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Statistical Data Processing under Interval Uncertainty: Algorithms and Computational Complexity

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Book cover Soft Methods for Integrated Uncertainty Modelling

Part of the book series: Advances in Soft Computing ((AINSC,volume 37))

Abstract

Why indirect measurements? In many real-life situations, we are interested in the value of a physical quantity y that is difficult or impossible to measure directly. Examples of such quantities are the distance to a star and the amount of oil in a given well. Since we cannot measure y directly, a natural idea is to measure yindirectly. Specifically, we find some easier-to-measure quantities x1,…,x n which are related to y by a known relation y = f (x1,…,x n ); this relation may be a simple functional transformation, or complex algorithm (e.g., for the amount of oil, numerical solution to an inverse problem).

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References

  1. J. Abellan and S. Moral, Range of entropy for credal sets, In: M. López-Diaz et al. (eds.), Soft Methodology and Random Information Systems, Springer, Berlin and Heidelberg, 2004, pp. 157–164.

    Google Scholar 

  2. J. Abellan and S. Moral, Difference of entropies as a nonspecificity function on credal sets, Intern. J. of General Systems, 2005, Vol. 34, No. 3, pp. 201–214.

    Article  MATH  MathSciNet  Google Scholar 

  3. J. Abellan and S. Moral, An algorithm that attains the maximum of entropy for order-2 capacities, Intern. J. of Uncertainty, Fuzziness, and Knowledge-Based Systems (to appear).

    Google Scholar 

  4. R. Aló, M. Beheshti, and G. Xiang, Computing Variance under Interval Uncertainty: A New Algorithm and Its Potential Application to Privacy in Statistical Databases, Proceedings of the International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems IPMU’06, Paris, France, July 2–7, 2006 (to appear).

    Google Scholar 

  5. J. B. Beck, V. Kreinovich, and B. Wu, Interval-Valued and Fuzzy-Valued Random Variables: From Computing Sample Variances to Computing Sample Covariances, In: M. Lopez, M. A. Gil, P. Grzegorzewski, O. Hrynewicz, and J. Lawry (eds.), Soft Methodology and Random Information Systems, Springer- Verlag, 2004, pp. 85–92.

    Google Scholar 

  6. D. Berleant, M.-P. Cheong, C. Chu, Y. Guan, A. Kamal, G. Sheblé, S. Ferson, and J. F. Peters, Dependable handling of uncertainty, Reliable Computing 9(6) (2003), pp. 407–418.

    Article  MATH  Google Scholar 

  7. D. Berleant, L. Xie, and J. Zhang, Statool: a tool for Distribution Envelope Determination (DEnv), an interval-based algorithm for arithmetic on random variables, Reliable Computing 9(2) (2003), pp. 91–108.

    Article  MATH  Google Scholar 

  8. D. Berleant and J. Zhang, Using Pearson correlation to improve envelopes around the distributions of functions, Reliable Computing, 10(2) (2004), pp. 139–161.

    Article  MATH  MathSciNet  Google Scholar 

  9. D. Berleant and J. Zhang, Representation and Problem Solving with the Distribution Envelope Determination (DEnv) Method, Reliability Engineering and System Safety, 85 (1–3) (July-Sept. 2004).

    Google Scholar 

  10. D. Berleant and J. Zhang, Using Pearson correlation to improve envelopes around the distributions of functions, Reliable Computing, 10(2) (2004), pp. 139–161.

    Article  MATH  MathSciNet  Google Scholar 

  11. E. Dantsin, V. Kreinovich, A. Wolpert, and G. Xiang, Population Variance under Interval Uncertainty: A New Algorithm, Reliable Computing, 2006, Vol. 12, No. 4, pp. 273–280.

    Article  MATH  MathSciNet  Google Scholar 

  12. E. Dantsin, A. Wolpert, M. Ceberio, G. Xiang, and V. Kreinovich, Detecting Outliers under Interval Uncertainty: A New Algorithm Based on Constraint Satisfaction, Proceedings of the International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems IPMU’06, Paris, France, July 2–7, 2006 (to appear).

    Google Scholar 

  13. S. Ferson, L. Ginzburg, and R. Akcakaya, Whereof One Cannot Speak: When Input Distributions Are Unknown, Applied Biomathematics Report, 2001.

    Google Scholar 

  14. S. Ferson, J. Hajagos, D. Berleant, J. Zhang, W. T. Tucker, L. Ginzburg, and W. Oberkampf, Dependence in Dempster-Shafer Theory and Probability Bounds Analysis, Technical Report SAND2004–3072, Sandia National Laboratory, 2004.

    Google Scholar 

  15. S. Ferson, V. Kreinovich, L. Ginzburg, D. S. Myers, and K. Sentz, Constructing Probability Boxes and Dempster-Shafer Structures, Sandia National Laboratories, Report SAND2002–4015, January 2003.

    Google Scholar 

  16. S. Ferson, L. Ginzburg, V. Kreinovich, L. Longpre, and M. Aviles, Exact Bounds on Finite Populations of Interval Data, Reliable Computing, 2005, Vol. 11, No. 3, pp. 207–233.

    Article  MATH  MathSciNet  Google Scholar 

  17. 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.

    Article  Google Scholar 

  18. L. Granvilliers, V. Kreinovich, and N. Mueller, Novel Approaches to Numerical Software with Result Verification, In: R. Alt, A. Frommer, R. B. Kearfott, and W. Luther (eds.), Numerical Software with Result Verification, International Dagstuhl Seminar, Dagstuhl Castle, Germany, January 19–24, 2003, Revised Papers, Springer Lectures Notes in Computer Science, 2004, Vol. 2991, pp. 274–305.

    Google Scholar 

  19. L. Jaulin, M. Kieffer, O. Didrit, and E. Walter, Applied interval analysis: with examples in parameter and state estimation, robust control and robotics, Springer Verlag, London, 2001.

    MATH  Google Scholar 

  20. G. J. Klir, Uncertainty and Information: Foundations of Generalized Information Theory, J. Wiley, Hoboken, New Jersey, 2005.

    Google Scholar 

  21. V. Kreinovich, Maximum entropy and interval computations, Reliable Computing, 1996, Vol. 2, No. 1, pp. 63–79.

    Article  MATH  MathSciNet  Google Scholar 

  22. V. Kreinovich, Probabilities, Intervals, What Next? Optimization Problems Related to Extension of Interval Computations to Situations with Partial Information about Probabilities, Journal of Global Optimization, 2004, Vol. 29, No. 3, pp. 265–280.

    Article  MATH  MathSciNet  Google Scholar 

  23. V. Kreinovich and S. Ferson, Computing Best-Possible Bounds for the Distribution of a Sum of Several Variables is NP-Hard, International Journal of Approximate Reasoning, 2006, Vol. 41, pp. 331–342.

    Article  MATH  MathSciNet  Google Scholar 

  24. V. Kreinovich, S. Ferson, and L. Ginzburg, Exact Upper Bound on the Mean of the Product of Many Random Variables With Known Expectations, Reliable Computing (to appear).

    Google Scholar 

  25. V. Kreinovich, A. Lakeyev, J. Rohn, P. Kahl, Computational complexity and feasibility of data processing and interval computations, Kluwer, Dordrecht, 1997.

    Google Scholar 

  26. V. Kreinovich and L. Longpre, Fast Quantum Algorithms for Handling Probabilistic and Interval Uncertainty, Mathematical Logic Quarterly, 2004, Vol. 50, No. 4/5, pp. 507–518.

    MathSciNet  Google Scholar 

  27. V. Kreinovich, L. Longpre, P. Patangay, S. Ferson, and L. Ginzburg, Outlier Detection Under Interval Uncertainty: Algorithmic Solvability and Computational Complexity, Reliable Computing, 2005, Vol. 11, No. 1, pp. 59–75.

    Article  MATH  MathSciNet  Google Scholar 

  28. V. Kreinovich, L. Longpré, S. A. Starks, G. Xiang, J. Beck, R. Kandathi, A. Nayak, S. Ferson, and J. Hajagos, ‘Interval Versions of Statistical Techniques, with Applications to Environmental Analysis, Bioinformatics, and Privacy in Statistical Databases’, Journal of Computational and Applied Mathematics (to appear).

    Google Scholar 

  29. V. Kreinovich, H. T. Nguyen, and B. Wu, On-Line Algorithms for Computing Mean and Variance of Interval Data, and Their Use in Intelligent Systems, Information Sciences (to appear).

    Google Scholar 

  30. V. Kreinovich, G. N. Solopchenko, S. Ferson, L. Ginzburg, and R. Alo, Probabilities, intervals, what next? Extension of interval computations to situations with partial information about probabilities, Proceedings of the 10th IMEKO TC7 International Symposium on Advances of Measurement Science, St. Petersburg, Russia, June 30-July 2, 2004, Vol. 1, pp. 137–142.

    Google Scholar 

  31. V. Kreinovich, G. Xiang, and S. Ferson, How the Concept of Information as Average Number of ‘Yes-No‘ Questions (Bits) Can Be Extended to Intervals, P-Boxes, and more General Uncertainty, Proceedings of the 24th International Conference of the North American Fuzzy Information Processing Society NAFIPS’2005, Ann Arbor, Michigan, June 22–25, 2005, pp. 80–85.

    Google Scholar 

  32. V. Kuznetsov, Interval Statistical Models (in Russian), Radio i Svyaz, Moscow, 1991.

    Google Scholar 

  33. W. A. Lodwick and K. D. Jamison, Estimating and validating the cumulative distribution of a function of random variables: toward the development of distribution arithmetic, Reliable Computing, 2003, Vol. 9, No. 2, pp. 127–141.

    Article  MATH  MathSciNet  Google Scholar 

  34. W. A. Lodwick, A. Neumaier, and F. Newman, Optimization under uncertainity: methods and applications in radiation therapy, Proc. 10th IEEE Int. Conf. Fuzzy Systems, December 2–5, 2001, Melbourne, Australia.

    Google Scholar 

  35. C. Manski, Partial Identification of Probability Distributions, Springer-Verlag, New York, 2003.

    MATH  Google Scholar 

  36. A. S. Moore, Interval risk analysis of real estate investment: a non-Monte-Carlo approach, Freiburger Intervall-Berichte 85/3, Inst. F. Angew. Math., Universitaet Freiburg I. Br., 23–49 (1985).

    Google Scholar 

  37. R. E. Moore, Automatic error analysis in digital computation, Technical Report Space Div. Report LMSD84821, Lockheed Missiles and Space Co., 1959.

    Google Scholar 

  38. R. E. Moore, Risk analysis without Monte Carlo methods, Freiburger Intervall- Berichte 84/1, Inst. F. Angew. Math., Universitaet Freiburg I. Br., 1–48 (1984).

    Google Scholar 

  39. A. Neumaier, Fuzzy modeling in terms of surprise, Fuzzy Sets and Systems 135, 2003, 21–38.

    Article  MATH  MathSciNet  Google Scholar 

  40. H. T. Nguyen, V. Kreinovich, and G. Xiang, Foundations of Statistical Processing of Set-Valued Data: Towards Efficient Algorithms, Proceedings of the Fifth International Conference on Intelligent Technologies InTech’04, Houston, Texas, December 2–4, 2004.

    Google Scholar 

  41. M. Orshansky, W.-S. Wang, M. Ceberio, and G. Xiang, Interval-Based Robust Statistical Techniques for Non-Negative Convex Functions, with Application to Timing Analysis of Computer Chips, Proceedings of the Symposium on Applied Computing SAC’06, Dijon, France, April 23–27, 2006, pp. 1645–1649.

    Google Scholar 

  42. M. Orshansky, W.-S. Wang, G. Xiang, and V. Kreinovich, Interval-Based Robust Statistical Techniques for Non-Negative Convex Functions, with Application to Timing Analysis of Computer Chips, Proceedings of the Second International Workshop on Reliable Engineering Computing, Savannah, Georgia, February 22–24, 2006, pp. 197–212.

    Google Scholar 

  43. H. Regan, S. Ferson and D. Berleant, Equivalence of methods for uncertainty propagation of real-valued random variables, International Journal of Approximate Reasoning, in press.

    Google Scholar 

  44. H.-P. Schröcker and J. Wallner, Geometric constructions with discretized random variables, Reliable Computing, 2006, Vol. 12, No. 3, pp. 203–223.

    Article  MATH  MathSciNet  Google Scholar 

  45. S. A. Starks, V. Kreinovich, L. Longpre, M. Ceberio, G. Xiang, R. Araiza, J. Beck, R. Kandathi, A. Nayak, and R. Torres, Towards combining probabilistic and interval uncertainty in engineering calculations, Proceedings of the Workshop on Reliable Engineering Computing, Savannah, Georgia, September 15- 17, 2004, pp. 193–213.

    Google Scholar 

  46. W. T. Tucker and S. Ferson, Probability Bounds Analysis in Environmental Risk Assessments, Applied Biomathematics Report.

    Google Scholar 

  47. P.Walley, Statistical Reasoning with Imprecise Probabilities, Chapman & Hall, N.Y., 1991.

    MATH  Google Scholar 

  48. B. Wu, H. T. Nguyen, and V. Kreinovich, Real-Time Algorithms for Statistical Analysis of Interval Data, Proceedings of the International Conference on Information Technology InTech’03, Chiang Mai, Thailand, December 17–19, 2003, pp. 483–490.

    Google Scholar 

  49. G. Xiang, Fast algorithm for computing the upper endpoint of sample variance for interval data: case of sufficiently accurate measurements, Reliable Computing, 2006, Vol. 12, No. 1, pp. 59–64.

    Article  MATH  MathSciNet  Google Scholar 

  50. G. Xiang, O. Kosheleva, and G. J. Klir, Estimating Information Amount under Interval Uncertainty: Algorithmic Solvability and Computational Complexity, Proceedings of the International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems IPMU’06, Paris, France, July 2–7, 2006 (to appear).

    Google Scholar 

  51. G. Xiang, S. A. Starks, V. Kreinovich, and L. Longpre, New Algorithms for Statistical Analysis of Interval Data, Proceedings of the Workshop on Stateof- the-Art in Scientific Computing PARA’04, Lyngby, Denmark, June 20–23, 2004, Vol. 1, pp. 123–129.

    Google Scholar 

  52. R. R. Yager and V. Kreinovich, Decision Making Under Interval Probabilities, International Journal of Approximate Reasoning, 1999, Vol. 22, No. 3, pp. 195–215.

    Article  MATH  MathSciNet  Google Scholar 

  53. J. Zhang and D. Berleant, Envelopes around cumulative distribution functions from interval parameters of standard continuous distributions, Proceedings, North American Fuzzy Information Processing Society (NAFIPS 2003), Chicago, pp. 407–412.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Texas at El Paso, 79968, El Paso, TX, USA

    Vladik Kreinovich

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  1. Vladik Kreinovich
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Editor information

Editors and Affiliations

  1. AI Group Department of Engineering Mathematics, University of Bristol, BS8 1TR, Bristol, UK

    Jonathan Lawry

  2. Statistics and Operations Research, Rey Juan Carlos University, C-Tulip˙n s/n MÛstoles28933, Spain

    Enrique Miranda

  3. Intelligent Systems Group Department of Electronics & Computer Science, University of Santiago de Compostela Santiago de Compostela, Spain

    Alberto Bugarin

  4. Department of Applied Mathematics, Beijing University of Technology, 100022, Beijing, P.R. China

    Shoumei Li

  5. Dpto. Estadistica e I.O y D.M. Calle Calvo Sotelo s/n, Universiad de Oviedo Fac. Ciencias, 33071, Oviedo, Spain

    Maria Angeles Gil

  6. Systems Research Institute Polish Academy of Sciences, Newelska 6, 01-447, Warsaw, Poland

    Przemys aw Grzegorzewski

  7. Systems Research Institute Polish Academy of Sciences, Newelska 6, 01-447, Warsaw, Poland

    Olgierd Hyrniewicz

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© 2006 Springer

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Kreinovich, V. (2006). Statistical Data Processing under Interval Uncertainty: Algorithms and Computational Complexity. In: Lawry, J., et al. Soft Methods for Integrated Uncertainty Modelling. Advances in Soft Computing, vol 37. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-34777-1_4

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  • DOI: https://doi.org/10.1007/3-540-34777-1_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-34776-7

  • Online ISBN: 978-3-540-34777-4

  • eBook Packages: EngineeringEngineering (R0)

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