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On Possibilistic Representations of Fuzzy Intervals

Author

Listed:
  • Luciano Stefanini

    (Department of Economics, Society & Politics, Università di Urbino "Carlo Bo)

  • Maria Letizia Guerra

    (Department of Mathematics, University of Bologna)

Abstract

It is well known that a fuzzy interval has two equivalent representations given in terms of the so called Left and Right sides of the membership function (LR-representation) or in terms of the Lower and Upper branches defining the endpoints of the -cuts (LU-representation). In this paper we suggest an additional representation of fuzzy intervals called ACF-representation (using an average cumulative function instead of the membership function), based on possibility theory. We illustrate how to build the new representation and we state its basic properties. The main result is that the Average Cumulative (AC) function can be uniquely defined for any fuzzy interval and it is possible to go from one representation to the others through appropriate transformations. An interesting link can be established between ACF-representation and quantile functions, with a possible statistical interpretation useful in real application. We also recommend a parametric form of the AC function.

Suggested Citation

  • Luciano Stefanini & Maria Letizia Guerra, 2016. "On Possibilistic Representations of Fuzzy Intervals," Working Papers 1602, University of Urbino Carlo Bo, Department of Economics, Society & Politics - Scientific Committee - L. Stefanini & G. Travaglini, revised 2016.
    • Handle: RePEc:urb:wpaper:16_02
    as

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    File URL: http://www.econ.uniurb.it/RePEc/urb/wpaper/WP_16_02.pdf
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    References listed on IDEAS

    as
    1. Dubois, Didier, 2006. "Possibility theory and statistical reasoning," Computational Statistics & Data Analysis, Elsevier, vol. 51(1), pages 47-69, November.
    2. De Rossi, Giuliano & Harvey, Andrew, 2009. "Quantiles, expectiles and splines," Journal of Econometrics, Elsevier, vol. 152(2), pages 179-185, October.
    3. Korzeń, Marcin & Jaroszewicz, Szymon, 2014. "PaCAL: A Python Package for Arithmetic Computations with Random Variables," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 57(i10).
    4. Koenker,Roger, 2005. "Quantile Regression," Cambridge Books, Cambridge University Press, number 9780521845731, September.
    5. Baudrit, C. & Dubois, D., 2006. "Practical representations of incomplete probabilistic knowledge," Computational Statistics & Data Analysis, Elsevier, vol. 51(1), pages 86-108, November.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    KEYWORDS: possibility distribution; parametric representations; fuzzy intervals; quantiles; average cumulative function.;
    All these keywords.

    JEL classification:

    • I31 - Health, Education, and Welfare - - Welfare, Well-Being, and Poverty - - - General Welfare, Well-Being
    • Q57 - Agricultural and Natural Resource Economics; Environmental and Ecological Economics - - Environmental Economics - - - Ecological Economics
    • E01 - Macroeconomics and Monetary Economics - - General - - - Measurement and Data on National Income and Product Accounts and Wealth; Environmental Accounts

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