Geodesic
Twofold integral and multi-step Choquet integral
Kybernetika, Tome 40 (2004) no. 1, pp. 39-50 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 03H05, 28E05, 28E10, 68T37
Keywords: aggregation; Choquet and Sugenointegrals; multi-step integral; twofold integral 
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Narukawa, Yasuo; Torra, Vicenç. Twofold integral and multi-step Choquet integral. Kybernetika, Tome 40 (2004) no. 1, pp. 39-50. http://geodesic.mathdoc.fr/item/KYB_2004_40_1_a3/

[1] Benvenuti P., Mesiar R.: A note on Sugeno and Choquet integrals. In: Proc. 8th Internat. Conference Information Processing and Management of Uncertainty in Knowledge-based Systems, 2000, pp. 582–585

[2] Benvenuti P., Mesiar, R., Vivona D.: Monotone set functions-based integrals. In: Handbook of Measure Theory (E. Pap, ed.), Elsevier, 2002 | MR | Zbl

[3] Calvo T., Mesiarová, A., Valášková L.: Construction of aggregation operators – new composition method. Kybernetika 39 (2003), 643–650 | MR

[4] Mesiar R., Vivona D.: Two-step integral with respect to fuzzy measure. Tatra Mt. Math. Publ. 16 (1999), 359–368 | MR | Zbl

[5] Murofushi T., Sugeno M.: An interpretation of fuzzy measures and the Choquet integral as an integral with respect to a fuzzy measure. Fuzzy Sets and Systems 29 (1989), 201–227 | MR | Zbl

[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

[7] Murofushi T., Narukawa Y.: A characterization of multi-level discrete Choquet integral over a finite set (in Japanese). In: Proc. 7th Workshop on Evaluation of Heart and Mind 2002, pp. 33–36

[8] Murofushi T., Sugeno M.: Fuzzy t-conorm integral with respect to fuzzy measures: generalization of Sugeno integral and Choquet integral. Fuzzy Sets and Systems 42 (1991), 57–71 | MR | Zbl

[9] Murofushi T., Sugeno, M., Fujimoto K.: Separated hierarchical decomposition of the Choquet integral. Internat. J. Uncertainty, Fuzziness and Knowledge-based Systems 5 (1997), 563–585 | DOI | MR | Zbl

[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 | MR | Zbl

[13] Ovchinnikov S.: Piecewise linear aggregation functions. Internat. J. of Uncertainty, Fuzziness and Knowledge-based Systems 10 (2002), 17–24 | DOI | MR | Zbl

[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.: Hierarchical decomposition of Choquet integral models. Internat. J. of Uncertainty, Fuzziness and Knowledge-based Systems 3 (1995), 1–15 | DOI | MR

[16] Torra V.: Twofold integral: A Choquet integral and Sugeno integral generalization. Butlletí de l’Associació Catalana d’Intel$\cdot $ligència Artificial 29 (2003), 14–20 (in Catalan). Preliminary version: IIIA Research Report TR-2003-08 (in English)