Defects and transformations of quasi-copulas

Michal Dibala; Susanne Saminger-Platz; Radko Mesiar; Erich Peter Klement

Kybernetika (2016)

  • Volume: 52, Issue: 6, page 848-865
  • ISSN: 0023-5954

Abstract

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Six different functions measuring the defect of a quasi-copula, i. e., how far away it is from a copula, are discussed. This is done by means of extremal non-positive volumes of specific rectangles (in a way that a zero defect characterizes copulas). Based on these defect functions, six transformations of quasi-copulas are investigated which give rise to six different partitions of the set of all quasi-copulas. For each of these partitions, each equivalence class contains exactly one copula being a fixed point of the transformation under consideration. Finally, an application to the construction of so-called imprecise copulas is given.

How to cite

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Dibala, Michal, et al. "Defects and transformations of quasi-copulas." Kybernetika 52.6 (2016): 848-865. <http://eudml.org/doc/287877>.

@article{Dibala2016,
abstract = {Six different functions measuring the defect of a quasi-copula, i. e., how far away it is from a copula, are discussed. This is done by means of extremal non-positive volumes of specific rectangles (in a way that a zero defect characterizes copulas). Based on these defect functions, six transformations of quasi-copulas are investigated which give rise to six different partitions of the set of all quasi-copulas. For each of these partitions, each equivalence class contains exactly one copula being a fixed point of the transformation under consideration. Finally, an application to the construction of so-called imprecise copulas is given.},
author = {Dibala, Michal, Saminger-Platz, Susanne, Mesiar, Radko, Klement, Erich Peter},
journal = {Kybernetika},
keywords = {copula; quasi-copula; transformation of quasi-copulas; imprecise copula},
language = {eng},
number = {6},
pages = {848-865},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Defects and transformations of quasi-copulas},
url = {http://eudml.org/doc/287877},
volume = {52},
year = {2016},
}

TY - JOUR
AU - Dibala, Michal
AU - Saminger-Platz, Susanne
AU - Mesiar, Radko
AU - Klement, Erich Peter
TI - Defects and transformations of quasi-copulas
JO - Kybernetika
PY - 2016
PB - Institute of Information Theory and Automation AS CR
VL - 52
IS - 6
SP - 848
EP - 865
AB - Six different functions measuring the defect of a quasi-copula, i. e., how far away it is from a copula, are discussed. This is done by means of extremal non-positive volumes of specific rectangles (in a way that a zero defect characterizes copulas). Based on these defect functions, six transformations of quasi-copulas are investigated which give rise to six different partitions of the set of all quasi-copulas. For each of these partitions, each equivalence class contains exactly one copula being a fixed point of the transformation under consideration. Finally, an application to the construction of so-called imprecise copulas is given.
LA - eng
KW - copula; quasi-copula; transformation of quasi-copulas; imprecise copula
UR - http://eudml.org/doc/287877
ER -

References

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  1. Alsina, C., Frank, M. J., Schweizer, B., 10.1142/9789812774200, World Scientific, Singapore 2006. Zbl1100.39023MR2222258DOI10.1142/9789812774200
  2. Alsina, C., Nelsen, R. B., Schweizer, B., 10.1016/0167-7152(93)90001-y, Statist. Probab. Lett. 17 (1993), 85-89. Zbl0798.60023MR1223530DOI10.1016/0167-7152(93)90001-y
  3. Alvoni, E., Papini, P. L., Spizzichino, F., 10.1016/j.fss.2008.03.025, Fuzzy Sets and Systems 160 (2009), 334-343. Zbl1175.62045MR2473107DOI10.1016/j.fss.2008.03.025
  4. Birkhoff, G., Lattice Theory., American Mathematical Society, Providence 1973. Zbl0537.06001MR0227053
  5. Clifford, A. H., 10.2307/2372706, Amer. J. Math. 76 (1954), 631-646. MR0062118DOI10.2307/2372706
  6. Baets, B. De, 10.1007/978-3-642-11960-6_6, In: Integrated Uncertainty Management and Applications. Selected Papers Based on the Presentations at the 2010 International Symposium on Integrated Uncertainty Managment and Applications (IUM 2010) (V.-N. Huynh, Y. Nakamori, J. Lawry, and M. Inuiguchi, eds.), Ishikawa 2010, p. 55. Springer, Berlin 2010. DOI10.1007/978-3-642-11960-6_6
  7. Baets, B. De, Meyer, H. De, D{í}az, S., 10.1016/j.fss.2008.04.001, Fuzzy Sets and Systems 160 (2009), 733-751. Zbl1175.62046MR2493272DOI10.1016/j.fss.2008.04.001
  8. Baets, B. De, Janssens, S., Meyer, H. De, 10.1016/j.ijar.2008.03.006, Internat. J. Approx. Reason. 50 (2009), 104-116. Zbl1191.68706MR2519040DOI10.1016/j.ijar.2008.03.006
  9. Schuymer, B. De, Meyer, H. De, Baets, B. De, 10.1016/j.jmva.2004.10.011, J. Multivariate Anal. 96 (2005), 352-373. Zbl1087.60018MR2204983DOI10.1016/j.jmva.2004.10.011
  10. D{í}az, S., Montes, S., Baets, B. De, 10.1109/tfuzz.2006.880004, IEEE Trans. Fuzzy Systems 15 (2007), 275-286. DOI10.1109/tfuzz.2006.880004
  11. Dolati, A., Mohseni, S., Úbeda-Flores, M., 10.1016/j.ins.2013.09.023, Inform. Sci. 257 (2014), 176-182. Zbl1321.62053MR3131786DOI10.1016/j.ins.2013.09.023
  12. Durante, F., Sarkoci, P., Sempi, C., 10.1016/j.jmaa.2008.11.064, J. Math. Anal. Appl. 352 (2009), 914-921. Zbl1160.60307MR2501937DOI10.1016/j.jmaa.2008.11.064
  13. Durante, F., Sempi, C., 10.1155/ijmms.2005.645 Zbl1078.62055DOI10.1155/ijmms.2005.645
  14. Durante, F., Sempi, C., 10.1201/b18674, CRC Press, Boca Raton 2015. MR3443023DOI10.1201/b18674
  15. Fuchs, S., Schmidt, K. D., 10.14736/kyb-2014-1-0109, Kybernetika 50 (2014), 109-125. MR3195007DOI10.14736/kyb-2014-1-0109
  16. Genest, C., Quesada-Molina, J. J., Rodríguez-Lallena, J. A., Sempi, C., 10.1006/jmva.1998.1809, J. Multivariate Anal. 69 (1999), 193-205. Zbl0935.62059MR1703371DOI10.1006/jmva.1998.1809
  17. Grabisch, M., Marichal, J.-L., Mesiar, R., Pap, E., 10.1017/cbo9781139644150, Cambridge University Press, Cambridge 2009. Zbl1206.68299MR2538324DOI10.1017/cbo9781139644150
  18. Hájek, P., Mesiar, R., 10.1007/s00500-008-0286-z, Soft Computing 12 (2008), 1239-1243. Zbl1152.03018DOI10.1007/s00500-008-0286-z
  19. Janssens, S., Baets, B. De, Meyer, H. De, 10.1016/j.fss.2004.03.015, Fuzzy Sets and Systems 148 (2004), 263-278. MR2100199DOI10.1016/j.fss.2004.03.015
  20. Kalická, J., 10.1016/j.fss.2008.06.017, Fuzzy Sets and Systems 160 (2009), 726-732. Zbl1175.62047MR2493271DOI10.1016/j.fss.2008.06.017
  21. Klement, E. P., Mesiar, R., Pap, E., 10.1007/978-94-015-9540-7, Kluwer Academic Publishers, Dordrecht 2000. Zbl1087.20041MR1790096DOI10.1007/978-94-015-9540-7
  22. Klement, E. P., Mesiar, R., Pap, E., Invariant copulas., Kybernetika 38 (2002), 275-285. Zbl1264.62045MR1944309
  23. Klement, E. P., Mesiar, R., Pap, E., Transformations of copulas., Kybernetika 41 (2005), 425-436. Zbl1243.62019MR2180355
  24. Ling, C. M., Representation of associative functions., Publ. Math. Debrecen 12 (1965), 189-212. Zbl0137.26401MR0190575
  25. Montes, I., Miranda, E., Montes, S., 10.1016/j.ejor.2013.09.013, European J. Oper. Res. 2342 (2014), 209-220. Zbl1305.91106MR3141610DOI10.1016/j.ejor.2013.09.013
  26. Montes, I., Miranda, E., Pelessoni, R., Vicig, P., 10.1016/j.fss.2014.10.007, Fuzzy Sets and Systems 278 (2015), 48-66. MR3383206DOI10.1016/j.fss.2014.10.007
  27. Nelsen, R. B., 10.1007/0-387-28678-0, Springer, New York 2006. MR2197664DOI10.1007/0-387-28678-0
  28. Nelsen, R. B., Quesada-Molina, J. J., Rodríguez-Lallena, J. A., Úbeda-Flores, M., 10.1007/978-94-017-0061-0_20, In: Distributions with Given Marginals and Statistical Modelling (C. M. Cuadras, J. Fortiana, and J. A. Rodríguez-Lallena, edis.), Kluwer Academic Publishers, Dordrecht 2002, pp. 187-194. Zbl1135.62339MR2058992DOI10.1007/978-94-017-0061-0_20
  29. Nelsen, R. B., Úbeda-Flores, M., 10.1016/j.crma.2005.09.026, Comptes Rendus Mathematique 341 (2005), 583-586. Zbl1076.62053MR2182439DOI10.1016/j.crma.2005.09.026
  30. Pelessoni, R., Vicig, P., Montes, I., Miranda, E., Imprecise copulas and bivariate stochastic orders., In: Proc. EUROFUSE 2013, Oviedo 2013, pp. 217-224. 
  31. Sainio, E., Turunen, E., Mesiar, R., 10.1016/j.fss.2007.09.018, Fuzzy Sets and Systems 159 (2008), 491-499. Zbl1176.03013MR2388105DOI10.1016/j.fss.2007.09.018
  32. Schweizer, B., Sklar, A., Associative functions and abstract semigroups., Publ. Math. Debrecen 10 (1963), 69-81. MR0170967
  33. Schweizer, B., Sklar, A., Probabilistic Metric Spaces., North-Holland, New York 1983. Zbl0546.60010MR0790314
  34. Sklar, A., Fonctions de répartition à n dimensions et leurs marges., Publ. Inst. Statist. Univ. Paris 8 (1959), 229-231. MR0125600
  35. Sklar, A., Random variables, joint distribution functions, and copulas., Kybernetika 9 (1973), 449-460. Zbl0292.60036MR0345164
  36. Walley, P., Statistical Reasoning with Imprecise Probabilities., Chapman and Hall, London 1991. Zbl0732.62004MR1145491

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