, Volume 94, Issue 2–4, pp 189–201 | Cite as

A generalization of p-boxes to affine arithmetic

  • Olivier BouissouEmail author
  • Eric Goubault
  • Jean Goubault-Larrecq
  • Sylvie Putot


We often need to deal with information that contains both interval and probabilistic uncertainties. P-boxes and Dempster–Shafer structures are models that unify both kind of information, but they suffer from the main defect of intervals, the wrapping effect. We present here a new arithmetic that mixes, in a guaranteed manner, interval uncertainty with probabilities, while using some information about variable dependencies, hence limiting the loss from not accounting for correlations. This increases the precision of the result and decreases the computation time compared to standard p-box arithmetic.


Affine arithmetic P-boxes Dempster–Shafer structures 

Mathematics Subject Classification (2010)

60A86 65G30 65G50 65C50 


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  1. 1.
    Berleant D, Goodman-Strauss C (1998) Bounding the results of arithmetic operations on random variables of unknown dependency using intervals. Reliab Comput 4(2): 147–165MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Busaba J, Suwan S, Kosheleva O (2010) A faster algorithm for computing the sum of p-boxes. J Uncertain Syst 4(4): 244–249Google Scholar
  3. 3.
    Comba JLD, Stolfi J (1993) Affine arithmetic and its applications to computer graphics. SEBGRAPI’93Google Scholar
  4. 4.
    Ferson S (2002) RAMAS Risk Calc 4.0 Software: risk assessment with uncertain numbers. Lewis Publishers, Boca RatonGoogle Scholar
  5. 5.
    Ferson S, Kreinovich V, Ginzburg L, Myers D, Sentz K (2003) Constructing probability boxes and Dempster–Shafer structures. Tech. Rep. SAND2002-4015Google Scholar
  6. 6.
    Ferson S, Nelsen R, Hajagos J, Berleant D, Zhang J, Tucker W, Ginzburg L, Oberkampf W (2004) Dependence in probabilistic modelling, Dempster–Shafer theory and probability bounds analysis. Tech. rep., Sandia National LaboratoriesGoogle Scholar
  7. 7.
    Frank MJ, Nelsen RB, Schweizer B (1987) Best-possible bounds for the distribution of a sum, a problem of Kolmogorov. Prob Theory Rel Fields 74: 199–211. doi: 10.1007/BF00569989 MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Ghorbal K, Goubault E, Putot S (2010) A logical product approach to zonotope intersection. In: CAV, LNCS, vol 6174Google Scholar
  9. 9.
    Goubault E, Putot S (2009) A zonotopic framework for functional abstractions. In: CoRR. abs/0910.1763Google Scholar
  10. 10.
    Goubault E, Putot S (2011) Static analysis of finite precision computations. In: VMCAI, LNCS, vol 6538, pp 232–247Google Scholar
  11. 11.
    Limbourg P, Savi R, Petersen J, Kochs HD (2007) Fault tree analysis in an early design stage using the Dempster–Shafer theory of evidence. In: ESREL 2007, pp 713–722Google Scholar
  12. 12.
    Makino K, Berz M (2003) Taylor models and other validated functional inclusion methods. Int J Pure Appl Math 4(4): 379–456MathSciNetzbMATHGoogle Scholar
  13. 13.
    Nelsen R (1999) An introduction to copulas. In: Lecture notes in statistics. Springer, BerlinGoogle Scholar
  14. 14.
    Regan HM, Ferson S, Berleant D (2004) Equivalence of methods for uncertainty propagation of real-valued random variables. Int J Approx Reason 36(1): 1–30MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Sanders WH, Meyer JF (2000) Stochastic activity networks: Formal definitions and concepts. In: European Educational Forum, pp 315–343Google Scholar
  16. 16.
    Shafer G (1976) A mathematical theory of evidence. Princeton University Press, PrincetonzbMATHGoogle Scholar
  17. 17.
    Sun J, Huang Y, Li J, Wang JM (2008) Chebyshev affine arithmetic based parametric yield prediction under limited descriptions of uncertainty. In: ASP-DAC ’08. IEEE Computer Society Press, Los Angeles, pp 531–536Google Scholar
  18. 18.
    Terejanu G, Singla P, Singh T, Scott PD (2010) Approximate interval method for epistemic uncertainty propagation using polynomial chaos and evidence theory. In: American Control ConferenceGoogle Scholar
  19. 19.
    Vignes J (1993) A stochastic arithmetic for reliable scientific computation. Math Comput Simul 35(3): 233–261MathSciNetCrossRefGoogle Scholar
  20. 20.
    Williamson RC, Downs T (1990) Probabilistic arithmetic I: numerical methods for calculating convolutions and dependency bounds. J Approx Reason 4(2): 89–158MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Zhang J, Berleant D (2005) Arithmetic on random variables: squeezing the envelopes with new joint distribution constraints. In: ISIPTA, pp 416–422Google Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Olivier Bouissou
    • 1
    Email author
  • Eric Goubault
    • 1
  • Jean Goubault-Larrecq
    • 2
  • Sylvie Putot
    • 1
  1. 1.CEA Saclay Nano-INNOVGif-sur-Yvette CedexFrance
  2. 2.LSV, ENS CachanCachanFrance

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