Twofold integral and multi-step Choquet integral

Yasuo Narukawa; Vicenç Torra

Kybernetika (2004)

  • Volume: 40, Issue: 1, page [39]-50
  • ISSN: 0023-5954

Abstract

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In this work we study some properties of the twofold integral and, in particular, its relation with the 2-step Choquet integral. First, we prove that the Sugeno integral can be represented as a 2-step Choquet integral. Then, we turn into the twofold integral studying some of its properties, establishing relationships between this integral and the Choquet and Sugeno ones and proving that it can be represented in terms of 2-step Choquet integral.

How to cite

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Narukawa, Yasuo, and Torra, Vicenç. "Twofold integral and multi-step Choquet integral." Kybernetika 40.1 (2004): [39]-50. <http://eudml.org/doc/33684>.

@article{Narukawa2004,
abstract = {In this work we study some properties of the twofold integral and, in particular, its relation with the 2-step Choquet integral. First, we prove that the Sugeno integral can be represented as a 2-step Choquet integral. Then, we turn into the twofold integral studying some of its properties, establishing relationships between this integral and the Choquet and Sugeno ones and proving that it can be represented in terms of 2-step Choquet integral.},
author = {Narukawa, Yasuo, Torra, Vicenç},
journal = {Kybernetika},
keywords = {aggregation; Choquet and Sugenointegrals; multi-step integral; twofold integral; aggregation; Choquet integral; Sugeno integral; multi-step integral; twofold integral},
language = {eng},
number = {1},
pages = {[39]-50},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Twofold integral and multi-step Choquet integral},
url = {http://eudml.org/doc/33684},
volume = {40},
year = {2004},
}

TY - JOUR
AU - Narukawa, Yasuo
AU - Torra, Vicenç
TI - Twofold integral and multi-step Choquet integral
JO - Kybernetika
PY - 2004
PB - Institute of Information Theory and Automation AS CR
VL - 40
IS - 1
SP - [39]
EP - 50
AB - In this work we study some properties of the twofold integral and, in particular, its relation with the 2-step Choquet integral. First, we prove that the Sugeno integral can be represented as a 2-step Choquet integral. Then, we turn into the twofold integral studying some of its properties, establishing relationships between this integral and the Choquet and Sugeno ones and proving that it can be represented in terms of 2-step Choquet integral.
LA - eng
KW - aggregation; Choquet and Sugenointegrals; multi-step integral; twofold integral; aggregation; Choquet integral; Sugeno integral; multi-step integral; twofold integral
UR - http://eudml.org/doc/33684
ER -

References

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  2. Benvenuti P., Mesiar, R., Vivona D., Monotone set functions-based integrals, In: Handbook of Measure Theory (E. Pap, ed.), Elsevier, 2002 Zbl1099.28007MR1954643
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  6. Murofushi T., Narukawa Y., A characterization of multi-step discrete Choquet integral, In: 6th Internat. Conference Fuzzy Sets Theory and Its Applications, Abstracts, 2002 p. 94 
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  9. Murofushi T., Sugeno, M., Fujimoto K., 10.1142/S0218488597000439, Internat. J. Uncertainty, Fuzziness and Knowledge-based Systems 5 (1997), 563–585 (1997) Zbl1232.28023MR1480752DOI10.1142/S0218488597000439
  10. Narukawa Y., Murofushi T., The n -step Choquet integral on finite spaces, In: Proc. 9th Internat. Conference Information Processing and Management of Uncertainty in Knowledge-based Systems, 2002, pp. 539–543 
  11. Narukawa Y., Torra V., Twofold integral: a graphical interpretation and its generalization to universal sets, In: EUSFLAT 2003, Zittau, Germany, pp. 718–722 
  12. Ovchinnikov S., Max-min representation of piecewise linear functions, Contributions to Algebra and Geometry 43 (2002), 297–302 Zbl0996.26007MR1913786
  13. Ovchinnikov S., 10.1142/S0218488502001314, Internat. J. of Uncertainty, Fuzziness and Knowledge-based Systems 10 (2002), 17–24 Zbl1070.91007MR1897837DOI10.1142/S0218488502001314
  14. Sugeno M., Theory of Fuzzy Integrals and Its Application, Ph.D. Thesis, Tokyo Institute of Technology, 1974 
  15. Sugeno M., Fujimoto, K., Murofushi T., 10.1142/S0218488595000025, Internat. J. of Uncertainty, Fuzziness and Knowledge-based Systems 3 (1995), 1–15 (1995) MR1321933DOI10.1142/S0218488595000025
  16. Torra V., Twofold integral: A Choquet integral and Sugeno integral generalization, Butlletí de l’Associació Catalana d’Intelligència Artificial 29 (2003), 14–20 (in Catalan). Preliminary version: IIIA Research Report TR-2003-08 (in English) 

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