Article

Full entry | PDF   (1.4 MB)
Keywords:
aggregation; Choquet and Sugenointegrals; multi-step integral; twofold integral
Summary:
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.
References:
[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 1954643 | Zbl 1099.28007
[3] Calvo T., Mesiarová, A., Valášková L.: Construction of aggregation operators – new composition method. Kybernetika 39 (2003), 643–650 MR 2042346
[4] Mesiar R., Vivona D.: Two-step integral with respect to fuzzy measure. Tatra Mt. Math. Publ. 16 (1999), 359–368 MR 1725307 | Zbl 0948.28015
[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 0980350 | Zbl 0662.28015
[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 1123577 | Zbl 0733.28014
[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 10.1142/S0218488597000439 | MR 1480752 | Zbl 1232.28023
[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 1913786 | Zbl 0996.26007
[13] Ovchinnikov S.: Piecewise linear aggregation functions. Internat. J. of Uncertainty, Fuzziness and Knowledge-based Systems 10 (2002), 17–24 DOI 10.1142/S0218488502001314 | MR 1897837 | Zbl 1070.91007
[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 10.1142/S0218488595000025 | MR 1321933
[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)

Partner of