Choquet-like integrals with respect to level-dependent capacities and ϕ -ordinal sums of aggregation function

Radko Mesiar; Peter Smrek

Kybernetika (2015)

  • Volume: 51, Issue: 3, page 420-432
  • ISSN: 0023-5954

Abstract

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In this study we merge the concepts of Choquet-like integrals and the Choquet integral with respect to level dependent capacities. For finite spaces and piece-wise constant level-dependent capacities our approach can be represented as a ϕ -ordinal sum of Choquet-like integrals acting on subdomains of the considered scale, and thus it can be regarded as extension method. The approach is illustrated by several examples.

How to cite

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Mesiar, Radko, and Smrek, Peter. "Choquet-like integrals with respect to level-dependent capacities and $\varphi $-ordinal sums of aggregation function." Kybernetika 51.3 (2015): 420-432. <http://eudml.org/doc/271612>.

@article{Mesiar2015,
abstract = {In this study we merge the concepts of Choquet-like integrals and the Choquet integral with respect to level dependent capacities. For finite spaces and piece-wise constant level-dependent capacities our approach can be represented as a $\varphi $-ordinal sum of Choquet-like integrals acting on subdomains of the considered scale, and thus it can be regarded as extension method. The approach is illustrated by several examples.},
author = {Mesiar, Radko, Smrek, Peter},
journal = {Kybernetika},
keywords = {Choquet integral; Choquet-like integral; level-dependent capacity; $\varphi $-ordinal sum of aggregation functions; Choquet integral; Choquet-like integral; level-dependent capacity; $\varphi $-ordinal sum of aggregation functions},
language = {eng},
number = {3},
pages = {420-432},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Choquet-like integrals with respect to level-dependent capacities and $\varphi $-ordinal sums of aggregation function},
url = {http://eudml.org/doc/271612},
volume = {51},
year = {2015},
}

TY - JOUR
AU - Mesiar, Radko
AU - Smrek, Peter
TI - Choquet-like integrals with respect to level-dependent capacities and $\varphi $-ordinal sums of aggregation function
JO - Kybernetika
PY - 2015
PB - Institute of Information Theory and Automation AS CR
VL - 51
IS - 3
SP - 420
EP - 432
AB - In this study we merge the concepts of Choquet-like integrals and the Choquet integral with respect to level dependent capacities. For finite spaces and piece-wise constant level-dependent capacities our approach can be represented as a $\varphi $-ordinal sum of Choquet-like integrals acting on subdomains of the considered scale, and thus it can be regarded as extension method. The approach is illustrated by several examples.
LA - eng
KW - Choquet integral; Choquet-like integral; level-dependent capacity; $\varphi $-ordinal sum of aggregation functions; Choquet integral; Choquet-like integral; level-dependent capacity; $\varphi $-ordinal sum of aggregation functions
UR - http://eudml.org/doc/271612
ER -

References

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  2. Denneberg, D., 10.1007/978-94-017-2434-0, In: Theory and Decision Library. Series B: Mathematical and Statistical Methods 27. Kluwer Academic Publishers Group, Dordrecht 1994. Zbl0968.28009MR1320048DOI10.1007/978-94-017-2434-0
  3. Grabisch, M., Marichal, J.-L., Mesiar, R., Pap, E., 10.1109/sisy.2008.4664901, Cambridge University Press, Cambridge 2009. Zbl1206.68299MR2538324DOI10.1109/sisy.2008.4664901
  4. Greco, S., Matarazzo, B., Giove, S., 10.1016/j.fss.2011.03.012, Fuzzy Sets and Systems 175 (2011), 1-35. Zbl1218.28014MR2803409DOI10.1016/j.fss.2011.03.012
  5. Mesiar, R., 10.1006/jmaa.1995.1312, J. Math. Anal. Appl. 194 (1995), 477-488. Zbl0845.28010MR1345050DOI10.1006/jmaa.1995.1312
  6. Mesiar, R., Baets, B. De, New construction methods for aggregation operators., In: Proc. IPMU'2000, Madrid, pp. 701-706. 
  7. Pap, E., ed., 10.1016/b978-044450263-6/50000-2, North-Holland, Amsterdam 2002. MR1953489DOI10.1016/b978-044450263-6/50000-2
  8. Pap, E., An integral generated by a decomposable measure., Zb. Rad. Prirod.-Mat. Fak. Ser. Mat. 20 (1990), 135-144. Zbl0754.28002MR1158414
  9. Sander, W., Siedekum, J., Multiplication, distributivity and fuzzy integral, I, II, III., Kybernetika 41 (2005) I: 397-422, II: 469-496, III: 497-518. MR2181427
  10. Schmeidler, D., 10.1090/s0002-9939-1986-0835875-8, Proc. Amer. Math. Soc. 97 (1986), 255-261. Zbl0687.28008MR0835875DOI10.1090/s0002-9939-1986-0835875-8
  11. Schmeidler, D., 10.2307/1911053, Econometrica 57 (1989), 571-87. Zbl0672.90011MR0999273DOI10.2307/1911053
  12. Sugeno, M., Theory of Fuzzy Integrals and its Applications., PhD Thesis, Tokyo Institute of Technology, 1974. 
  13. Sugeno, M., Murofushi, T., 10.1016/0022-247x(87)90354-4, J. Math. Anal. Appl. 122 (1987), 197-222. Zbl0611.28010MR0874969DOI10.1016/0022-247x(87)90354-4
  14. Vitali, G., 10.1007/BF02409934, Ann. Mat. Pura Appl. 2 (1925), 111-121. English translation: On the definition of integral of functions of one variable. Rivista di Matematica per le Scienze Sociali 20 (1997), 159-168. MR1553076DOI10.1007/BF02409934
  15. Wang, Z., Klir, G. J., 10.1007/978-0-387-76852-6, Springer, 2009. Zbl1184.28002MR2453907DOI10.1007/978-0-387-76852-6

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