Advertisement

Description and Properties of Curve-Based Monotone Functions

  • Mikel Sesma-SaraEmail author
  • Laura De Miguel
  • Antonio Francisco Roldán López de Hierro
  • Jana Špirková
  • Radko Mesiar
  • Humberto Bustince
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 981)

Abstract

Curve-based monotonicity is one of the lately introduced relaxations of monotonicity. As directional monotonicity regards monotonicity along fixed rays, which are given by real vectors, curve-based monotonicity studies the increase of functions with respect to a general curve \(\alpha \). In this work we study some theoretical properties of this type of monotonicity and we relate this concept with previous relaxations of monotonicity.

Keywords

Curve-based monotonicity Weak monotonicity Directional monotonicity Aggregation function 

Notes

Acknowledgements

This work is supported by the research group FQM268 of Junta de Andalucía, by the project TIN2016-77356-P (AEI/FEDER, UE), by the Slovak Scientific Grant Agency VEGA no. 1/0093/17 Identification of risk factors and their impact on products of the insurance and savings schemes, by Slovak grant APVV-14-0013, and by Czech Project LQ1602 “IT4Innovations excellence in science”.

References

  1. 1.
    Beliakov, G., Calvo, T., Wilkin, T.: Three types of monotonicity of averaging functions. Knowl.-Based Syst. 72, 114–122 (2014).  https://doi.org/10.1016/j.knosys.2014.08.028Google Scholar
  2. 2.
    Beliakov, G., Špirková, J.: Weak monotonicity of Lehmer and Gini means. Fuzzy Sets Syst. 299, 26–40 (2016).  https://doi.org/10.1016/j.fss.2015.11.006Google Scholar
  3. 3.
    Beliakov, G., Calvo, T., Wilkin, T.: On the weak monotonicity of Gini means and other mixture functions. Inf. Sci. 300, 70–84 (2015).  https://doi.org/10.1016/j.ins.2014.12.030Google Scholar
  4. 4.
    Bustince, H., Barrenechea, E., Sesma-Sara, M., Lafuente, J., Dimuro, G.P., Mesiar, R., Kolesárová, A.: Ordered directionally monotone functions. Justification and application. IEEE Trans. Fuzzy Syst. 26(4), 2237–2250 (2018).  https://doi.org/10.1109/TFUZZ.2017.2769486Google Scholar
  5. 5.
    Bustince, H., Fernandez, J., Kolesárová, A., Mesiar, R.: Directional monotonicity of fusion functions. Eur. J. Oper. Res. 244(1), 300–308 (2015).  https://doi.org/10.1016/j.ejor.2015.01.018Google Scholar
  6. 6.
    Elkano, M., Sanz, J.A., Galar, M., Pekala, B., Bentkowska, U., Bustince, H.: Composition of interval-valued fuzzy relations using aggregation functions. Inf. Sci. 369, 690–703 (2016).  https://doi.org/10.1016/j.ins.2016.07.048Google Scholar
  7. 7.
    Gagolewski, M.: Data fusion: theory, methods, and applications. Institute of Computer Science Polish Academy of Sciences (2015)Google Scholar
  8. 8.
    García-Lapresta, J., Martínez-Panero, M.: Positional voting rules generated by aggregation functions and the role of duplication. Int. J. Intell. Syst. 32(9), 926–946 (2017).  https://doi.org/10.1002/int.21877Google Scholar
  9. 9.
    Roldán López de Hierro, A.F., Sesma-Sara, M., Špirková, J., Lafuente, J., Pradera, A., Mesiar, R., Bustince, H.: Curve-based monotonicity: a generalization of directional monotonicity. Int. J. General Syst. (2019).  https://doi.org/10.1080/03081079.2019.1586684
  10. 10.
    Lucca, G., Sanz, J., Dimuro, G., Bedregal, B., Asiain, M.J., Elkano, M., Bustince, H.: CC-integrals: Choquet-like copula-based aggregation functions and its application in fuzzy rule-based classification systems. Knowl.-Based Syst. 119, 32–43 (2017).  https://doi.org/10.1016/j.knosys.2016.12.004Google Scholar
  11. 11.
    Lucca, G., Sanz, J.A., Dimuro, G.P., Bedregal, B., Mesiar, R., Kolesárová, A., Bustince, H.: Preaggregation functions: construction and an application. IEEE Trans. Fuzzy Syst. 24(2), 260–272 (2016).  https://doi.org/10.1109/TFUZZ.2015.2453020Google Scholar
  12. 12.
    Lucca, G., Sanz, J.A., Dimuro, G.P., Bedregal, B., Bustince, H., Mesiar, R.: CF-integrals: a new family of pre-aggregation functions with application to fuzzy rule-based classification systems. Inf. Sci. 435, 94–110 (2018).  https://doi.org/10.1016/j.ins.2017.12.029Google Scholar
  13. 13.
    Mesiar, R., Kolesárová, A., Stupňanová, A.: Quo vadis aggregation? Int. J. General Syst. 47(2), 97–117 (2018).  https://doi.org/10.1080/03081079.2017.1402893Google Scholar
  14. 14.
    Paternain, D., Fernandez, J., Bustince, H., Mesiar, R., Beliakov, G.: Construction of image reduction operators using averaging aggregation functions. Fuzzy Sets Syst. 261, 87–111 (2015).  https://doi.org/10.1016/j.fss.2014.03.008Google Scholar
  15. 15.
    Rousseeuw, P.J., Leroy, A.M.: Robust Regression and Outlier Detection, vol. 589. Wiley, Hoboken (2005)Google Scholar
  16. 16.
    Sesma-Sara, M., Lafuente, J., Roldán, A., Mesiar, R., Bustince, H.: Strengthened ordered directionally monotone functions. Links between the different notions of monotonicity. Fuzzy Sets Syst. 357, 151–172 (2019).  https://doi.org/10.1016/j.fss.2018.07.007Google Scholar
  17. 17.
    Sesma-Sara, M., Bustince, H., Barrenechea, E., Lafuente, J., Kolesárová, A., Mesiar, R.: Edge detection based on ordered directionally monotone functions. In: Advances in Fuzzy Logic and Technology 2017, pp. 301–307. Springer (2017).  https://doi.org/10.1007/978-3-319-66827-7_27
  18. 18.
    Sesma-Sara, M., Mesiar, R., Bustince, H.: Weak and directional monotonicity of functions on Riesz spaces to fuse uncertain data. Fuzzy Sets Syst. (in press).  https://doi.org/10.1016/j.fss.2019.01.019
  19. 19.
    Wilkin, T., Beliakov, G.: Weakly monotonic averaging functions. Int. J. Int. Syst. 30(2), 144–169 (2015).  https://doi.org/10.1002/int.21692Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Mikel Sesma-Sara
    • 1
    • 2
    Email author
  • Laura De Miguel
    • 1
    • 2
  • Antonio Francisco Roldán López de Hierro
    • 3
  • Jana Špirková
    • 4
  • Radko Mesiar
    • 5
    • 6
  • Humberto Bustince
    • 1
    • 2
  1. 1.Public University of NavarraPamplonaSpain
  2. 2.Institute of Smart Cities (UPNA)PamplonaSpain
  3. 3.University of GranadaGranadaSpain
  4. 4.Matej Bel UniversityBanská BystricaSlovakia
  5. 5.Slovak University of Technology in BratislavaBratislavaSlovakia
  6. 6.University of OstravaOstravaCzech Republic

Personalised recommendations