Ordered Weighted Averages on Intervals and the Sub/Super-Additivity

Yuji Yoshida

doi:10.20965/jaciii.2013.p0520

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JACIII Vol.17 No.4 pp. 520-525

doi: 10.20965/jaciii.2013.p0520

(2013)

Paper:

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Ordered Weighted Averages on Intervals and the Sub/Super-Additivity

Yuji Yoshida

Faculty of Economics and Business Administration, University of Kitakyushu, 4-2-1 Kitagata, Kokuraminami, Kitakyushu 802-8577, Japan

Received:

February 28, 2013

Accepted:

April 12, 2013

Published:

July 20, 2013

Keywords:

OWA, subadditivity, truncation weight, monotone weight, value-at-risk

Abstract

This paper deals with continuous Ordered Weighted Averages (OWA) on a closed interval and investigates their fundamental properties. In this paper, we focus on OWA with a truncation weight and derive the subadditivity of a top-concentrated average. We then deal with OWA from the bottom and investigate their relations. The subadditivity for OWA with monotone weights is also discussed, then OWA based on probability are demonstrated and value-at-risks are explained as an example.

Cite this article as:

Y. Yoshida, “Ordered Weighted Averages on Intervals and the Sub/Super-Additivity,” J. Adv. Comput. Intell. Intell. Inform., Vol.17 No.4, pp. 520-525, 2013.

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References

[1] G. Beliakov, A. Pradera, and T. Calvo, “Aggregation Functions: A Guide for Practitioners,” Springer, 2007.
[2] T. Calvo, A. Kolesárová, M. Komorníková, and R. Mesiar, “Aggregation operators: Basic concepts, issues and properties,” in: T. Calvo, G. Mayor, and R. Mesiar (Eds.), Aggregation Operators: New Trends and Applications, pp. 3-104, Phisica-Verlag, Springer, 2002.
[3] R. R. Yager, “Ordered weighted averaging aggregation operators in multi-criteria decision making,” IEEE Trans. on Systems, Man and Cybernetics, Int. J. of Intel. Syst., Vol.18, pp. 183-190, 1988.
[4] R. R. Yager, “Families of OWA operators,” Fuzzy Sets and Systems, Vol.59, pp. 125-148, 1993.
[5] R. R. Yager and D. P. Filev, “Parameterized and-like and or-like OWA opera-tors,” Int. J. of General Systems, Vol.22, pp. 297-316, 1994.
[6] R. R. Yager, “OWA aggregation over a continuous interval argument with application to decision making,” IEEE Trans. on Systems, Man, and Cybern., Part B: Cybernetics, Vol.34, pp. 1952-1963, 2004.
[7] R. R. Yager and Z. Xu, “The continuous order weighted geometric operator and its application to decision making,” Fuzzy Sets and Systems, Vol.157, pp. 1393-1402, 2006.
[8] V. Torra, “The weighted OWA operator,” Int. J. of Intel. Syst., Vol.12, pp. 153-166, 1997.
[9] V. Torra and Y. Narukawa, “Modeling Decisions – Information Fusion and Aggregation Operators,” Springer, 2002.
[10] V. Torra and L. Godo, “Continuous WOWA operators with application to defuzzification,” Aggregation operators: New trends and applications, pp. 159-176, Physica-Verlag, Springer, 2002.
[11] Y. Narukawa and V. Torra, “Continuous OWA Operator and its Calculation,” CD-ROM Proc., IFSA-EUSFLAT 2009, pp. 1132-1134, 2009.
[12] Y. Narukawa, V. Torra, and M. Sugeno, “Choquet integral of a function on the real line,” in: V. Torra, Y. Narukawa, and M. Daumas (Eds.), Modeling Decisions for Artificial Intelligence 2010, CDROM Proc., MDAI 2010, pp. 24-33, 2010.
[13] Y. Yoshida, “An ordered weighted average with a truncation weight on intervals,” in: V. Torra, Y. Narukawa, B. López, and M. Villaret (Eds.), MDAI 2012, LNAI, Vol.7647, pp. 45-55, Springer, Nov. 2012.
[14] C. Dellacherie, “Quelques commentarires sur les prolongements de capacités,” Séminare de Probabilites 1969/1970, Strasbourg, Lecture Notes in Artificial Intelligence, Vol.191, pp. 77-81, Springer, 1971.
[15] D. Renneberg, “Non Additive Measure and Integral,” Kluwer Academic Publ., Dordrecht, 1994.
[16] P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath, “Thinking coherently,” Risk, Vol.10, pp. 68-71, 1997.
[17] P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath, “Coherent measures of risk,” Mathematical Finance, Vol.9, pp. 203-228, 1999.
[18] A. Meucci, “Risk and Asset Allocation,” Springer-Verlag, Heidelberg, 2005.
[19] Y. Yoshida, “A perception-based portfolio under uncertainty: Minimization of average rates of falling,” in: V. Torra, Y. Narukawa, and M. Inuiguchi (Eds.), MDAI 2009, LNAI, Vol.5861, pp. 149-160, Springer, Nov. 2009.
[20] Y. Yoshida, “An average value-at-risk portfolio model under uncertainty: A perception-based approach by fuzzy random variables,” J. of Advanced Computational Intelligence and Intelligent Informatics, Vol.15, pp. 56-62, 2011.
[21] S. Kusuoka, “On law-invariant coherent risk measures,” Advances in Mathematical Economics, Vol.3, pp. 83-95, 2001.

This article is published under a Creative Commons Attribution-NoDerivatives 4.0 Internationa License.

[1] [1] G. Beliakov, A. Pradera, and T. Calvo, “Aggregation Functions: A Guide for Practitioners,” Springer, 2007.

[2] [2] T. Calvo, A. Kolesárová, M. Komorníková, and R. Mesiar, “Aggregation operators: Basic concepts, issues and properties,” in: T. Calvo, G. Mayor, and R. Mesiar (Eds.), Aggregation Operators: New Trends and Applications, pp. 3-104, Phisica-Verlag, Springer, 2002.

[3] [3] R. R. Yager, “Ordered weighted averaging aggregation operators in multi-criteria decision making,” IEEE Trans. on Systems, Man and Cybernetics, Int. J. of Intel. Syst., Vol.18, pp. 183-190, 1988.

[4] [4] R. R. Yager, “Families of OWA operators,” Fuzzy Sets and Systems, Vol.59, pp. 125-148, 1993.

[5] [5] R. R. Yager and D. P. Filev, “Parameterized and-like and or-like OWA opera-tors,” Int. J. of General Systems, Vol.22, pp. 297-316, 1994.

[6] [6] R. R. Yager, “OWA aggregation over a continuous interval argument with application to decision making,” IEEE Trans. on Systems, Man, and Cybern., Part B: Cybernetics, Vol.34, pp. 1952-1963, 2004.

[7] [7] R. R. Yager and Z. Xu, “The continuous order weighted geometric operator and its application to decision making,” Fuzzy Sets and Systems, Vol.157, pp. 1393-1402, 2006.

[8] [8] V. Torra, “The weighted OWA operator,” Int. J. of Intel. Syst., Vol.12, pp. 153-166, 1997.

[9] [9] V. Torra and Y. Narukawa, “Modeling Decisions – Information Fusion and Aggregation Operators,” Springer, 2002.

[10] [10] V. Torra and L. Godo, “Continuous WOWA operators with application to defuzzification,” Aggregation operators: New trends and applications, pp. 159-176, Physica-Verlag, Springer, 2002.

[11] [11] Y. Narukawa and V. Torra, “Continuous OWA Operator and its Calculation,” CD-ROM Proc., IFSA-EUSFLAT 2009, pp. 1132-1134, 2009.

[12] [12] Y. Narukawa, V. Torra, and M. Sugeno, “Choquet integral of a function on the real line,” in: V. Torra, Y. Narukawa, and M. Daumas (Eds.), Modeling Decisions for Artificial Intelligence 2010, CDROM Proc., MDAI 2010, pp. 24-33, 2010.

[13] [13] Y. Yoshida, “An ordered weighted average with a truncation weight on intervals,” in: V. Torra, Y. Narukawa, B. López, and M. Villaret (Eds.), MDAI 2012, LNAI, Vol.7647, pp. 45-55, Springer, Nov. 2012.

[14] [14] C. Dellacherie, “Quelques commentarires sur les prolongements de capacités,” Séminare de Probabilites 1969/1970, Strasbourg, Lecture Notes in Artificial Intelligence, Vol.191, pp. 77-81, Springer, 1971.

[15] [15] D. Renneberg, “Non Additive Measure and Integral,” Kluwer Academic Publ., Dordrecht, 1994.

[16] [16] P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath, “Thinking coherently,” Risk, Vol.10, pp. 68-71, 1997.

[17] [17] P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath, “Coherent measures of risk,” Mathematical Finance, Vol.9, pp. 203-228, 1999.

[18] [18] A. Meucci, “Risk and Asset Allocation,” Springer-Verlag, Heidelberg, 2005.

[19] [19] Y. Yoshida, “A perception-based portfolio under uncertainty: Minimization of average rates of falling,” in: V. Torra, Y. Narukawa, and M. Inuiguchi (Eds.), MDAI 2009, LNAI, Vol.5861, pp. 149-160, Springer, Nov. 2009.

[20] [20] Y. Yoshida, “An average value-at-risk portfolio model under uncertainty: A perception-based approach by fuzzy random variables,” J. of Advanced Computational Intelligence and Intelligent Informatics, Vol.15, pp. 56-62, 2011.

[21] [21] S. Kusuoka, “On law-invariant coherent risk measures,” Advances in Mathematical Economics, Vol.3, pp. 83-95, 2001.