Reliable Computing

, Volume 11, Issue 3, pp 207–233 | Cite as

Exact Bounds on Finite Populations of Interval Data

  • Scott Ferson
  • Lev Ginzburg
  • Vladik Kreinovich
  • Luc Longpré
  • Monica Aviles
Article

Abstract

In this paper, we start research into using intervals to bound the impact of bounded measurement errors on the computation of bounds on finite population parameters (“descriptive statistics”). Specifically, we provide a feasible (quadratic time) algorithm for computing the lower bound \(\underline{\sigma^2}\) on the finite population variance function of interval data. We prove that the problem of computing the upper bound \(\bar{\sigma}^2\) is, in general, NP-hard. We provide a feasible algorithm that computes \(\bar{\sigma}^2\) under reasonable easily verifiable conditions, and provide preliminary results on computing other functions of finite populations.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Cormen, Th. H., Leiserson, C. E., Rivest, R. L., and Stein, C.: Introduction to Algorithms, MIT Press, Cambridge, and Mc-Graw Hill Co., N. Y., 2001.Google Scholar
  2. 2.
    Ferson, S.: RAMAS Risk Calc: Risk Assessment with Uncertain Numbers, Applied Biomathematics, Setauket, 1999.Google Scholar
  3. 3.
    Ferson, S., Ginzburg, L., Kreinovich, V., Longpré, L., and Aviles, M.: Computing Variance for Interval Data Is NP-Hard, ACM SIGACT News 33 (2)(2002), pp. 108–118.CrossRefGoogle Scholar
  4. 4.
    Fuller, W. A.: Measurement Error Models, J. Wiley & Sons, New York, 1987.Google Scholar
  5. 5.
    Garey, M. R. and Johnson, S. D.: Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H.Freeman and Company, New York, 1979.Google Scholar
  6. 6.
    Hansen, E.: Sharpness in Interval Computations, Reliable Computing 3 (1)(1997), pp. 17–29.CrossRefGoogle Scholar
  7. 7.
    Kearfott, R. B.: Rigorous Global Search: Continuous Problems, Kluwer Academic Publishers, Dordrecht, 1996.Google Scholar
  8. 8.
    Kreinovich, V., Lakeyev, A., Rohn, J., and Kahl, P.: Computational Complexity and Feasibility of Data Processing and Interval Computations, Kluwer Academic Publishers, Dordrecht, 1997.Google Scholar
  9. 9.
    Kuznetsov, V. P.: Interval Statistical Models, Radio i Svyaz Publ., Moscow, 1991 (in Russian).Google Scholar
  10. 10.
    Nivlet, P., Fournier, F., and Royer, J.: A New Methodology to Account for Uncertainties in 4-D Seismic Interpretation, in: Proceedings of the 71st Annual International Meeting of the Society of Exploratory Geophysics SEG’2001, San Antonio, Texas, September 9–14, 2001, pp. 1644–1647.Google Scholar
  11. 11.
    Nivlet, P., Fournier, F., and Royer, J.: Propagating Interval Uncertainties in Supervised Pattern Recognition for Reservoir Characterization, in: Proceedings of the 2001 Society of Petroleum Engineers Annual Conference SPE’2001, New Orleans, Louisiana, September 30–October 3, 2001, paper SPE–71327.Google Scholar
  12. 12.
    Osegueda, R., Kreinovich, V., Potluri, L., and Aló, R.: Non-Destructive Testing of Aerospace Structures: Granularity and Data Mining Approach, in: Proceedings of FUZZ–IEEE’2002, Honolulu, Hawaii, May 12–17, 2002, pp. 685–689.Google Scholar
  13. 13.
    Papadimitriou, C. H.: Computational Complexity, Addison Wesley, San Diego, 1994.Google Scholar
  14. 14.
    Rabinovich, S.: Measurement Errors: Theory and Practice, American Institute of Physics, New York, 1993.Google Scholar
  15. 15.
    Vavasis, S. A.: Nonlinear Optimization: Complexity Issues, Oxford University Press, N. Y., 1991.Google Scholar
  16. 16.
    Wadsworth, Jr., H. M. (ed.): Handbook of Statistical Methods for Engineers and Scientists, McGraw-Hill Publishing Co., New York, 1990.Google Scholar
  17. 17.
    Walley, P.: Statistical Reasoning with Imprecise Probabilities, Chapman and Hall, N. Y., 1991.Google Scholar
  18. 18.
    Walster, G. W.: Philosophy and Practicalities of Interval Arithmetic, in: Moore, R. E.(ed.), Reliability in Computing, 1988, pp. 307–323.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  • Scott Ferson
    • 1
  • Lev Ginzburg
    • 1
  • Vladik Kreinovich
    • 1
  • Luc Longpré
    • 2
  • Monica Aviles
    • 2
  1. 1.Applied BiomathematicsSetauketUSA
  2. 2.Computer Science DepartmentUniversity of Texas at El PasoEl PasoUSA

Personalised recommendations