Random Fuzzy Sets as Ill-Perceived Random Variables

  • Inés CousoEmail author
  • Didier Dubois
  • Luciano Sánchez
Part of the SpringerBriefs in Applied Sciences and Technology book series (BRIEFSAPPLSCIENCES)


The concept of fuzzy random variable, that extends the classical definition of random variable, was introduced by Féron [37] in 1976. Later on, several authors, and especially Kwakernaak [54], Puri and Ralescu [62], Kruse and Meyer [53], Diamond and Kloeden [26], proposed other variants. More recently Krätschmer [51] surveyed all of these definitions and proposed a unified formal approach. In all of these papers, a fuzzy random variable is defined as a function that assigns a fuzzy subset to each possible output of a random experiment. Just like in the particular case of random sets considered in Chap. 1, the different definitions in the literature disagree on the measurability conditions imposed to this mapping, and in the properties of the output space, but all of them intend to model situations that combine fuzziness and randomness.


Probability Measure Fuzzy Subset Possibility Distribution Fuzzy Random Variable Possibility Measure 
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  1. 1.
    J. Aumann, Integral of set valued functions. J. Math. Anal. Appl. 12, 1–12 (1965)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    J.-P. Auray, H. Prade. Robert Féron: a pioneer in soft methods for probability and statistics. In D. Dubois et al. (eds.), Soft Methods for Handling Variability and Imprecision (Proc. SMPS 2008), Advances in Soft Computing, 48, pp. 27–32. Springer (2008)Google Scholar
  3. 3.
    A. Bardossy, I. Bogardi, W.E. Kelly, Imprecise fuzzy information in geostatistics. Math. Geol. 20, 287–311 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    A. Bardossy, I. Bogardi, W.E. Kelly, Kriging with imprecise (fuzzy) variograms. I: theory. Math. Geol. 22, 63–79 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    A. Bardossy, I. Bogardi, W.E. Kelly, Kriging with imprecise (fuzzy) variograms. II: application. Math. Geol. 22, 81–94 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    C. Baudrit, D. Dubois, D. Guyonnet, H. Fargier. Joint treatment of imprecision and randomness in uncertainty propagation. In: Modern Information Processing: From Theory to Applications. B. Bouchon-Meunier, G. Coletti, R.R. Yager (Eds.), Elsevier, 37–47 (2006)Google Scholar
  7. 7.
    C. Baudrit, I. Couso, D. Dubois, Joint propagation of probability and possibility in risk analysis: towards a formal framework. Int. J. Approximate Reasoning 45, 82–105 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    L. Boyen, G. de Cooman, E. E. Kerre. On the extension of P-consistent mappings. In: G. de Cooman, D. Ruan, E. E. Kerre (Eds.), Foundations and Applications of Possibility Theory-Proceedings of FAPT’95, World Scientific (Singapore, 1995) 88–98.Google Scholar
  9. 9.
    J. Casillas L. Sánchez. Knowledge extraction from fuzzy data for estimating consumer behavior models. In: Proceedings of 2006 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE’06). Vancouver, Canada, pp. 572–578 (2006)Google Scholar
  10. 10.
    A. Colubi, R. Coppi, P. D’Urso, M.A. Gil. Statistics with fuzzy random variables, METRON–Int. J. Stat. vol. LXV, 277–303 (2007)Google Scholar
  11. 11.
    I. Couso, D. Dubois, On the variability of the concept of variance for fuzzy random variables. IEEE Trans. Fuzzy Syst. 17, 1070–1080 (2009)CrossRefGoogle Scholar
  12. 12.
    I. Couso, D. Dubois, S. Montes, L. Sánchez. On various definitions of the variance of a fuzzy random variable, in: Proceedings of Fifth International Symposium on Imprecise Probabilities, Theory and Applications (ISIPTA 07) Prague, Czech Republic, pp. 135–144 (2007)Google Scholar
  13. 13.
    I. Couso, E. Miranda, G. de Cooman, A possibilistic interpretation of the expectation of a fuzzy random variable, in Soft Methodology and Random Information Systems, ed. by M. López-Daz, M.A. Gil, P. Grzegorzewski, O. Hryniewicz, J. Lawry (Springer, Berlin, 2004), pp. 133–140CrossRefGoogle Scholar
  14. 14.
    I. Couso, S. Montes and P. Gil. The necessity of the strong alpha-cuts of a fuzzy set, Int. J. Unc., Fuzz. Knowledge-Based Syst. 9, 249–262 (2001)Google Scholar
  15. 15.
    I. Couso, S. Montes P. Gil. Second-order possibility measure induced by a fuzzy random variable, in C. Bertoluzza, M. A. Gil, and D. A. Ralescu (Eds.) Statistical Modeling, Analysis and Management of Fuzzy data, Springer, Heidelberg pp. 127–144 (2002)Google Scholar
  16. 16.
    I. Couso, L. Sánchez, Higher order models for fuzzy random variables. Fuzzy Sets and Syst. 159, 237–258 (2008)CrossRefzbMATHGoogle Scholar
  17. 17.
    I. Couso, L. Sánchez, P. Gil, Imprecise distribution function associated to a random set. Inf. Sci. 159, 109–123 (2004)CrossRefzbMATHGoogle Scholar
  18. 18.
    G. de Cooman, A behavioural model for vague probability assessments. Fuzzy Sets Syst. 154, 305–358 (2005)CrossRefzbMATHGoogle Scholar
  19. 19.
    G. de Cooman, D. Aeyels, Supremum preserving upper probabilities. Inf. Sci. 118, 173–212 (1999)CrossRefzbMATHGoogle Scholar
  20. 20.
    G. de Cooman, P. Walley, An imprecise hierarchical model for behaviour under uncertainty. Theor. Decis. 52, 327–374 (2002)CrossRefzbMATHGoogle Scholar
  21. 21.
    T. Denœux, Modeling vague beliefs using fuzzy-valued belief structures. Fuzzy Sets Syst. 116, 167–199 (2000)CrossRefzbMATHGoogle Scholar
  22. 22.
    T. Denœux, Maximum likelihood estimation from fuzzy data using the EM algorithm. Fuzzy Sets Syst. 183, 72–91 (2011)CrossRefzbMATHGoogle Scholar
  23. 23.
    P. Diamond, Fuzzy least squares. Inf. Sci. 46, 141–157 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    P. Diamond, Interval-valued random functions and the kriging of intervals. Math. Geol. 20, 145–165 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    P. Diamond, Fuzzy kriging. Fuzzy Sets Syst. 33, 315–332 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    P. Diamond, P. Kloeden. Metric Spaces of Fuzzy Sets, World Scientic (Singapore, 1994)Google Scholar
  27. 27.
    D. Dubois, L. Foulloy, G. Mauris, H. Prade, Probability-possibility transformations, triangular fuzzy sets, and probabilistic inequalities. Reliable Comput. 10, 273–297 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    D. Dubois, J. Lang, H. Prade. Possibilistic logic. In D.M. Gabbay, C.J. Hogger, J.A. Robinson, D. Nute (eds.) Handbook of Logic in Artificial Intelligence and Logic Programming, Vol. 3, Oxford University Press, pp. 439–513 (1994)Google Scholar
  29. 29.
    D. Dubois, H. Prade, Evidence measures based on fuzzy information. Automatica 21, 547–562 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    D. Dubois, H. Prade, The mean value of a fuzzy number. Fuzzy Sets Syst. 24, 279–300 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    D. Dubois, H. Prade, Possibility Theory (PLenum Press, New-York, 1988)CrossRefzbMATHGoogle Scholar
  32. 32.
    D. Dubois, H. Prade, When upper probabilities are possibility measures. Fuzzy Sets Syst. 49, 65–74 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    D. Dubois, H. Prade, Fuzzy sets—a convenient fiction for modeling vagueness and possibility. IEEE Trans. Fuzzy Syst. 2, 16–21 (1994)CrossRefGoogle Scholar
  34. 34.
    D. Dubois, H. Prade, The three semantics of fuzzy sets. Fuzzy Sets Syst. 90, 141–150 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    D. Dubois, H. Prade, Gradual elements in a fuzzy set. Soft Comput. 12, 165–175 (2008)CrossRefzbMATHGoogle Scholar
  36. 36.
    D. Dubois, H. Prade, Gradualness, uncertainty and bipolarity: making sense of fuzzy sets. Fuzzy Sets Syst. 192, 3–24 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    R. Féron. Ensembles aléatoires flous, C.R. Acad. Sci. Paris Ser. A 282, 903–906 (1976)Google Scholar
  38. 38.
    S. Ferson, L. Ginzburg, V. Kreinovich, H.T. Nguyen, S.A. Starks. Uncertainty in risk analysis: towards a general second-order approach combining interval, probabilistic, and fuzzy techniques, In Proceedings of 2002 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE’02) 1342–1347. Honolulu, Hawaii, USA (2002)Google Scholar
  39. 39.
    S. Ferson, V. Kreinovich, L. Ginzburg, K. Sentz, D.S. Myers, Constructing probability boxes and Dempster-Shafer structures, Sandia National Laboratories, SAND2002-4015 (Albuquerque, NM, USA, 2003)Google Scholar
  40. 40.
    Y. Feng, L. Hu, H. Shu, The variance and covariance of fuzzy random variables and their applications. Fuzzy Sets Syst. 120, 487–497 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  41. 41.
    M.B. Ferraro, R. Coppi, G. González, A. Rodríguez, A. Colubi, A linear regression model for imprecise response. Int. J. Approximate Reasoning 51, 759–770 (2010)CrossRefzbMATHGoogle Scholar
  42. 42.
    M.A. Gil, M.A. Lubiano, M. Montenegro, M.T. López, Least squares fitting of an affine function and strength of association for interval-valued data. Metrika 56, 97–111 (2002)CrossRefMathSciNetGoogle Scholar
  43. 43.
    G. González-Rodríguez, A. Blanco, A. Colubi, M.A. Lubiano, Estimation of a simple linear regression model for fuzzy random variables. Fuzzy Sets Syst. 160, 357–370 (2009)CrossRefzbMATHGoogle Scholar
  44. 44.
    G. González-Rodríguez, A. Colubi, M.A. Gil, Fuzzy data treated as functional data. A one-way ANOVA test approach. Comput. Stat. Data Anal. 56, 943–955 (2012)CrossRefzbMATHGoogle Scholar
  45. 45.
    J.A. Herencia, Graded sets and points: A stratified approach to fuzzy sets and points. Fuzzy Sets Syst. 77, 191–202 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  46. 46.
    A. Herzig, J. Lang, P. Marquis. Action representation and partially observable planning using epistemic logic. In: Proceedings of the International Joint Conference on Artificial Intelligence (IJCAI-03), Acapulco, Morgan Kaufmann, San Francisco, pp. 1067–1072 (2003)Google Scholar
  47. 47.
    E. Hüllermeier. Learning from imprecise and fuzzy observations: data disambiguation through generalized loss minimization. Int. J. Approximate Reasoning (2013)
  48. 48.
    M. Ishizuka, K.S. Fu, J.T.P. Yao, Inferences procedures and uncertainty for the problem-reduction method. Inf. Sci. 28, 179–206 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  49. 49.
    E.P. Klement, M.L. Puri, D.A. Ralescu, Limit theorems for fuzzy random variables. Proc. Roy. Soc. London A 407, 171–182 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  50. 50.
    R. Körner, On the variance of fuzzy random variables. Fuzzy Sets Syst. 92, 83–93 (1997)CrossRefzbMATHGoogle Scholar
  51. 51.
    V. Krätschmer, A unified approach to fuzzy random variables. Fuzzy Sets Syst. 123, 1–9 (2001)CrossRefzbMATHGoogle Scholar
  52. 52.
    R. Kruse, On the variance of random sets. J. Math. Anal. Appl. 122, 469–473 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  53. 53.
    R. Kruse, K.D. Meyer, Statistics with vague data (D. Reidel Publishing Company, Dordrecht, 1987)CrossRefzbMATHGoogle Scholar
  54. 54.
    H. Kwakernaak, Fuzzy random variables definition and theorems. Inf. Sci. 15, 1–29 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  55. 55.
    K. Loquin, D. Dubois, A fuzzy interval analysis approach to kriging with ill-known variogram and data. Soft Computing, Special issue on Knowledge extraction from low quality data: theoretical, methodological and practical issues 16, 769–784 (2012)Google Scholar
  56. 56.
    M.A. Lubiano. Variation measures for imprecise random elements, Ph.D. Thesis, Universidad de Oviedo, Spain (1999). (In Spanish)Google Scholar
  57. 57.
    E. Miranda, G. de Cooman, I. Couso, Lower previsions induced by multi-valued mappings. J. Stat. Plan. Infer. 133, 173–197 (2005)CrossRefzbMATHGoogle Scholar
  58. 58.
    E. Miranda, I. Couso and P. Gil. Study of the probabilistic information of a random set. In Proceedings of Third International Symposium on Imprecise Probabilities and Their Applications (ISIPTA’03). Lugano, Switzerland (2003)Google Scholar
  59. 59.
    E. Miranda, I. Couso, P. Gil, Random sets as imprecise random variables. J. Math. Anal. Appl. 307, 32–47 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  60. 60.
    S. Moral. Comments on “Statistical reasoning with set-valued information: ontic versus epistemic view” by Inés Couso and Didier Dubois, Int. J. Approximate Reasoning (2014)
  61. 61.
    H.T. Nguyen, On random sets and belief functions. J. Math. Anal. Appl. 65, 531–542 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  62. 62.
    M. Puri, D. Ralescu, Fuzzy random variables. J. Math. Anal. Appl. 114, 409–422 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  63. 63.
    E. Ruspini, New experimental results in fuzzy clustering. Inf. Sci. 6, 273–284 (1973)CrossRefGoogle Scholar
  64. 64.
    L. Sánchez, I. Couso, Advocating the use of imprecisely observed data in genetic fuzzy systems. IEEE Trans. Fuzzy Syst. 15, 551–562 (2007)CrossRefGoogle Scholar
  65. 65.
    L. Sánchez, I. Couso, J. Casillas, in A Multiobjective Genetic Fuzzy System with Imprecise Probability Fitness for Vague Data, 2nd International Symposium on Evolving Fuzzy Systems 2006 (EFS06) (Ambleside, Lake District, UK, 2006), pp. 131–136Google Scholar
  66. 66.
    L. Sánchez, I. Couso, J. Casillas, Advocating the use of imprecisely observed data in genetic fuzzy systems. IEEE Trans. Fuzzy Syst. 15, 551–562 (2007)CrossRefGoogle Scholar
  67. 67.
    L. Sánchez, J. Otero and J. R. Villar. Boosting of fuzzy models for high-dimensional imprecise datasets. In: Proceedings of 11th Information Processing and Management of Uncertainty in Knowledge-Based Systems Conference (IPMU). Paris, France (2006)Google Scholar
  68. 68.
    L. Sánchez, M.R. Suárez, I. Couso. A fuzzy definition of Mutual Information with application to the design of Genetic Fuzzy Classifiers. In: Proceedings of International Conference on Machine Intelligence (ACIDCA-ICMI05). Tozeur, Tunisia, pp. 602–609 (2005)Google Scholar
  69. 69.
    L. Sánchez, M.R. Suárez, J.R. Villar, I. Couso, Mutual information-based feature selection and partition design in fuzzy rule-based classifiers from vague data. Int. J. Approximate Reasoning 49, 607–622 (2008)CrossRefGoogle Scholar
  70. 70.
    G.L.S. Shackle, Decision, Order, and Time, 2nd edn. (Cambridge University Press, Cambridge, 1969)Google Scholar
  71. 71.
    P. Smets, The degree of belief in a fuzzy event. Inf. Sci. 25, 1–19 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  72. 72.
    SMIRE Research Group at the University of Oviedo. A distance-based statistical analysis of fuzzy number-valued data. Int. J. Approx. Reason. (2013)
  73. 73.
    P. Walley, Statistical reasoning with imprecise probabilities (Chapman and Hall, London, 1991)CrossRefzbMATHGoogle Scholar
  74. 74.
    P. Walley, Statistical inferences based on a second-order possibility distribution. Int. J. Gen. Syst. 26, 337–384 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  75. 75.
    R.R. Yager, Generalized probabilities of fuzzy events from fuzzy belief structures. Inf. Sci. 28, 45–62 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  76. 76.
    J. Yen. Generalizing the Dempster-Shafer theory to fuzzy sets, IEEE Trans. Syst. Man Cybern. 20, 559–570 (1990)Google Scholar
  77. 77.
    J. Yen, Computing generalized belief functions for continuous fuzzy sets. Int. J. Approximate Reasoning 6, 1–31 (1992)CrossRefzbMATHGoogle Scholar
  78. 78.
    L.A. Zadeh, Probability measures of fuzzy events. J. Math. Anal. Appl. 23, 421–427 (1968)Google Scholar
  79. 79.
    L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning, Information Sciences, Part 1: 8, 199–249; Part 2: 8, 301–357. Part 3(9), 43–80 (1975)Google Scholar
  80. 80.
    L.A. Zadeh. Fuzzy sets and information granularity, Advances in Fuzzy Set Theory and Applications. In: Gupta M.M., Ragade R.K. and Yager R.R. (eds.), North-Holland, Amsterdam, 3–18 (1979).Google Scholar
  81. 81.
    L.A. Zadeh, Fuzzy probabilities. Inf. Proc. Manag. 20, 363–372 (1984)CrossRefzbMATHGoogle Scholar
  82. 82.
    L.A. Zadeh, Toward a generalized theory of uncertainty (GTU)-an outline. Inf. Sci. 172, 1–40 (2005)CrossRefzbMATHMathSciNetGoogle Scholar

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© The Author(s) 2014

Authors and Affiliations

  1. 1.Departamento de Estadística e I. O. y D. M.Universidad de OviedoOviedoSpain
  2. 2.IRIT, CNRSUniversité Paul SabatierToulouse Cedex 09France
  3. 3.Departamento de InformáticaUniversidad de OviedoOviedoSpain

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