A generalization of p-boxes to affine arithmetic

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.

[1]  M. Berz,et al.  TAYLOR MODELS AND OTHER VALIDATED FUNCTIONAL INCLUSION METHODS , 2003 .

[2]  M. J. Frank,et al.  Best-possible bounds for the distribution of a sum — a problem of Kolmogorov , 1987 .

[3]  Eric Goubault,et al.  The Zonotope Abstract Domain Taylor1+ , 2009, CAV.

[4]  Jean Vignes,et al.  A stochastic arithmetic for reliable scientific computation , 1993 .

[5]  Eric Goubault,et al.  Towards an Industrial Use of FLUCTUAT on Safety-Critical Avionics Software , 2009, FMICS.

[6]  Vladik Kreinovich,et al.  Absolute Bounds on the Mean of Sum, Product, Max, and Min: A Probabilistic Extension of Interval Arithmetic , 2002 .

[7]  Robert C. Williamson,et al.  Probabilistic arithmetic. I. Numerical methods for calculating convolutions and dependency bounds , 1990, Int. J. Approx. Reason..

[8]  Scott Ferson,et al.  Constructing Probability Boxes and Dempster-Shafer Structures , 2003 .

[9]  P. Limbourg,et al.  Fault tree analysis in an early design stage using the Dempster-Shafer theory of evidence , 2007 .

[10]  Olga Kosheleva,et al.  A Faster Algorithm for Computing the Sum of p-Boxes , 2010 .

[11]  Jun Li,et al.  Chebyshev Affine-Arithmetic-Based Parametric Yield Prediction Under Limited Descriptions of Uncertainty , 2008, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[12]  Eric Goubault,et al.  Static Analysis of Finite Precision Computations , 2011, VMCAI.

[13]  Daniel Berleant,et al.  Bounding the Results of Arithmetic Operations on Random Variables of Unknown Dependency Using Intervals , 1998, Reliab. Comput..

[14]  Glenn Shafer,et al.  A Mathematical Theory of Evidence , 2020, A Mathematical Theory of Evidence.

[15]  Eric Goubault,et al.  A Logical Product Approach to Zonotope Intersection , 2010, CAV.

[16]  Anna M. Bonner,et al.  Acknowledgments , 2019, The Neurodiagnostic journal.

[17]  T Singh,et al.  Approximate interval method for epistemic uncertainty propagation using Polynomial Chaos and evidence theory , 2010, Proceedings of the 2010 American Control Conference.

[18]  Jean Goubault-Larrecq.  Prevision Domains and Convex Powercones , 2008, FoSSaCS.

[19]  J. Stolfi,et al.  Aane Arithmetic and Its Applications to Computer Graphics , 1990 .

[20]  David Monniaux.  Abstraction of Expectation Functions Using Gaussian Distributions , 2003, VMCAI.

[21]  Arnold Neumaier.  Clouds, Fuzzy Sets, and Probability Intervals , 2004, Reliab. Comput..

[22]  William H. Sanders,et al.  Stochastic Activity Networks: Formal Definitions and Concepts , 2002, European Educational Forum: School on Formal Methods and Performance Analysis.

[23]  Scott Ferson,et al.  Dependence in Dempster-Shafer theory and probability bounds analysis* , 2004 .

[24]  Daniel Berleant,et al.  Arithmetic on Random Variables: Squeezing the Envelopes with New Joint Distribution Constraints , 2005, ISIPTA.

[25]  Helen M. Regan,et al.  Equivalence of methods for uncertainty propagation of real-valued random variables , 2004, Int. J. Approx. Reason..

[26]  Eric Goubault,et al.  Static Analysis of Numerical Algorithms , 2006, SAS.

[27]  Jun Li,et al.  Chebyshev affine arithmetic based parametric yield prediction under limited descriptions of uncertainty , 2008, ASP-DAC 2008.

[28]  R. Nelsen.  An Introduction to Copulas , 1998 .