From Interval Computations to Constraint-Related Set Computations: Towards Faster Estimation of Statistics and ODEs under Interval and p-Box Uncertainty

  • Vladik Kreinovich
Part of the IUTAM Bookseries book series (IUTAMBOOK, volume 27)


Interval computations estimate the uncertainty of the result of data processing in situations in which we only know the upper bounds Δ on the measurement errors. In this case, based on the measurement result \(\widetilde{x}\), we can only conclude that the actual (unknown) value x of the desired quantity is in the interval  \([\widetilde{x}-\Delta, \widetilde{x}+\Delta]\). In interval computations, at each intermediate stage of the computation, we have intervals of possible values of the corresponding quantities. As a result, we often have bounds with excess width. To remedy this problem, in our previous papers, we proposed an extension of interval technique to set computations, where on each stage, in addition to intervals of possible values of the quantities, we also keep sets of possible values of pairs (triples, etc.). In this paper, we show that in several practical problems, such as estimating statistics (variance, correlation, etc.) and solutions to ordinary differential equations (ODEs) with given accuracy, this new formalism enables us to find estimates in feasible (polynomial) time.


Arithmetic Operation Interval Arithmetic Interval Uncertainty Interval Computation Interval Linear System 
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  1. 1.
    M. Ceberio, S. Ferson, V. Kreinovich, S. Chopra, G. Xiang, A. Murguia, and J. Santillan, “How To Take Into Account Dependence Between the Inputs: From Interval Computations to Constraint-Related Set Computations”, Proc. 2nd Int’l Workshop on Reliable Engineering Computing, Savannah, Georgia, February 22–24, 2006, pp. 127–154; final version in Journal of Uncertain Systems, 2007, Vol. 1, No. 1, pp. 11–34. Google Scholar
  2. 2.
    S. Ferson, RAMAS Risk Calc 4.0. CRC Press, Boca Raton, Florida, 2002. Google Scholar
  3. 3.
    S. Ferson, L. Ginzburg, V. Kreinovich, L. Longpré, and M. Aviles, “Computing Variance for Interval Data is NP-Hard”, ACM SIGACT News, 2002, Vol. 33, No. 2, pp. 108–118. CrossRefGoogle Scholar
  4. 4.
    L.Jaulin, M. Kieffer, O. Didrit, and E. Walter, Applied Interval Analysis, Springer, London, 2001. zbMATHGoogle Scholar
  5. 5.
    V. Kreinovich, A. Lakeyev, J. Rohn, and P. Kahl, Computational complexity and feasibility of data processing and interval computations, Kluwer, Dordrecht, 1997. Google Scholar
  6. 6.
    S. P. Shary, “Parameter partitioning scheme for interval linear systems with constraints”, Proceedings of the International Workshop on Interval Mathematics and Constraint Propagation Methods (ICMP’03), July 8–9, 2003, Novosibirsk, Akademgorodok, Russia, pp. 1–12 (in Russian). Google Scholar
  7. 7.
    S. P. Shary, “Solving tied interval linear systems”, Siberian Journal of Numerical Mathematics, 2004, Vol. 7, No. 4, pp. 363–376 (in Russian). zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.University of Texas at El PasoEl PasoUSA

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