Advertisement

Comparing the magnitude of fuzzy intervals and fuzzy random variables from the standpoint of gradual numbers

  • 110 Accesses

Abstract

Gradual numbers have been introduced to separate fuzziness (understood as gradation) from uncertainty in so-called fuzzy numbers. Then a fuzzy number can be viewed as a standard interval of functions, each interpreted as a gradual number. Gradual numbers are naturally met when representing probabilities of fuzzy events, midpoints of fuzzy intervals, etc. They can be viewed as a non-monotonic generalization of cumulative probability distributions. This paper presents three methods for comparing gradual numbers that generalize stochastic orderings to such non-monotonic functions. Then it proposes joint extensions of stochastic dominance and statistical preference to random fuzzy intervals when the fuzzy intervals are understood as intervals of gradual numbers. This approach, which combines known probabilistic orderings with known forms of interval orderings, can be viewed as a systematic way of constructing methods for ranking fuzzy random variables.

This is a preview of subscription content, log in to check access.

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 99

This is the net price. Taxes to be calculated in checkout.

Notes

  1. 1.

    In particular, \(\le _1 \ne \lnot (>_1)\).

  2. 2.

    It assumes a continuous density, i.e., with no mass concentrated at a point.

  3. 3.

    The joint distribution of the random set is obtained from a joint probability space and the corresponding multimappings. It is not equivalent to taking the joint of probabilities, one in each credal set.

  4. 4.

    If \(\mu _{\tilde{A}}\) is not continuous, these results hold with strict \(\alpha \)-cuts \( \tilde{A}^\alpha = \{ r : \mu _{\tilde{A}}(r) > \alpha \}, \alpha < 1\).

  5. 5.

    Defining a fuzzy set by the \(\alpha \)-cut multimapping is at the origin of fuzzy sets (Zadeh 1965) and has been studied especially by Negoita and Ralescu (1975), and more recently extended under the popular name “soft set”, dropping the nestedness assumption (Molodtsov 1999). However, soft sets are often only general multimappings or relations between two sets, without direct connection to fuzzy sets.

  6. 6.

    Namely 6 possible interval comparisons, and 7 stochastic gradual number comparisons as the combination of sign dominance with probabilistic orderings leads to one method appearing in Table 3.

References

  1. Aiche F, Dubois D (2010) An extension of stochastic dominance to fuzzy random variables. In: Hüllermeier E et al (eds) Proceedings of international conference on information processing and management of uncertainty in knowledge-based systems (IPMU 2010), LNAI, vol 6178. Springer, pp 159–168

  2. Aiche F, Abbas M, Dubois D (2013) Chance-constrained programming with fuzzy stochastic coefficients. Fuzzy Optim Decis Mak 12(2):125–152

  3. Arrow KJ, Hurwicz L (1977) An optimality criterion for decision making under ignorance. In: Arrow KJ, Hurwicz L (eds) Studies in resource allocation processes. Cambridge University Press, Cambridge

  4. Boukezzoula R, Galichet S, Foulloy L, Elmasry M (2014) Extended gradual interval (EGI) arithmetic and its application to gradual weighted averages. Fuzzy Sets Syst 257:67–84

  5. Campos L, Munoz A (1989) A subjective approach for ranking fuzzy numbers. Fuzzy Sets Syst 29:145–153

  6. Chanas S, Nowakowski M (1988) Single value simulation of fuzzy variable. Fuzzy Sets Syst 25:43–57

  7. Chanas S, Zielinski P (1999) Ranking fuzzy real intervals in the setting of random sets—further results. Inf Sci 117:191–200

  8. Chanas S, Delgado M, Verdegay JL, Vila MA (1993) Ranking fuzzy real intervals in the setting of random sets. Inf Sci 69:201–217

  9. Chateauneuf A, Cohen M, Tallon J-M (2009) Decision under risk: the classical expected utility model. In: Bouyssou D (ed) Decision-making process, chap 8. ISTE and Wiley, London, pp 363–382

  10. Couso I, Dubois D (2009) On the variability of the concept of variance for fuzzy random variables. IEEE Trans Fuzzy Syst 17:1070–1080

  11. Couso I, Dubois D (2012) An imprecise probability approach to joint extensions of stochastic and interval orderings. In: Proceedings of international conference on information processing and management of uncertainty in knowledge-based systems, IPMU 2012, vol 3, pp 388–399

  12. Couso I, Sánchez L (2011) Upper and lower probabilities induced by a fuzzy random variable. Fuzzy Sets Syst 165(1):1–23

  13. Couso I, Dubois D, Sánchez L (2014) Random sets and random fuzzy sets as ill-perceived random variables. Springer briefs in computational intelligence. Springer, Berlin

  14. Couso I, Moral S, Sánchez L (2015) The behavioral meaning of the median. Inf Sci 294:127–138

  15. David H (1963) The method of paired comparisons. Griffin’s statistical monographs & courses, vol 12. Griffin, London

  16. De Baets B, De Meyer H (2008) On the cycle-transitive comparison of artificially coupled random variables. Int J Approx Reason 47:306–322

  17. Delgado M, Martín-Bautista MJ, Sánchez D, Vila MA (2002) A probabilistic definition of a non convex fuzzy cardinality. Fuzzy Sets Syst 126(2):41–54

  18. Denoeux T (2009) Extending stochastic order to belief functions on the real line. Inf Sci 179:1362–1376

  19. Destercke S, Couso I (2015) Ranking of fuzzy intervals seen through the imprecise probabilistic lens. Fuzzy Sets Syst 278:20–39

  20. Dubois D (2006) Possibility theory and statistical reasoning. Comput Stat Data Anal 51:47–69

  21. Dubois D (2011) The role of fuzzy sets in decision sciences: old techniques and new directions. Fuzzy Sets Syst 184(1):3–28

  22. Dubois D, Prade H (1983) Ranking fuzzy numbers in the setting of possibility theory. Inf Sci 30:183–224

  23. Dubois D, Prade H (1987) The mean value of a fuzzy number. Fuzzy Sets Syst 24:279–300

  24. Dubois D, Prade H (1988) Possibility theory. Plenum Press, New York

  25. Dubois D, Prade H (1991) Random sets and fuzzy interval analysis. Fuzzy Sets Syst 42:87–101

  26. Dubois D, Prade H (2008) Gradual elements in a fuzzy set. Soft Comput 12:165–175

  27. Dubois D, Kerre E, Mesiar R, Prade H (2000) Fuzzy interval analysis. In: Dubois D, Prade H (eds) Fundamentals of fuzzy sets, The handbooks of fuzzy sets series. Kluwer, Boston, pp 483–581

  28. Ferson S, Ginzburg LR (1996) Different methods are needed to propagate ignorance and variability. Reliab Eng Syst Saf 54:133–144

  29. Ferson S, Hajagos JG (2004) Arithmetic with uncertain numbers: rigorous and (often) best possible answers. Reliab Eng Syst Saf 85:135–152

  30. Ferson S, Ginzburg L, Kreinovich V, Myers DM, Sentz K (2003) Constructing probability boxes and Dempster–Shafer structures. Technical report, Sandia National Laboratories, USA

  31. Fishburn P (1987) Interval orderings. Wiley, New York

  32. Fortemps P, Roubens M (1996) Ranking and defuzzification methods based on area compensation. Fuzzy Sets Syst 82:319–330

  33. Fortin J, Dubois D, Fargier H (2008) Gradual numbers and their application to fuzzy interval analysis. IEEE Trans Fuzzy Syst 16:388–402

  34. Friedman M, Ma M, Kandel A (1998) Fuzzy linear systems. Fuzzy Sets Syst 96:201–209

  35. Gil MA, López-Díaz M, Ralescu DA (2006) Overview on the development of fuzzy random variables. Fuzzy Sets Syst 157(19):2546–2557

  36. Herencia JA (1996) Graded sets and points: a stratified approach to fuzzy sets and points. Fuzzy Sets Syst 77:191–202

  37. Kruse R, Meyer KD (1987) Statistics with vague data. D. Reidel, Dordrecht

  38. Kwakernaak H (1978) Fuzzy random variables I. Definitions and theorems. Inf Sci 15:1–29

  39. Martin TP, Azvine B (2013) The X-mu approach: fuzzy quantities, fuzzy arithmetic and fuzzy association rules. In: IEEE symposium on foundations of computational intelligence (FOCI), Singapore, pp 24–29

  40. Montes I, Destercke S (2017) Comonotonicity for sets of probabilities. Fuzzy Sets Syst 328:1–34

  41. Montes I, Miranda E, Montes S (2017) Imprecise stochastic orders and fuzzy rankings. Fuzzy Optim Decis Mak 16(3):297–327

  42. Molodtsov D (1999) Soft set theory—first results. Comput Math Appl 37(4/5):19–31

  43. Negoita CV, Ralescu DA (1975) Applications of fuzzy sets to systems analysis. Birkhauser, Basel

  44. Ogura Y, Li S-M, Ralescu DA (2001) Set defuzzification and Choquet integral. Int J Uncertain Fuzziness Knowl Based Syst 9(1):1–12

  45. Puri ML, Ralescu D (1986) Fuzzy random variables. J Math Anal Appl 114:409–420

  46. Rocacher D, Bosc P (2005) The set of fuzzy rational numbers and flexible querying. Fuzzy Sets Syst 155(3):317–339

  47. Sánchez D, Delgado M, Vila MA, Chamorro-Martinez J (2012) On a non-nested level-based representation of fuzziness. Fuzzy Sets Syst 192:159–175

  48. Smets P (2005) Belief functions on real numbers. Int J of Approx Reason 40:181–223

  49. Wang X, Kerre E (2001) Reasonable properties for the ordering of fuzzy quantities (2 parts). Fuzzy Sets Syst 118:375–406

  50. Williamson RC, Downs T (1990) Probabilistic arithmetic I. Numerical methods for calculating convolutions and dependency bounds. Int J Approx Reason 4(2):89–158

  51. Yager RR (1978) Ranking fuzzy subsets over the unit interval. In: Proceedings of IEEE international conference on decision and control, pp 1435–1437

  52. Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353

Download references

Acknowledgements

The first author benefited from visit scholarships granted by University of Tizi-Ouzou, and E.N.P.E.I., Algiers.

Author information

Correspondence to Didier Dubois.

Ethics declarations

Conflict of interest

The authors declare that none of them has any conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by A. Di Nola.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Aiche, F., Dubois, D. Comparing the magnitude of fuzzy intervals and fuzzy random variables from the standpoint of gradual numbers. Soft Comput 23, 5975–5990 (2019). https://doi.org/10.1007/s00500-018-3526-x

Download citation

Keywords

  • Gradual numbers
  • Stochastic dominance
  • Statistical preference
  • Probability
  • Interval orderings
  • Fuzzy intervals
  • Fuzzy random variables