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Towards Combining Probabilistic and Interval Uncertainty in Engineering Calculations: Algorithms for Computing Statistics under Interval Uncertainty, and Their Computational Complexity

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Reliable Computing

Abstract

In many engineering applications, we have to combine probabilistic and interval uncertainty. For example, in environmental analysis, we observe a pollution level x(t) in a lake at different moments of time t, and we would like to estimate standard statistical characteristics such as mean, variance, autocorrelation, correlation with other measurements. In environmental measurements, we often only measure the values with interval uncertainty. We must therefore modify the existing statistical algorithms to process such interval data.

In this paper, we provide a survey of algorithms for computing various statistics under interval uncertainty and their computational complexity. The survey includes both known and new algorithms.

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References

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

  2. Berleant, D.: Automatically Verified Arithmetic on Probability Distributions and Intervals, in: Kearfott, R. B. and Kreinovich, V (eds), Applications of Interval Computations,Kluwer Aca demic Publishers, Dordrecht, 1996.

  3. Berleant, D.: Automatically Verified Arithmetic with Both Intervals and Probability Density Functions, Interval Computations2 (1993), pp. 48–70.

    MATH  MathSciNet  Google Scholar 

  4. Berleant, D. and Goodman-Strauss, C.: Bounding the Results of Arithmetic Operations on Random Variables of Unknown Dependency Using Intervals, Reliable Computing 4(2) (1998), pp. 147–165.

    Article  MATH  MathSciNet  Google Scholar 

  5. Berleant, D., Xie, L., and Zhang, J.: Statool: A Tool for Distribution Envelope Determination (DEnv), an Interval-Based Algorithm for Arithmetic on Random Variables, Reliable Computing9 (2) (2003), pp. 91–108.

    Article  MATH  Google Scholar 

  6. Gormen, Th. H., Leiserson, C. E., Rivest, R. L., and Stein, C.: Introduction to Algorithms,MIT Press, Cambridge, 2001.

  7. Person, S.: RAMAS Risk Calc 4.0: Risk Assessment with Uncertain Numbers,CRC Press, Boca Raton, 2002.

  8. Person, S., Ginzburg, L., Kreinovich, V, and Aviles, M.: Exact Bounds on Sample Variance of Interval Data, in: Extended Abstracts of the 2002 SIAM Workshop on Validated Computing,Toronto, 2002, pp. 67–69.

  9. Person, S., Ginzburg, L., Kreinovich, V., Longpre, L., and Aviles, M.: Computing Variance for Interval Data Is NP-Hard, ACM SIGACT News33 (2) (2002), pp. 108–118.

    Article  Google Scholar 

  10. Person, S., Ginzburg, L., Kreinovich, V., Longpre, L., and Aviles, M.: Exact Bounds on Finite Populations of Interval Data, Reliable Computing11 (3) (2005), pp. 207–233.

    Article  MathSciNet  Google Scholar 

  11. Person, S., Myers, D., and Berleant, D.: Distribution-Free Risk Analysis: I. Range, Mean, and Variance,Technical Report, Applied Biomathematics, 2001.

  12. Garey, M. E. and Johnson, D. S.: Computers and Intractability: A Guide to the Theory of NF’-Completeness,Freeman, San Francisco, 1979.

  13. Garling, D. J. H.: A Book Review, American Mathematical Monthly112 (6) (2005), pp. 575–579.

    Article  Google Scholar 

  14. Granvilliers, L., Kreinovich, V., and Miiller, N.: Novel Approaches to Numerical Software with Result Verification, in: Alt, R., Frommer, A., Kearfott, R. B., and Luther, W. (eds), Numerical Software with Result Verification,(International Dagstuhl Seminar, Dagstuhl Cas tle, Germany, January 19–24, 2003), Springer Lectures Notes in Computer Science2991, 2004, pp. 274–305.

  15. Hardy, G. H., Littlewood, J. E., and Polya, G.: Inequalities,Cambridge University Press, 1988.

  16. Huber, P. J.: Robust Statistics,Wiley, New York, 2004.

  17. Jaja, J.: An Introduction to Parallel Algorithms,Addison-Wesley, Reading, 1992.

  18. Kreinovich, V.: Probabilities, Intervals, WhatNext? Optimization Problems Related to Extension of Interval Computations to Situations with Partial Information about Probabilities, Journal of Global Optimization29 (3) (2004), pp. 265–280.

    Article  MATH  MathSciNet  Google Scholar 

  19. Kreinovich, V, Lakeyev, A., Rohn, J., and Kahl, P.: Computational Complexity and Feasibil ity of Data Processing and Interval Computations,Kluwer Academic Publishers, Dordrecht, 1997.

  20. Kreinovich, V and Longpre, L.: Computational Complexity and Feasibility of Data Processing and Interval Computations, with Extension to Cases When We Have Partial Information about Probabilities, in: Brattka, V, Schroeder, M., Weihrauch, K., and Zhong, N. (eds), Proc. Conf. on Computability and Complexity in Analysis CCA’2003,Cincinnati, Ohio, USA, August 28–30, 2003, pp. 19–54.

  21. Kreinovich, V and Longpre, L.: Fast Quantum Algorithms for Handling Probabilistic and Interval Uncertainty, Mathematical Logic Quarterly50 (4/5) (2004), pp. 507–518.

    MathSciNet  Google Scholar 

  22. Kreinovich, V, Longpre, L., Person, S., and Ginzburg, L.: Computing Higher Central Moments for Interval Data,University of Texas at El Paso, Department of Computer Science, Technical Report UTEP-CS-03–14b, 2004, http://www.cs.utep.edu/vladik/2003/tr03–14b.pdf Kreinovich, V, Longpre, L., Patangay, P., Person, S., and Ginzburg, L.: Outlier Detection under Interval Uncertainty: Algorithmic Solvability and Computational Complexity, in: Lirkov, L, Margenov, S., Wasniewski, J., and Yalamov, P. (eds), Large-Scale Scientific Computing,Pro ceedings of the 4-th International Conference LSSC’2003, Sozopol, Bulgaria, June 4–8, 2003, Springer Lecture Notes in Computer Science2907, 2004, pp. 238–245.

  23. Kreinovich, V, Longpre, L., Patangay, P., Person, S., and Ginzburg, L.: Outlier Detection under Interval Uncertainty: Algorithmic Solvability and Computational Complexity, Reliable Computing11 (1) (2005), pp. 59–76.

    Article  MATH  MathSciNet  Google Scholar 

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

  25. Kreinovich, V, Patangay, P., Longpre, L., Starks, S. A., Campos, C., Person, S., and Ginzburg, L.: Outlier Detection Under Interval and Fuzzy Uncertainty: Algorithmic Solvability and Compu tational Complexity, in: Proceedings of the 22nd International Conference of the North Amer ican Fuzzy Information Processing Society NAFIPS’2003,Chicago, Illinois, July 24–26, 2003, pp. 401 106.

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

  27. Lodwick, W. A. and Jamison, K. D.: Estimating and Validating the Cumulative Distribution of a Function of Random Variables: Toward the Development of Distribution Arithmetic, Reliable Computing 9(2) (2003), pp. 127–141.

    Article  MATH  MathSciNet  Google Scholar 

  28. Martinez, M., Longpre, L., Kreinovich, V., Starks, S. A., and Nguyen, H. T.: Fast Quantum Algorithms for Handling Probabilistic, Interval, and Fuzzy Uncertainty, in: Proceedings of the 22nd International Conference of the North American Fuzzy Information Processing Society NAFIPS’2003,Chicago, Illinois, July 24–26, 2003, pp. 395 00.

  29. Moore, R. E. and Lodwick, W. A.: Interval Analysis and Fuzzy Set Theory, Fuzzy Sets and Systems135 (1) (2003), pp. 5–9.

    Article  MATH  MathSciNet  Google Scholar 

  30. Morgenstein, D. and Kreinovich, V.: Which Algorithms Are Feasible and Which Are Not Depends on the Geometry of Space-Time, Geombinatorics4 (3) (1995), pp. 80–97.

    MATH  Google Scholar 

  31. Nguyen, H. T. and Kreinovich, V.: Nested Intervals and Sets: Concepts, Relations to Fuzzy Sets, and Applications, in: Kearfott, R. B. and Kreinovich, V. (eds), Applications of Interval Computations,Kluwer Academic Publishers, Dordrecht, 1996, pp. 245–290.

  32. Nguyen, H. T. and Walker, E. A.: First Course in Fuzzy Logic,CRC Press, Boca Raton, 1999.

  33. Nguyen, H. T, Wang, T., and Kreinovich, V., Towards Foundations of Processing Imprecise Data: From Traditional Statistical Techniques of Processing Crisp Data to Statistical Processing of Fuzzy Data, in: Liu, Y., Chen, G., Ying, M., and Cai, K.-Y (eds), Proceedings of the International Conference on Fuzzy Information Processing: Theories and Applications FIP’2003,Beijing, China, March 1 , 2003, Vol. II, pp. 895–900.

  34. Nguyen, H. T., Wu, B., and Kreinovich, V.: Shadows of Fuzzy Sets—A Natural Approach Towards Describing 2-D and Multi-D Fuzzy Uncertainty in Linguistic Terms, in: Proc. 9th IEEE Int’l Conference on Fuzzy Systems FUZZ-IEEE’2000,San Antonio, Texas, May 7–10, 2000, Vol. 1, pp. 340–345.

  35. Nivlet, P., Fournier, F., and Royer, J.: A New Methodology to Account for Uncertainties in 4-D Seismic Interpretation, in: Proc. 71st Annual Int’l Meeting ofSoc. of Exploratory Geophysics SEG’2001,San Antonio, TX, September 9–14, 2001, pp. 1644–1647.

  36. Nivlet, P., Fournier, F., and Royer, J.: Propagating Interval Uncertainties in Supervised Pat tern Recognition for Reservoir Characterization, in: Proc. 2001 Society of Petroleum Engineers Annual Conf. SPE’2001,New Orleans, LA, September 30-October 3, 2001, paper SPE-71327.

  37. Osegueda, R., Kreinovich, V, Potluri, L., A16, R.: Non-Destructive Testing of Aerospace Struc tures: Granularity and Data Mining Approach, in: Proc. FUZZ-IEEE’2002,Honolulu, HI, May 12–17, 2002, Vol. 1, pp. 685–689.

  38. Rabinovich, S.: Measurement Errors: Theory and Practice,American Institute of Physics, New York, 1993.

  39. Regan, H., Person, S., and Berleant, D.: Equivalence of Five Methods for Bounding Uncertainty, Journal of Approximate Reasoning36 (1) (2004), pp. 1–30.

    Article  MATH  MathSciNet  Google Scholar 

  40. Rowe, N. C.: Absolute Bounds on the Mean and Standard Deviation of Transformed Data for Constant-Sign-Derivative Transformations, SIAM Journal of Scientific Statistical Computing 9(1988), pp. 1098–1113.

    Article  MATH  MathSciNet  Google Scholar 

  41. Shmulevich, I. and Zhang, W: Binary Analysis and Optimization-Based Normalization of Gene Expression Data, Bioinformatics 18(4) (2002), pp. 555–565.

    Article  Google Scholar 

  42. Starks, S. A., Kreinovich, V, Longpre, L., Ceberio, M., Xiang, G., Araiza, R., Beck, J., Kan- dathi, R., Nayak, A., and Torres, R.: Towards Combining Probabilistic and Interval Uncertainty in Engineering Calculations, in: Proceedings of the Workshop on Reliable Engineering Computing,Savannah, Georgia, September 15–17, 2004, pp. 193–213.

  43. Wadsworth, H. M., Jr. (ed.): Handbook of Statistical Methods for Engineers and Scientists,McGraw-Hill Publishing, New York, 1990.

  44. Walley, P.: Statistical Reasoning with Imprecise Probabilities,Chapman & Hall, New York, 1991.

  45. Walster, G. W: Philosophy and Practicalities of Interval Arithmetic, in: Reliability in Computing,Academic Press, New York, 1988, pp. 309–323.

  46. Walster, G. W. and Kreinovich, V: For Unknown-but-Bounded Errors, Interval Estimates Are Often Better Than Averaging, ACM SIGNUMNewsletter 31(2) (1996), pp. 6–19.

    Google Scholar 

  47. Williamson, R. and Downs, T.: Probabilistic Arithmetic I: Numerical Methods for Calculat ing Convolutions and Dependency Bounds, International Journal of Approximate Reasoning 4(1990), pp. 89–158.

    Article  MATH  MathSciNet  Google Scholar 

  48. Wu, B., Nguyen, H. T., and Kreinovich, V.: Real-Time Algorithms for Statistical Analysis of Interval Data, in: Proceedings of the International Conference on Information Technology InTech’03,Chiang Mai, Thailand, December 17–19, 2003, pp. 483 90.

  49. Xiang, G.: Fast Algorithm for Computing the Upper Endpoint of Sample Variance for Inter val Data: Case of Sufficiently Accurate Measurements, Reliable Computing12 (1) (2006), pp. 59–64.

    Article  MATH  MathSciNet  Google Scholar 

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

  51. Zhang, W., Shmulevich, L, and Astola, J.: Microarray Quality Control,Wiley, Hoboken, 2004.

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Authors and Affiliations

  1. NASA Pan-American Center for Earth and Environmental Studies (PACES), University of Texas, El Paso, TX, 79968, USA

    Vladik Kreinovich, Gang Xiang, Scott A. Starks, Luc Longpré, Martine Ceberio, Roberto Araiza, Jan Beck, Raj Kandathi, Asis Nayak & Roberto Torres

  2. Applied Biomathematics, 100 North Country Road, Setauket, NY, 11733, USA

    Janos G. Hajagos

  3. Dept. of Ecology and Evolution, State University of New York, Stony Brook, NY, 11794, USA

    Janos G. Hajagos

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  1. Vladik Kreinovich
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  2. Gang Xiang
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  3. Scott A. Starks
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Correspondence to Vladik Kreinovich.

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Kreinovich, V., Xiang, G., Starks, S.A. et al. Towards Combining Probabilistic and Interval Uncertainty in Engineering Calculations: Algorithms for Computing Statistics under Interval Uncertainty, and Their Computational Complexity. Reliable Comput 12, 471–501 (2006). https://doi.org/10.1007/s11155-006-9015-4

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