Skip to main content

Types and Measures of Uncertainty

  • Chapter
Consensus Under Fuzziness

Part of the book series: International Series in Intelligent Technologies ((ISIT,volume 10))

Abstract

This paper is an overview of the current state of affairs in the area of measuring uncertainty. Three basic types of uncertainty are introduced: nonspecificity and conflict, which result from information deficiency, and fuzziness, which results from linguistic imprecision. Well-justified measures of these types of uncertainty in fuzzy set theory, possibility theory, Dempster-Shafer, theory as well as in classical set theory and probability theory are overviewed.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. J. Aczél and Z. Daróczy. On Measures of Information and their Characterizations. Academic Press, New York, 1975.

    Google Scholar 

  2. I. Z. Batyrshin. On fuzziness measures of entropy on Kleene algebras. Fuzzy Sets and Systems, 34(1):47–60, 1990.

    Article  Google Scholar 

  3. S. R. Chakravarty and T. Roy. Measurement of fuzziness: a general approach. Theory and Decision, 19(1):163–169, 1985.

    Article  Google Scholar 

  4. C. W. R. Chau, P. Lingras, and S. K. M. Wong. Upper and lower entropies of belief functions using compatible probability functions. In J. Komorowski and Z. W. Raś, editors, Methodologies for Intelligent Systems, Proceedings of 7th International Symposium ISMIS’93, pages 306–315. Springer-Verlag, 1993.

    Google Scholar 

  5. A. de Luca and S. Termini. A definition of a nonprobabilistic entropy in the setting of fuzzy set theory. Information and Control, 20(4):301–312, 1972.

    Article  Google Scholar 

  6. A. de Luca and S. Termini. Entropy of L-fuzzy sets. Information and Control, 24(1):55–73, 1974.

    Article  Google Scholar 

  7. D. Dubois and H. Prade. A note on measures of specificity for fuzzy sets. International Journal of General Systems, 10(4):279–283, 1985.

    Article  Google Scholar 

  8. D. Dubois and H. Prade. Possibility Theory. Plenum Press, New York, 1988.

    Book  Google Scholar 

  9. J. F. Geer and G. J. Klir. Discord in possibility theory. International Journal of General Systems, 19(2):119–132, 1991.

    Article  Google Scholar 

  10. T. George and N. R. Pal. Quantification of conflict in Dempster-Shafer framework: a new approach. International Journal of General Systems, 24(4):407–423, 1996.

    Article  Google Scholar 

  11. M. A. Gil. Fuzziness and loss of information in statistical problems. IEEE Transaction on Systems, Man, and Cybernetics, 17(6):1016–1025, 1987.

    Google Scholar 

  12. D. Harmanec. Toward a characterization of uncertainty measure for the Dempster-Shafer theory. In P. Besnard and S. Hanks, editors, Proceedings of the Eleventh Conference on Uncertainty in Artificial Intelligence, pages 255–261, San Mateo, California, 1995. Morgan Kaufmann Publishers.

    Google Scholar 

  13. D. Harmanec and G. J. Klir. Measuring total uncertainty in Dempster-Shafer theory: A novel approach. International Journal of General Systems, 22(4):405–419, 1994.

    Article  Google Scholar 

  14. D. Harmanec and G. J. Klir. On information-preserving transformations. International Journal of General Systems, 1996. (Submitted).

    Google Scholar 

  15. D. Harmanec, G. Resconi, G. J. Klir, and Y. Pan. On the computation of the uncertainty measure for the Dempster-Shafer theory. International Journal of General Systems, 25(2), 1996. (to appear).

    Google Scholar 

  16. R. V. Hartley. Transmission of information. Bell System Technical Journal, 7:535–563, 1928.

    Google Scholar 

  17. M. Higashi and G. J. Klir. On measures of fuzzines and fuzzy complements. International Journal of General Systems, 8(3):169–180, 1982.

    Article  Google Scholar 

  18. M. Higashi and G. J. Klir. Measures of uncertainty and information based on possibility distributions. International Journal of General Systems, 9(1):43–58, 1983.

    Article  Google Scholar 

  19. A. Kaufmann. Introduction to the theory of fuzzy subsets. Academic Press, New York, 1985.

    Google Scholar 

  20. G. J. Klir. Develop ments in uncertainty — based information. In M. C. Yovits, editor, Advances in Computers, volume 36, pages 255–332. Academic Press, San Diego, 1993.

    Google Scholar 

  21. G. J. Klir. Principles of uncertainty: What are they? Why do we need them? Fuzzy Sets and Systems, 74(1):15–31, 1995.

    Article  Google Scholar 

  22. G. J. Klir and D. Harmanec. On some bridges to possibility theory. In G. de Cooman, D. Ruan, and E. E. Kerre, editors, Foundations and Applications of Possibility Theory (Proceedings of FAPT’95), volume 8 of Advances in Fuzzy Systems — Applications and Theory, pages 3–19, New Jersey, 1995. World Scientific.

    Google Scholar 

  23. G. J. Klir and M. Mariano. On the uniqueness of possibilistic measure of uncertainty and information. Fuzzy Sets and Systems, 24(2):197–219, 1987.

    Article  Google Scholar 

  24. G. J. Klir and B. Parviz. A note on measure of discord. In D. Dubois et al., editors, Proceedings of the Eighth Conference on Uncertainty in Artificial Intelligence, pages 138–141, San Mateo, California, 1992. Morgan Kaufman.

    Google Scholar 

  25. G. J. Klir and A. Ramer. Uncertainty in the Dempster-Shafer theory: A critical re-examination. International Journal of General Systems, 18(2):155–166, 1990.

    Article  Google Scholar 

  26. G. J. Klir and B. Yuan. On measures of conflict among set-valued statements. In Proceedings of 1993 World Congress on Neural Networks, Portland, Oregon, 1993.

    Google Scholar 

  27. G. J. Klir and B. Yuan. Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice Hall PTR, Upper Saddle River, NJ, 1995.

    Google Scholar 

  28. G. J. Klir and B. Yuan. On nonspecificity of fuzzy sets with continuous membership functions. In Proceedings of 1995 IEEE International Conference on Systems, Man, and Cybernetics, Vancouver, 1995.

    Google Scholar 

  29. J. Knopfmacher. On measures of fuzziness. Journal of Mathematical Analysis and Applications, 49:529–534, 1975.

    Article  Google Scholar 

  30. R. Körner and W. Näther. On the specificity of evidences. Fuzzy Sets and Systems, 71:183–196, 1995.

    Article  Google Scholar 

  31. Y. Maeda and H. Ichihashi. An uncertainty measure with monotonicity under the random set inclusion. International Journal of General Systems, 21(4):379–392, 1993.

    Article  Google Scholar 

  32. I. Maung. Two characterizations of a minimum-information principle for possibilistic reasoning. International Journal of Approximate Reasoning, 12(2):133–156, 1995.

    Article  Google Scholar 

  33. A. Meyerowitz, F. Richman, and E. A. Walker. Calculating maximum-entropy probability densities for belief functions. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 2(4):377–389, 1994.

    Article  Google Scholar 

  34. N. R. Pal and J. C. Bezdek. Measuring fuzzy uncertainty. IEEE Transactions on Fuzzy Systems, 2(2): 107–118, 1994.

    Article  Google Scholar 

  35. N. R. Pal, J. C. Bezdek, and R. Hemashina. Uncertainty measures for evidential reasoning i: A review. International Journal of Approximate Reasoning, 7(3,4):165–183, 1992.

    Article  Google Scholar 

  36. N. R. Pal, J. C. Bezdek, and R. Hemashina. Uncertainty measures for evidential reasoning ii: A new measure of total uncertainty. International Journal of Approximate Reasoning, 8(1):1–16, 1993.

    Article  Google Scholar 

  37. A. Ramer. Uniqueness of information measure in the theory of evidence. Fuzzy Sets and Systems, 24(2): 183–196, 1987.

    Article  Google Scholar 

  38. A. Ramer and G. J. Klir. Measures of discord in the Dempster-Shafer theory. Information Sciences, 67(1-2):35–50, 1993.

    Article  Google Scholar 

  39. A. Rényi. Probability Theory, chapter IX, Introduction to Information Theory, pages 540–616. North-Holland, Amsterdam, 1970.

    Google Scholar 

  40. T. L. Saaty. Measuring of fuzziness of sets. Journal of Cybernetics, 4:53–61, 1974.

    Article  Google Scholar 

  41. G. Shafer. A Mathematical Theory of Evidence. Princeton University Press, Princeton, 1976.

    Google Scholar 

  42. C. E. Shannon. The mathematical theory of communication. The Bell Technical Journal, 27 (July, October):379–423, 623-656, 1948.

    Google Scholar 

  43. J. Vejnarová. A few remarks on measures of uncertainty in Dempster-Shafer theory. In Proceedings of Workshop on Uncertainty in Expert Systems, Prague, 1991. Czechoslovak Academy of Sciences.

    Google Scholar 

  44. J. Vejnarová and G. J. Klir. Measure of strife in Dempster-Shafer theory. International Journal of General Systems, 22(1):25–42, 1993.

    Article  Google Scholar 

  45. R. R. Yager. On the measure of fuzziness and negation. Part I: Membership in the unit interval. International Journal of General Systems, 5(4):221–229, 1979.

    Article  Google Scholar 

  46. R. R. Yager. On the measure of fuzziness and negation. Part II: Lattices. Information and Control, 44(3):236–260, 1980.

    Article  Google Scholar 

  47. R. R. Yager. Measures of fuzziness based on t-norms. Stochastica, 6(1):207–229, 1982.

    Google Scholar 

  48. R. R. Yager. Ordinal measures of specificity. International Journal of General Systems, 17(1):57–72, 1990.

    Article  Google Scholar 

  49. R. R. Yager. Similarity based specificity measures. International Journal of General Systems, 19(2):91–105, 1991.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer Science+Business Media New York

About this chapter

Cite this chapter

Klir, G.J., Harmanec, D. (1997). Types and Measures of Uncertainty. In: Kacprzyk, J., Nurmi, H., Fedrizzi, M. (eds) Consensus Under Fuzziness. International Series in Intelligent Technologies, vol 10. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-6333-4_3

Download citation

  • DOI: https://doi.org/10.1007/978-1-4615-6333-4_3

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-7908-9

  • Online ISBN: 978-1-4615-6333-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics