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Copula as a Bridge Between Probability Theory and Fuzzy Logic

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Uncertainty Modeling

Part of the book series: Studies in Computational Intelligence ((SCI,volume 683))

Abstract

This work shows how dependence in many-valued logic and probability theory can be fused into one concept by using copulas and marginal probabilities. It also shows that the t-norm concept used in fuzzy logic is covered by this approach. This leads to a more general statement that axiomatic probability theory covers logic structure of fuzzy logic. This paper shows the benefits of using structures that go beyond the simple concepts of classical logic and set theory for the modeling of dependences.

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References

  1. B. Kovalerchuk, and D.Shapiro, “On the relation of the probability theory and the fuzzy sets theory, Foundation Computers and Artificial Intelligence, Vol. 7, pp. 385–396, 1988.

    Google Scholar 

  2. D. Dubois, and H. Prade “Fuzzy sets, probability and measurement,” Europ. J. of Operations Research, Vol. 40, pp. 135–154, 1989.

    Google Scholar 

  3. D.I. Blockley, “Fuzziness and probability: a discussion of Gaines’ axioms,” Civ. Engng Syst., Vol. 2, pp. 195–200, 1985.

    Google Scholar 

  4. P. Cheeseman, “Probabilistic versus fuzzy reasoning,” in Uncertainty in Artificial Intelligence 1, L. Kanal, and J. Lemmer, Eds., North-Holland, Amsterdam, 1988, pp. 85–102

    Google Scholar 

  5. B.R. Gaines, “Fuzzy and probability uncertainty logics,” Information and Control, Vol. 38, pp. 154–169, 1978.

    Google Scholar 

  6. G.J. Klir, and B. Parviz, “Probability-possibility transformations: a comparison,” Int. J. of General Systems, Vol. 21, pp. 291–310, 1992.

    Google Scholar 

  7. B. Kosko, “Fuzziness vs. probability,” Int. J. of General Systems, Vol. 17, pp. 211–240, 1990.

    Google Scholar 

  8. G. Resconi, B. Kovalerchuk: Agents’ model of uncertainty. Knowl. Inf. Syst. 18(2): 213–229, 2009.

    Google Scholar 

  9. G. Resconi, B. Kovalerchuk: Agents in neural uncertainty. IJCNN 2009, pp. 2649–2656

    Google Scholar 

  10. G. Resconi, B. Kovalerchuk: Fusion in agent-based uncertainty theory and neural image of uncertainty. IJCNN 2008: pp. 3538–3544, 2007

    Google Scholar 

  11. G. Resconi, B. Kovalerchuk: Explanatory Model for the Break of Logic Equivalence by Irrational Agents in Elkan’s Paradox. EUROCAST 2007: 26–33

    Google Scholar 

  12. G. Resconi, C. J. Hinde: Introduction to Active Sets and Unification. T. Computational Collective Intelligence 8: 1–36, 2012.

    Google Scholar 

  13. Graadel, E., Vaananen, J.: Dependence and independence. Studia Logica, 101(2), pp. 399–410, 2013.

    Google Scholar 

  14. J.Vaananen, Dependence Logic - A New Approach to Independence Friendly Logic, Cambridge University Press, 2007.

    Google Scholar 

  15. J. Ebbing, A. Hella, J-S. Meier, J. Muller, J. Virtema and H. Vollmer, Extended modal dependence logic, in: WoLLIC, 2013, pp. 126–137.

    Google Scholar 

  16. J. Ebbing, P. Lohmann, Complexity of model checking for modal dependence logic, in: M. Bieliková, G. Friedrich, G. Gottlob, S. Katzenbeisser and G. Turán, editors, SOFSEM, LNCS 7147, 2012, pp. 226–237.

    Google Scholar 

  17. J. Ebbing, P. Lohmann and F. Yang, Model checking for modal intuitionistic dependence logic, in: G. Bezhanishvili, S. Lobner, V. Marra and F. Richter, eds., Logic, Language, and Computation, Lecture Notes in Computer Science 7758, Springer, 2013 pp. 231–256.

    Google Scholar 

  18. P. Galliani, The dynamification of modal dependence logic, Journal of Logic, Language and Information 22 (2013), pp. 269–295.

    Google Scholar 

  19. J. Hintikka, and G. Sandu, Informational independence as a semantical phenomenon, in: Logic, methodology and philosophy of science, VIII (Moscow, 1987), Stud. Logic Found. Math. 126, North-Holland, Amsterdam, 1989 pp. 571–589.

    Google Scholar 

  20. J.-S. Müller, and H. Vollmer, Model checking for modal dependence logic: An approach through post’s lattice, in: L. Libkin, U. Kohlenbach and R. Queiroz, editors, Logic, Language, Information, and Computation, Lecture Notes in Computer Science 8071, 2013 pp. 238–250.

    Google Scholar 

  21. F. Yang, “On Extensions and Variants of Dependence Logic,” Ph.D. thesis, University of Helsinki, 2014.

    Google Scholar 

  22. I.R. Goodman, Fuzzy sets as equivalence classes of random sets, In: Yager, R., et al. (eds.), Fuzzy Sets and Possibility Theory, Oxford, UK: Pergamon Press, 327–432, 1982.

    Google Scholar 

  23. H. T. Nguyen, V. Kreinovich, “How to Fully Represent Expert Information about Imprecise Properties in a Computer System–Random Sets, Fuzzy Sets, and Beyond: An Overview”, International Journal of General Systems, 2014, 43(5–6), pp. 586–609.

    Google Scholar 

  24. P. Embrechts, F.Lindskog, A. McNeil: Modelling dependence with copulas and applications to risk management. In Handbook of heavy tailed distributions in finance, Rachev ST, ed. Elsevier/North-Holland, Amsterdam, 2003.

    Google Scholar 

  25. P. Jaworski, F. Durante, W. K. Härdle, T. Rychlik (Eds), Copula Theory and Its Applications, Lecture Notes in Statistics, Springer, 2010.

    Google Scholar 

  26. J. C. Fodor and M. Roubens: Fuzzy Preference Modelling and Multicriteria Decision Support, Kluwer, Dordrecht 1994.

    Google Scholar 

  27. O. Hadžić, E. Pap: Fixed Point Theory in Probabilistic Metric Spaces. Kluwer, Dordrecht 2001.

    Google Scholar 

  28. B. Schweizer, A. Sklar: Probabilistic Metric Spaces. North-Holland, New York, 1983.

    Google Scholar 

  29. E. Klement, R. Mesiar and E. Pap: Triangular Norms. Kluwer, Dordrecht 2000.

    Google Scholar 

  30. E. Klement; R. Mesiar; E. Pap, Invariant copulas, Kybernetika, Vol. 38 2002, No. 3, 275–286.

    Google Scholar 

  31. Many-Valued Logic, Stanford Encyclopedia of Philosophy, 2015, http://plato.stanford.edu/entries/logic-manyvalued/#TNorBasSys

  32. P. Hájek, Metamathematics of Fuzzy Logic, Dordrecht: Kluwer, 1998.

    Google Scholar 

  33. K. Aas, C. Czado, A. Frigessi, and H. Bakken. Pair-copula constructions of multiple dependencies. Insurance: Mathematics and Economics, 44:182–198, 2009.

    Google Scholar 

  34. R. Accioly and F. Chiyoshi. Modeling dependence with copulas: a useful tool for field development decision process. Journal of Petroleum Science and Engineering, 44:83–91, 2004.

    Google Scholar 

  35. A. Sklar. Fonctions de repartition a n dimensions et leurs marges. Publications de l’Institut de Statistique de L’Universite de Paris, 8:229–231, 1959.

    Google Scholar 

  36. A. Sklar: Random variables, joint distribution functions, and copulas. Kybernetika 9, 1973, 449–460.

    Google Scholar 

  37. J.-F. Mai, M. Scherer, “Simulating Copulas: Stochastic Models, Sampling Algorithms, and Applications”, World Scientific, 2012, http://www.worldscientific.com/doi/suppl/10.1142/p842/suppl_file/p842_chap01.pdf

  38. M.S. Tenney, “Introduction to Copulas”, Enterprise Risk Management Symposium, 2003, Casualty Actuarial Society, http://www.casact.org/education/rcm/2003/ERMHandouts/tenney1.pdf

  39. C. Klüppelberg, C. and S. Resnick, The Pareto copula, aggregation of risk and the Emperor’s socks. Preprint, Technical University of Munich, 2009.

    Google Scholar 

  40. T. Mikosch, Copulas: tales and facts. Extremes, Vol. 9, Issue 1, pp. 3–20, Springer, 2006.

    Google Scholar 

  41. B. Kovalerchuk, E. Vityaev, J. Ruiz, Consistent and Complete Data and “Expert” Mining in Medicine, In: Medical Data Mining and Knowledge Discovery, Springer, 2001, pp. 238–280.

    Google Scholar 

  42. G. Beliakov, A. Pradera and T. Calvo, Aggregation Functions: A Guide for Practitioners, Springer, Heidelberg, Berlin, New York, 2007.

    Google Scholar 

  43. R. Bellman, M. Giertz. “On the Analytic Formalism of the Theory of Fuzzy Sets.” Information Sciences 5 (1973): 149–156.

    Google Scholar 

  44. M. Navara, Triangular norms and conorms, Scholarpedia, 2(3):2398, 2007. http://www.scholarpedia.org/article/Triangular_norms_and_conorms

  45. S. Sriboonchitta, J. Liu, V. Kreinovich, and H.T. Nguyen, “Vine Copulas as a Way to Describe and Analyze Multi-Variate Dependence in Econometrics: Computational Motivation and Comparison with Bayesian Networks and Fuzzy Approaches”, In: V.-N. Huynh, V. Kreinovich, and S. Sriboonchitta eds.), Modeling Dependence in Econometrics, Springer, Berlin, 2014, pp. 169–187.

    Google Scholar 

  46. E. Jouini and C. Napp, Conditional comonotonicity, Decisions in Economics and Finance, 27 (2):153–166, 2004.

    Google Scholar 

  47. A. Patton, Applications of Copula Theory in Financial Econometrics, Dissertation, University of California, San Diego, 2002.

    Google Scholar 

  48. J. Bell, “On the Einstein Podolsky Rosen Paradox”. Physics 1 (3): 195–200, 1964

    Google Scholar 

  49. D’Espagnat Conceptual Foundations of Quantum Mechanics, 2nd ed. Addison Wesley

    Google Scholar 

  50. R. Feynman, R., Leighton, and M. Sands, The Feynman lectures on physics: Mainly mechanics, radiation, and heat, volume 3. Basic Books, NY, 2011.

    Google Scholar 

  51. J.A. de Barros, G. Oas and P. Supper, Negative probabilities and counterfactual reasoning on the double – slit experiment, [quant-ph] 16 Dec 2014 arXiv:1412.4888v1.

  52. P. Suppes, M. Zanotti, Existence of hidden variables having only upper probabilities Foundations of Physics 21 (12):1479–1499 (1991)

    Google Scholar 

  53. J. A. de Barros, P. Suppes, “Probabilistic Inequalities and Upper Probabilities in Quantum Mechanical Entanglement,” Manuscrito, v. 33, pp. 55–71 (2010).

    Google Scholar 

  54. P. Suppes, S. Hartmann, Entanglement, upper probabilities and decoherence in quantum mechanics, In: M. Suaráz et al (ed.), EPSA Philosophical Issues in the Sciences: Launch of the European Philosophy of Science Association, Springer, pp. 93–103, 2010

    Google Scholar 

  55. P. Feynman, QED: The Strange Theory of Light and Matter. Princeton University Press, 1988.

    Google Scholar 

  56. Double-slit experiment, Wikipedia, http://en.wikipedia.org/wiki/Double-slit_experiment, 2015.

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Authors and Affiliations

  1. Department of Mathematics and Physics, Catholic University, 25121, Brescia, Italy

    Germano Resconi

  2. Department of Computer Science, Central Washington University, Ellensburg, WA, 98926-7520, USA

    Boris Kovalerchuk

Authors
  1. Germano Resconi
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  2. Boris Kovalerchuk
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Correspondence to Germano Resconi .

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Editors and Affiliations

  1. University of Texas at El Paso, Department of Computer Science University of Texas at El Paso, El Paso, TX, USA

    Vladik Kreinovich

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Resconi, G., Kovalerchuk, B. (2017). Copula as a Bridge Between Probability Theory and Fuzzy Logic. In: Kreinovich, V. (eds) Uncertainty Modeling. Studies in Computational Intelligence, vol 683. Springer, Cham. https://doi.org/10.1007/978-3-319-51052-1_15

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  • DOI: https://doi.org/10.1007/978-3-319-51052-1_15

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