Combinatorial Structure of the Polytope of 2-Additive Measures
| dc.contributor.author | Miranda Menéndez, Pedro | |
| dc.contributor.author | García Segador, Pedro | |
| dc.date.accessioned | 2023-06-17T08:56:08Z | |
| dc.date.available | 2023-06-17T08:56:08Z | |
| dc.date.issued | 2020-11 | |
| dc.description.abstract | In this paper we study the polytope of 2-additive measures, an important subpolytope of theIn this paper we study the polytope of 2-additive measures, an important subpolytope of the polytope of fuzzy measures. For this polytope, we obtain its combinatorial structure, namely the adjacency structure and the structure of 2-dimensional faces, 3-dimensional faces, and so on. Basing on this information, we build a triangulation of this polytope satisfying that all simplices in the triangulation have the same volume. As a consequence, this allows a very simple and appealing way to generate points in a random way in this polytope, an interesting problema arising in the practical identi_cation of 2-additive measures. Finally, we also derive the volume, the centroid, and some properties concerning the adjacency graph of this polytope. polytope of fuzzy measures. For this polytope, we obtain its combinatorial structure, namely the adjacency structure and the structure of 2-dimensional faces, 3-dimensional faces, and so on. Basing on this information, we build a triangulation of this polytope satisfying that all simplices in the triangulation have the same volume. As a consequence, this allows a very simple and appealing way to generate points in a random way in this polytope, an interesting problema arising in the practical identi_cation of 2-additive measures. Finally, we also derive the volume, the centroid, and some properties concerning the adjacency graph of this polytope. | |
| dc.description.department | Depto. de Estadística e Investigación Operativa | |
| dc.description.faculty | Fac. de Ciencias Matemáticas | |
| dc.description.refereed | FALSE | |
| dc.description.status | pub | |
| dc.eprint.id | https://eprints.ucm.es/id/eprint/63271 | |
| dc.identifier.citation | Miranda P, Garcia-Segador P. Combinatorial Structure of the Polytope of 2-Additive Measures. IEEE Trans Fuzzy Syst 2020; 28: 2864–2874. [DOI: 10.1109/TFUZZ.2019.2945243] | |
| dc.identifier.doi | 10.1109/TFUZZ.2019.2945243 | |
| dc.identifier.issn | 1063-6706 | |
| dc.identifier.officialurl | http://www.ieee.org/index.html | |
| dc.identifier.relatedurl | https://ieeexplore.ieee.org/document/8854864 | |
| dc.identifier.uri | https://hdl.handle.net/20.500.14352/7574 | |
| dc.issue.number | 11 | |
| dc.journal.title | IEEE Transactions on Fuzzy Systems | |
| dc.language.iso | eng | |
| dc.publisher | Institute of Electrical and Electronics Engineers | |
| dc.rights.accessRights | open access | |
| dc.rights.uri | https://creativecommons.org/licenses/by/4.0/ | |
| dc.subject.cdu | 519.83 | |
| dc.subject.cdu | 528.331 | |
| dc.subject.keyword | Transforms | |
| dc.subject.keyword | Additives | |
| dc.subject.keyword | Game theory | |
| dc.subject.keyword | Complexity theory | |
| dc.subject.keyword | Fuzzy systems | |
| dc.subject.keyword | Games | |
| dc.subject.keyword | 2-additive measures | |
| dc.subject.keyword | combinatorial structure | |
| dc.subject.keyword | fuzzy measures | |
| dc.subject.keyword | random generation | |
| dc.subject.keyword | triangulation | |
| dc.subject.keyword | Teoría de los juegos | |
| dc.subject.keyword | Triangulación | |
| dc.subject.ucm | Matemáticas (Matemáticas) | |
| dc.subject.ucm | Estadística aplicada | |
| dc.subject.ucm | Estadística matemática (Matemáticas) | |
| dc.subject.unesco | 12 Matemáticas | |
| dc.subject.unesco | 1209 Estadística | |
| dc.title | Combinatorial Structure of the Polytope of 2-Additive Measures | |
| dc.type | journal article | |
| dc.volume.number | 28 | |
| dcterms.references | [1] T. Calvo and B. De Baets. Aggregation operators de_ned by k-order additive/maxitive fuzzy measures. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, (6):533-550, 1998. [2] I. Barany and G. Furedi. Computing the volume is di_cult. Discrete Comput. Geom., 2(4):31-326, 1987. [3] A. Chateauneuf and J.-Y. Ja_ray. Some characterizations of lower probabilities and other monotone capacities through the use of M�obius inversion. Mathematical Social Sciences, (17):263-283,1989. [4] G. Choquet. Theory of capacities. Annales de l'Institut Fourier, (5):131-295, 1953. [5] E. F. Combarro and P. Miranda. Identi_cation of fuzzy measures from sample data with genetic algorithms. Computers and Operations Research, 33(10):3046{3066, 2006. [6] E. F. Combarro and P. Miranda. Adjacency on the order polytope with applications to the theory of fuzzy measures. Fuzzy Sets and Systems, 180:384-398, 2010. [7] L. Devroye. Non-uniform random variate generation. Springer-Verlag, New York, 1986. [8] M. Grabisch. On equivalence classes of fuzzy connectives-the case of fuzzy integrals. IEEE Transactions on Fuzzy Systems,3 (1):96-109, 1995. [9] M. Grabisch. Alternative representations of discrete fuzzy measures for decision making. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 5:587-607, 1997. [10] M. Grabisch. k-order additive discrete fuzzy measures and their representation. Fuzzy Sets and Systems, (92):167-189, 1997. [11] M. Grabisch. A graphical interpretation of the Choquet integral. IEEE Transactions on Fuzzy Systems,8 (5):627-631, 2000. [12] M. Grabisch. Ensuring the boundedness of the core of games with restricted cooperation. Annals of Operations Research, (191):137-154, 2011. [13] M. Grabisch. Set functions, games and capacities in Decision Making, volume 46 of Theory and Decision Library. Springer, 2016. [14] M. Grabisch and C. Labreuche. Monotone decomposition of 2-additive generalized additive independence models. Mathematical Social Sciences, (92):64-73, 2018. [15] M. Grabisch and Raufaste An empirical study of statistical properties of Choquet and Sugeno integrals. IEEE Transactions on Fuzzy Systems,16 (4):839-850, 2008. [16] P. Gritzmann and V. Klee. On the complexity of some basic problems in computational convexity, in Polytopes: Abstract, Convex and Computational. Kluwer Academic, Dordrecht, 374-466, 1994. [17] M. Grabisch and M. Roubens. Application of the Choquet integral in Multicriteria Decision Making. In M. Grabisch, T. Murofushi, and M. Sugeno, editors, Fuzzy Measures and Integrals, number 40 in Studies in Fuzziness and Soft Computing, pages 348-375. Physica-Verlag, 2000. [18] J. C. Harsanyi. A simpli_ed bargaining model for the n-person cooperative game. International Economic Review, 4:194-220, 1963. [19] S.G. Krantz, J.E. McCarthy and H.R. Parks. Geometric characterizations of centroids of simplices. Journal of Mathematical Analysis and Applications, 316:87-109, 2006. [20] Ch. Labreuche and M. Grabisch. Using multiple reference levels in multi-criteria decision aid: the generalized-additive independence model and the Choquet integral approaches. Eur. J. of Operational Research, to appear. [21] J. Leydold and W. H�ormann. A Sweep-Plane Algorithm for Generating Random Tuples in Simple Polytopes. J. Math. Comp., (67):1617-1635, 1998. [22] J.-L. Marichal. Tolerant or intolerant character of interacting criteria in aggregation by the Choquet integral. European Journal of Operational Research, 155(3):771-791, 2004. [23] Jiri Matousek. Lectures on Discrete Geometry. Springer-Verlag New York, Inc., Secaucus, NJ, USA, 2002. [24] P. Miranda and E.F. Combarro. On the structure of some families of fuzzy measures. IEEE Transactions on Fuzzy Systems, 18(6): 679-690, 2007. [25] P. Miranda, E.F. Combarro, and P. Gil. Extreme points of some families of non-additive measures. European Journal of Operational Research, 33(10):3046-3066, 2006. [26] P. Miranda, M. Grabisch, and P. Gil. p-symmetric fuzzy measures. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 10 (Suppl.):105-123, 2002. [27] J. Murillo, S. Guillaume, and P. Bulacio. k-maxitive fuzzy measures: A scalable approach to model interactions. Fuzzy Sets and Systems, 324:33-48, 2017. [28] D. Naddef and W. R. Pulleyblank. Hamiltonicity and Combinatorial Polyhedra. Journal of Combinatorial Theory, Series B, 31:297-312, 1981. [29] G. Owen. Game Theory. Academic Press, 1995. [30] G. C. Rota. On the foundations of combinatorial theory I. Theory of M�obius functions. Zeitschrift f�ur Wahrscheinlichkeitstheorie und Verwandte Gebiete, (2):340-368, 1964. [31] R.Y. Rubinstein and D.P. Kroese. Simulation and the Monte Carlo method. Wiley-Interscience, 2007. [32] G. Shafer. A Mathematical Theory of Evidence. Princeton University Press, Princeton, New Jersey, (USA), 1976. [33] R. Stanley. Two poset polytopes. Discrete Comput. Geom., 1(1):9-23, 1986. [34] M. Sugeno. Theory of fuzzy integrals and its applications. PhD thesis, Tokyo Institute of Technology, 1974. [35] D.B. West. Introduction to Graph Theory. Pearson Education, 2001. [36] R.R. Yager Uncertainty representation using fuzzy measures IEEE Transactions on Cybernetics, 32(1):13-20, 2002. [37] G. Ziegler. Lectures on Polytopes. Springer-Verlag, 1995. | |
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