Statistical Reasoning with Set-Valued Information: Ontic vs. Epistemic Views

Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 285)

Abstract

Sets, hence fuzzy sets, may have a conjunctive or a disjunctive reading. In the conjunctive reading a (fuzzy) set represents an object of interest for which a (gradual rather than Boolean) composite description makes sense. In contrast disjunctive (fuzzy) sets refer to the use of sets as a representation of incomplete knowledge. They do not model objects or quantities, but partial information about an underlying object or a precise quantity. In this case the fuzzy set captures uncertainty, and its membership function is a possibility distribution.We call epistemic such fuzzy sets, since they represent states of incomplete knowledge. Distinguishing between ontic and epistemic fuzzy sets is important in information-processing tasks because there is a risk of misusing basic notions and tools, such as distance between fuzzy sets, variance of a fuzzy random variable, fuzzy regression, etc. We discuss several examples where the ontic and epistemic points of view yield different approaches to these concepts.

Keywords

Interval Data Statistical Reasoning Belief Function Possibility Distribution Fuzzy Random Variable 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Baudrit, C., Couso, I., Dubois, D.: Joint propagation of probability and possibility in risk analysis: Towards a formal framework. Int. J. Approx. Reas. 45, 82–105 (2007)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bertoluzza, A.S.C., Salas, A., Corral, N.: On a new class of distances between fuzzy numbers. Mathware and Soft Comp. 2, 71–84 (1995)MathSciNetMATHGoogle Scholar
  3. 3.
    Boukezzoula, R., Galichet, S., Bisserier, A.: A Midpoint-Radius approach to regression with interval data. Int. J. Approx. Reasoning 52, 1257–1271 (2011)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Bouyssou, D., Dubois, D., Pirlot, M., Prade, H. (eds.): Decision-making Process — Concepts and Methods. ISTE London & Wiley, New York (2009)Google Scholar
  5. 5.
    Colubi, A.: Statistical inference about the means of fuzzy random variables: Applications to the analysis of fuzzy- and real-valued data. Fuzzy Sets Syst. 160(3), 344–356 (2009)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Colubi, A., González-Rodríguez, G., Gil, M.A., Trutschnig, W.: Nonparametric criteria for supervised classification of fuzzy data. Int. J. Approx. Reas. 52, 1272–1282 (2011)MATHCrossRefGoogle Scholar
  7. 7.
    Couso, I., Dubois, D.: On the Variability of the Concept of Variance for Fuzzy Random Variables. IEEE Trans. Fuzzy Syst. 17, 1070–1080 (2009)CrossRefGoogle Scholar
  8. 8.
    Couso, I., Sánchez, L.: Upper and lower probabilities induced by a fuzzy random variable. Fuzzy Sets Syst. 165, 1–23 (2011)MATHCrossRefGoogle Scholar
  9. 9.
    De Campos, L.M., Lamata, M.T., Moral, S.: The concept of conditional fuzzy measure. Int. J. of Intell. Syst. 5, 237–246 (1990)MATHCrossRefGoogle Scholar
  10. 10.
    De Cooman, G., Walley, P.: An imprecise hierarchical model for behaviour under uncertainty. Theory and Decision 52, 327–374 (2002)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Dempster, A.P.: Upper and lower probabilities induced by a multivalued mapping. Ann. Math. Stat. 38, 325–339 (1967)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Diamond, P.: Fuzzy least squares. Inform. Sci. 46, 141–157 (1988)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Diamond, P., Kloeden, P.: Metric spaces of fuzzy sets. World Scientific, Singapore (1994)MATHGoogle Scholar
  14. 14.
    Dubois, D.: Possibility theory and statistical reasoning. Comp. Stat. & Data Anal. 51, 47–69 (2006)MATHCrossRefGoogle Scholar
  15. 15.
    Dubois, D.: The role of fuzzy sets in decision sciences: Old techniques and new directions. Fuzzy Sets Syst. 184, 3–28 (2011)MATHCrossRefGoogle Scholar
  16. 16.
    Dubois, D., Prade, H.: Possibility Theory. Plenum Press, New York (1988)MATHCrossRefGoogle Scholar
  17. 17.
    Dubois, D., Prade, H.: Incomplete conjunctive information. Comp. & Math. Appl. 15, 797–810 (1988)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Dubois, D., Prade, H.: When upper probabilities are possibility measures. Fuzzy Sets Syst. 49, 65–74 (1992)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Dubois, D., Prade, H.: The three semantics of fuzzy sets. Fuzzy Sets Syst. 90, 141–150 (1997)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Dubois, D., Prade, H.: Formal representations of uncertainty. In: [4], ch. 3, pp. 85–156 (2009)Google Scholar
  21. 21.
    Dubois, D., Prade, H.: Gradualness, uncertainty and bipolarity: Making sense of fuzzy sets. Fuzzy Sets Syst. 192, 3–24 (2012)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Dubois, D., Foulloy, L., Mauris, G., Prade, H.: Probability-possibility transformations, triangular fuzzy sets, and probabilistic inequalities. Reliable Computing 10, 273–297 (2004)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Fagin, R., Halpern, J.Y.: A new approach to updating beliefs. In: Bonissone, P.P., Henrion, M., Kanal, L.N., Lemmer, J.F. (eds.) Uncertainty in Artificial Intelligence (UAI 1991), pp. 347–374. Elsevier, New York (1991)Google Scholar
  24. 24.
    Ferraro, M.B., Coppi, R., González-Rodríguez, G., Colubi, A.: A linear regression model for imprecise response. Int. J. Approx. Reas. 51, 759–770 (2010)MATHCrossRefGoogle Scholar
  25. 25.
    Ferson, S., Ginzburg, L., Kreinovich, V., Longpre, L., Aviles, M.: Computing variance for interval data is NP-hard. ACM SIGACT News 33, 108–118 (2002)CrossRefGoogle Scholar
  26. 26.
    González-Rodríguez, G., Blanco, A., Colubi, A., Lubiano, M.A.: Estimation of a simple linear regression model for fuzzy random variables. Fuzzy Sets and Systems 160(3), 357–370 (2009)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    González-Rodríguez, G., Colubi, A., Gil, M.A.: Fuzzy data treated as functional data. A one-way ANOVA test approach. Comp. Stat. and Data Anal. 56(4), 943–955 (2012)MATHCrossRefGoogle Scholar
  28. 28.
    Körner, R.: On the variance of fuzzy random variables. Fuzzy Sets Syst. 92, 83–93 (1997)MATHCrossRefGoogle Scholar
  29. 29.
    Halpern, J.Y., Fagin, R., Moses, Y., Vardi, M.Y.: Reasoning About Knowledge. MIT Press, Cambridge (2003)Google Scholar
  30. 30.
    Herzig, A., Lang, J., Marquis, P.: Action representation and partially observable planning using epistemic logic. In: Proc. Int. Joint Conf. on Artificial Intelligence (IJCAI 2003), pp. 1067–1072. Morgan Kaufmann, San Francisco (2003)Google Scholar
  31. 31.
    Kendall, D.G.: Foundations of a theory of random sets. In: Harding, E.F., Kendall, D.G. (eds.) Stochastic Geometry, pp. 322–376. J. Wiley & Sons, New York (1974)Google Scholar
  32. 32.
    Kruse, R., Meyer, K.: Statistics with Vague Data. D. Reidel, Dordrecht (1987)MATHCrossRefGoogle Scholar
  33. 33.
    Kwakernaak, H.: Fuzzy random variables — I. definitions and theorems. Inform. Sci. 15, 1–29 (1978)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Kwakernaak, H.: Fuzzy random variables — II. Algorithms and examples for the discrete case. Inform. Sci. 17, 253–278 (1979)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Lindley, D.V.: Scoring rules and the inevitability of probability. Int. Statist. Rev. 50, 1–26 (1982)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Loquin, K., Dubois, D.: Kriging and Epistemic Uncertainty: A Critical Discussion. In: Jeansoulin, R., Papini, O., Prade, H., Schockaert, S. (eds.) Methods for Handling Imperfect Spatial Information. STUDFUZZ, vol. 256, pp. 269–305. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  37. 37.
    Loquin, K., Dubois, D.: Kriging with Ill-Known Variogram and Data. In: Deshpande, A., Hunter, A. (eds.) SUM 2010. LNCS (LNAI), vol. 6379, pp. 219–235. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  38. 38.
    Matheron, G.: Random Sets and Integral Geometry. J. Wiley & Sons, New York (1975)MATHGoogle Scholar
  39. 39.
    Moore, R.: Methods and Applications of Interval Analysis. SIAM Studies in Applied Mathematics. SIAM, Philadelphia (1979)MATHCrossRefGoogle Scholar
  40. 40.
    Nguyen, H.T.: On random sets and belief functions. J. Math. Anal. Appl. 65, 531–542 (1978)MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Pichon, F., Dubois, D., Denoeux, T.: Relevance and truthfulness in information correction and fusion. Int. J. Approx. Reas. 53, 159–175 (2012)MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    Prade, H., Serrurier, M.: Maximum-likelihood principle for possibility distributions viewed as families of probabilities. In: Proc. IEEE Int. Conf. on Fuzzy Systems (FUZZ-IEEE 2011), pp. 2987–2993. IEEE Press, Piscataway (2011)Google Scholar
  43. 43.
    Puri, M., Ralescu, D.: Fuzzy random variables. J. Math. Anal. Appl. 114, 409–422 (1986)MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Ramos-Guajardo, A.B., Lubiano, M.A.: K-sample tests for equality of variances of random fuzzy sets. Comp. Stat. & Data Anal. 56, 956–966 (2012)MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    Shackle, G.L.S.: Decision, Order and Time in Human Affairs, 2nd edn. Cambridge University Press (1961)Google Scholar
  46. 46.
    Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press (1976)Google Scholar
  47. 47.
    Shafer, G., Tversky, A.: Languages and designs for probability. Cogn. Sci. 9, 309–339 (1985)CrossRefGoogle Scholar
  48. 48.
    Spadoni, M., Stefanini, L.: Computing the variance of interval and fuzzy data. Fuzzy Sets Syst. 165, 24–36 (2011)MathSciNetMATHCrossRefGoogle Scholar
  49. 49.
    Smets, P.: The normative representation of quantified beliefs by belief functions. Artif. Intell. 92, 229–242 (1997)MathSciNetMATHCrossRefGoogle Scholar
  50. 50.
    Tanaka, H., Guo, P.: Possibilistic Data Analysis for Operations Research. Physica-Verlag, Heidelberg (1999)MATHGoogle Scholar
  51. 51.
    Trutschnig, W., González-Rodríguez, G., Colubi, A., Gil, M.A.: A new family of metrics for compact, convex (fuzzy) sets based on a generalized concept of mid and spread. Inform. Sci. 179, 3964–3972 (2009)MathSciNetMATHCrossRefGoogle Scholar
  52. 52.
    Walley, P.: Statistical Reasoning with Imprecise Probabilities. Chapman and Hall (1991)Google Scholar
  53. 53.
    Yager, R.R.: Set-based representations of conjunctive and disjunctive knowledge. Inform. Sci. 41, 1–22 (1987)MathSciNetMATHCrossRefGoogle Scholar
  54. 54.
    Yen, J.: Generalizing the Dempster-Shafer theory to fuzzy sets. IEEE Trans. Syst. Man and Cybern. 20, 559–569 (1990)MATHCrossRefGoogle Scholar
  55. 55.
    Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning, part I. Inform. Sci. 8, 199–249 (1975)MathSciNetMATHCrossRefGoogle Scholar
  56. 56.
    Zadeh, L.A.: Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst. 1, 1–28 (1978)MathSciNetCrossRefGoogle Scholar
  57. 57.
    Zadeh, L.A.: PRUF — a meaning representation language for natural languages. Int. J. Man-Mach. Stud. 10, 395–460 (1978)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.IRIT, CNRS and Université de ToulouseToulouseFrance

Personalised recommendations