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Directional and Ordered Directional Monotonicity of Mixture Functions

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Book cover Aggregation Functions in Theory and in Practice (AGOP 2017)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 581))

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Abstract

In this contribution, we discuss the concepts of so-called fusion functions, pre-aggregation functions and their directional and ordered directional monotonicity in the context of mixture functions. Mixture functions represent a special class of weighted averaging functions whose weights are determined by continuous weighting functions which depend on the input values. They need not be monotone, in general. If they are monotone increasing, they also belong to the important class of aggregation functions. If the are directionally monotone, they belong to the class of pre-aggregation functions.

Currently there is increased interest in studying generalized forms of monotonicity such as weak, directional or ordered directional monotonicity due to their possible application in fields such classification or image processing.

This paper discusses properties of selected mixture functions with special emphasis on their directional and ordered directional monotonicity. The concept of directional and ordered directional monotonicity of mixture functions is investigated with respect to linear and quadratic weighting functions.

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Notes

  1. 1.

    The term “increasing” is understood in a non-strict sense.

References

  1. Beliakov, G., Pradera, A., Calvo, T.: A Guide for Practitioners. Springer, Berlin (2007)

    Google Scholar 

  2. Beliakov, G., Calvo, T., Wilkin, T.: Three types of monotonicity of averaging functions. Knowl.-Based Syst. 72, 114–122 (2014)

    Article  Google Scholar 

  3. Bustince, H., Barrenechea, E., Calvo, T., James, S., Beliakov, G.: Consensus in multi-expert decision making problems using penalty functions defined over a Cartesian product of lattices. Inf. Fusion 17, 56–64 (2014)

    Article  Google Scholar 

  4. Beliakov, G., Špirková, J.: Weak monotonicity of Lehmer and Gini means. Fuzzy Sets Syst. 299, 26–40 (2016)

    Article  MathSciNet  Google Scholar 

  5. Bustince, H., Fernandez, J., Kolesárová, A., Mesiar, R.: Directional monotonicity of fusion functions. Eur. J. Oper. Res. 244(1), 300–308 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bustince, H., Fernandez, J., Mesiar, R., Kolesárová, Bedregal B., Dimuro, G. P., Barrenechea, E.: Directional monotonicity and ordered directional monotonicity. In: International Symposium on Aggregation and Structures, ISAS 2016, Book of Abstracts, vol. 18 (2016)

    Google Scholar 

  7. Bustince, H., Barrenechea, L. J., Mesiar, R., Kolesárová, A.: Ordered directionally monotone functions. Justification and application. IEEE Trans. Fuzzy Syst. (2016, submitted)

    Google Scholar 

  8. Grabisch, J., Marichal, J., Mesiar, R., Pap, E.: Aggregation Functions. Cambridge University Press, Cambridge (2009)

    Book  MATH  Google Scholar 

  9. Jurio, A., Bustince, H., Pagola, M., Pradera, A., Yager, R.: Some properties of overlap and grouping functions and their application to image thresholding. Fuzzy Sets Syst. 229, 69–90 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. 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)

    Article  Google Scholar 

  11. Marques Pereira, R.A., Pasi, G.: On non-monotonic aggregation: mixture operators. In: Proceedings of the 4th Meeting of the EURO Working Group on Fuzzy Sets (EUROFUSE 1999) and 2nd International Conference on Soft and Intelligent Computing (SIC 1999), Budapest, Hungary, pp. 513–517 (1999)

    Google Scholar 

  12. Ribeiro, R.A., Marques Pereira, R.A.: Generalized mixture operators using weighting functions: a comparative study with WA and OWA. Eur. J. Oper. Res. 145, 329–342 (2003)

    Article  MATH  Google Scholar 

  13. Ribeiro, R.A., Pereira, R.A.M.: Aggregation with generalized mixture operators using weighting functions. Fuzzy Sets Syst. 137, 43–58 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Mesiar, R., Špirková, J.: Weighted means and weighting functions. Kybernetika 42(2), 151–160 (2006)

    MathSciNet  MATH  Google Scholar 

  15. Mesiar, R., Špirková, J., Vavríková, L.: Weighted aggregation operators based on minimization. Inf. Sci. 17(4), 1133–1140 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Špirková, J.: Weighted operators based on dissimilarity function. Inf. Sci. 281, 172–181 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Špirková, J.: Induced weighted operators based on dissimilarity functions. Inf. Sci. 294, 530–539 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wilkin, T., Beliakov, G., Calvo, T.: Weakly monotone averaging functions. In: Laurent, A., Strauss, O., Bouchon-Meunier, B., Yager, R.R. (eds.) Information Processing and Management of Uncertainty in Knowledge-Based Systems. Communications in Computer and Information Science, vol. 444, pp. 364–373. Springer, Cham (2014)

    Google Scholar 

  19. Wilkin, T., Beliakov, G.: Weakly monotonic averaging functions. Int. J. Intell. Syst. 30(2), 144–169 (2015)

    Article  Google Scholar 

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Acknowledgements

Jana Špirková has been supported by the Project VEGA no. 1/0093/17 Identification of risk factors and their impact on products of the insurance and savings schemes.

Humberto Bustince and Javier Fernández have been supported by the Spanish Government grant TIN2016-77356-P.

Author information

Authors and Affiliations

  1. Faculty of Economics, Matej Bel University, Tajovského 10, 975 90, Banská Bystrica, Slovakia

    Jana Špirková

  2. School of Information Technology, Deakin University, 221 Burwood Hwy, Burwood, 3125, Australia

    Gleb Beliakov

  3. Dept. de Automática y Computación, Universidad Pública de Navarra, Campus Arrosadia, s/n, 31.006, Pamplona, Spain

    Humberto Bustince & Javier Fernández

  4. Institute of Smart Cities, Universidad Pública de Navarra, Campus Arrosadia, s/n, 31.006, Pamplona, Spain

    Humberto Bustince & Javier Fernández

Authors
  1. Jana Špirková
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  2. Gleb Beliakov
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  3. Humberto Bustince
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  4. Javier Fernández
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Corresponding author

Correspondence to Jana Špirková .

Editor information

Editors and Affiliations

  1. School of Informatics, University of Skövde, Skövde, Sweden

    Vicenç Torra

  2. Department of Mathematics and Descriptive Geometry, Slovak University of Technology, Bratislava, Slovakia

    Radko Mesiar

  3. KERMIT Research Unit Knowledge-based Systems, Ghent University, Ghent, Belgium

    Bernard De Baets

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Špirková, J., Beliakov, G., Bustince, H., Fernández, J. (2018). Directional and Ordered Directional Monotonicity of Mixture Functions. In: Torra, V., Mesiar, R., Baets, B. (eds) Aggregation Functions in Theory and in Practice. AGOP 2017. Advances in Intelligent Systems and Computing, vol 581. Springer, Cham. https://doi.org/10.1007/978-3-319-59306-7_10

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  • DOI: https://doi.org/10.1007/978-3-319-59306-7_10

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-59305-0

  • Online ISBN: 978-3-319-59306-7

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