossRefGoogle Scholar
  3. 3.
    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, 781–797 (2011)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Agahi H., Mesiar R., Ouyang Y.: General Minkowski type inequalities for Sugeno integrals. Fuzzy Sets Syst. 161, 708–715 (2010)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Agahi H., Mesiar R., Ouyang Y.: New general extensions of Chebyshev type inequalities for Sugeno integrals. Int. J. Approx. Reason. 51, 135–140 (2009)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Bassan B., Spizzichino F.: Relations among univariate aging, bivariate aging and dependence for exchangeable lifetimes. J. Multivar. Anal. 93, 313–339 (2005)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Benvenuti P., Mesiar R., Vivona D.: Monotone set functions-based integrals. In: Pap, E. (eds) Handbook of Measure Theory, vol. II, pp. 1329–1379. Elsevier, Amsterdam (2002)CrossRefGoogle Scholar
  8. 8.
    Choquet, G.: Theory of capacities. Ann. Inst. Fourier (Grenoble) 5, 131–292 (1953–1954)Google Scholar
  9. 9.
    Dellacherie C., Quelques commentaires sur les prolongements de capacités. In: Seminaire de Probabilites (1969/1970), Strasbourg. Lecture Notes in Mathematics, vol. 191. pp. 77–81. Springer, Berlin (1970)Google Scholar
  10. 10.
    Durante F., Sempi C.: Semicopulae. Kybernetika 41, 315–328 (2005)MathSciNetMATHGoogle Scholar
  11. 11.
    Flores-Franulič A., Román-Flores H.: A Chebyshev type inequality for fuzzy integrals. Appl. Math. Comput. 190, 1178–1184 (2007)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Grabisch, M., Murofushi, T., Sugeno, M. (eds): Fuzzy measures and integrals Theory and Applications.. Physica-Verlag, Heidelberg (2000)Google Scholar
  13. 13.
    Klement E.P., Mesiar R., Pap E.: Triangular norms, Trends in Logic. Studia Logica Library, vol. 8. Kluwer, Dodrecht (2000)CrossRefGoogle Scholar
  14. 14.
    Klement E.P., Mesiar R., Pap E.: A universal integral as common frame for Choquet and Sugeno integral. IEEE Trans. Fuzzy Syst. 18(1), 178–187 (2010)CrossRefGoogle Scholar
  15. 15.
    Klement E.P., Ralescu D.A.: Nonlinearity of the fuzzy integral. Fuzzy Sets Syst. 11, 309–315 (1983)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Mesiar R.: Choquet-like integrals. J. Math. Anal. Appl. 194, 477–488 (1995)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Mesiar R., Mesiarov’a A.: Fuzzy integrals and linearity. Int. J. Approx. Reason. 47, 352–358 (2008)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Mesiar R., Ouyang Y.: General Chebyshev type inequalities for Sugeno integrals. Fuzzy Sets Syst. 160, 58–64 (2009)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Ouyang Y., Fang J., Wang L.: Fuzzy Chebyshev type inequality. Int. J. Approx. Reason. 48, 829–835 (2008)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Ouyang Y., Mesiar R., Agahi H.: An inequality related to Minkowski type for Sugeno integrals. Inf. Sci. 180, 2793–2801 (2010)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Ouyang Y., Mesiar R.: On the Chebyshev type inequality for seminormed fuzzy integral. Appl. Math. Lett. 22, 1810–1815 (2009)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Pap E.: Null-Additive Set Functions. Kluwer, Dordrecht (1995)MATHGoogle Scholar
  23. 23.
    Pap E.: Handbook of Measure Theory. Elsevier, Amsterdam (2002)MATHGoogle Scholar
  24. 24.
    Román-Flores H., Flores-Franulič A., Chalco-Cano Y.: A Jensen type inequality for fuzzy integrals. Inf. Sci. 177, 3192–3201 (2007)MATHCrossRefGoogle Scholar
  25. 25.
    Saminger S., Mesiar R., Bodenhofer U.: Domination of aggregation operators and preservation of transitivity. Int. J. Uncertain. Fuzziness Knowl. Based Syst. 10(Suppl.), 11–36 (2002)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Shilkret N.: Maxitive measure and integration. Indag. Math. 33, 109–116 (1971)MathSciNetGoogle Scholar
  27. 27.
    Suárez García F., Gil Álvarez P.: Two families of fuzzy integrals. Fuzzy Sets Syst. 18, 67–81 (1986)MATHCrossRefGoogle Scholar
  28. 28.
    Sugeno, M.: Theory of fuzzy integrals and its applications. PhD Dissertation. Tokyo Institute of Technology (1974)Google Scholar
  29. 29.
    Sugeno M., Murofushi T.: Pseudo-additive measures and integrals. J. Math. Anal. Appl. 122, 197–222 (1987)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Wang Z., Klir G.J.: Fuzzy Measure Theory. Plenum Press, New York (1992)MATHCrossRefGoogle Scholar
  31. 31.
    Weber S.: Two integrals and some modified versions: critical remarks. Fuzzy Sets Syst. 20, 97–105 (1986)MATHCrossRefGoogle Scholar
  32. 32.
    Wu L., Sun J., Ye X., Zhu L.: Hölder type inequality for Sugeno integrals. Fuzzy Sets Syst. 161, 2337–2347 (2010)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  • Hamzeh Agahi
    • 1
    • 2
  • A. Mohammadpour
    • 1
  • S. Mansour Vaezpour
    • 1
  1. 1.Department of Mathematics and Computer ScienceAmirkabir University of Technology (Tehran Polytechnic)TehranIran
  2. 2.Statistical Research and Training Center (SRTC)TehranIran

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