Computer arithmetic for probability distribution variables

Reliability Engineering and System Safety

Volumn 85, Issue 1-3, 2004, Pages 191-209

  Li, Weiye ^a   Hyman, James Mac ^a  

^a LOS ALAMOS NATIONAL LABORATORY   (United States)

Author keywords

Dependency bound; Dependency tracking; Epistemic uncertainty; Interval arithmetic; Interval computation; Monte Carlo; PDV arithmetic; Probabilistic arithmetic; Probabilistic discretization; Probability bound; Probability box; Uncertainty computation

Indexed keywords

COMPUTER PROGRAM LISTINGS; CORRELATION METHODS; MATHEMATICAL MODELS; MATHEMATICAL TECHNIQUES; MONTE CARLO METHODS; RELIABILITY;

EPISTEMIC UNCERTAINTY; INTERVAL ARITHMETIC; INTERVAL COMPUTATION; MONTE CARLO; PDV ARITHMETIC; PROBABILISTIC ARITHMETIC; PROBABILISTIC DISCRETIZATION; PROBABILITY BOUND; PROBABILITY BOX; UNCERTAINTY COMPUTATION;

PROBABILITY DISTRIBUTIONS;

EID: 2642543293     PISSN: 09518320     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.ress.2004.03.012     Document Type: Conference Paper

References (47)

1. Alefeld G., Herzberger J. Introduction to interval computations. 1983;Academic Press, New York.
2. Berleant D. Automatically verified reasoning with both intervals and probability density functions. Interval Comput. 2:1993;48-70.
3. Berleant D., Cheng H. A software tool for automatically verified operations on intervals and probability distributions. Reliab Comput. 4:(1):1998;71-82.
4. Berleant D., Goodman-Strauss C. Bounding the results of arithmetic operations on random variables of unknown dependency using intervals. Reliab Comput. 4:(2):1998;147-165.
5. Berleant D., Xie L., Zhang J. Statool: a tool for distribution envelope determination (DEnv), an interval-based algorithm for arithmetic on random variables. Reliab Comput. 9:(2):2003;91-108.
6. Berleant D, Zhang J. Derivation of the correlation constraint, manuscript. http://class.ee.iastate.edu/berleanthome/Research/Pdfs/PSERC/index. htm .
7. Burkill J.C. Functions of intervals. Proc Lond Math Soc. 22:1924;375-446.
8. Colomba A.G., Jaarsma R.J. A powerful numerical method to combine random variables. IEEE Trans Reliab. R-29:(2):1980;126-129.
9. Cossette H., Denuit M., Marceau É. Distributional bounds for functions of dependent risks. Bull Swiss Assoc Actuaries. 2002;45-65.
10. Dempster A.P. Upper and lower probability induced by a multivalued mapping. Ann Math Stat. 38:(2):1967;325-339.
11. Dempster A.P. Upper and lower probability inferences based on a sample from a finite univariate population. Biometrika. 54:(3-4):1967;515-528.
12. Feller W. An introduction to probability theory and its applications. New York: Wiley; 1957-1971.
13. Ferson S. What Monte Carlo methods cannot do. J Hum Ecol Risk Assess. 2:(4):1996;990-1007.
14. Ferson S. RAMAS Risk Calc 4.0: risk assessment with uncertain numbers. 2002;Lewis Press, Boca Raton, FL.
15. Ferson S, Hajagos J. Don't open that envelope: solutions to the Sandia problems using probability boxes. Poster, presented at Sandia, 2002 [40], http://www.sandia.gov/epistemic/Papers/ferson.pdf .
16. Frank M.J., Nelson R.B., Schweizer B. Best possible bounds for the distribution of a sum-a problem of Kolmogorov. Probab Theor Relat Fields. 74:1987;199-211.
17. Gunst R.F., Mason R.L. Regression analysis and its application: a data-oriented approach. 1980;Marcel Dekker, New York.
18. Hyman JM. FORSIG: an extension of FORTRAN with significance arithmetic. Los Alamos National Laboratory Report, LA-9448-MS; 1982.
19. Ingram G.E., Welker E.L., Herrmann C.R. Designing for reliability based on probabilistic modeling using remote access computer systems. In: Proceedings of Seventh Reliability and Maintainability Conference, American Society of Mechanical Engineers. 1968;. p. 492-500.
20. Jaulin L., Kieffer M., Didrit O., Walter É. Applied interval analysis. 2001;Springer, Berlin.
21. Kaplan S. On the method of discrete probability distributions in risk and reliability calculations, applications to seismic risk assessment. Risk Anal. 1:(3):1981;189-196.
22. Karloff H. Linear programming. 1991;Birkhauser, Boston.
23. Kazakov DA. Interval arithmetic for ADA, version 1.0. http://www.dmitry-kazakov.deada/intervals.htm .
24. Kearfott R.B. Interval computations: introduction, uses, and resources. Euromath Bull. 2:(1):1996;95-112.
25. Kearfott R.B. Algorithm 763: INTERVAL_ARITHMETIC: a Fortran 90 module for an interval data type. ACM Trans Math Software. 22:(4):1996;385-392.
26. Kearfott R.B., Kreinovich V. Application of interval computations. 1996;Kluwer, Dordrecht.
27. Klir G., Wierman M.J. Uncertainty-based information, 2nd ed Elements of generalized information theory. 1999;Physica-Verlag, Heidelberg. p. 24.
28. Korn G.A., Korn T.M. Mathematical handbook for scientists and engineers: definitions, theorems, and formulas for reference and review. 1968;McGraw-Hill, New York.
29. Kreinovich V., Lakeyev A., Rohn A., Kahl P. Computational complexity and feasibility of data processing and interval computations. 1997;Kluwer, Dordrecht.
30. Li W, Hyman JM. PDVFOR77: PDV Arithmetic implementation in Fortran 77, version 1.0. http://math.lanl.gov/~liw .
31. Li W, Hyman JM. Theory and implementation of PDV arithmetic for uncertainty computation, in review. http://math.lanl.gov/~liw/ .
32. Moore RE. Interval arithmetic and automatic error analysis in digitalcomputing. PhD Thesis. Stanford University; October 1962.
33. Moore R.E. Interval analysis. 1966;Prentice-Hall, Englewood Cliffs, NJ.
34. Moore R.E. Methods and applications of interval analysis. 1979;SIAM Studies in Applied Mathematics, Philadelphia.
35. Moore R.E. Risk analysis without Monte Carlo methods. Freiburger Intervall-Berichte. 84:(1):1984;1-48.
36. Nelson R.B. An introduction to copulas. Lecture notes in statistics 139. 1999;Springer, Berlin.
37. Nickel K.L.E. Interval mathematics. 1980;Academic Press, New York.
38. Oberkampf WL, Helton JC, Wojtkiewicz SF, Joslyn CA, Ferson S. Challenge problems: uncertainty in system response given uncertain parameters. Reliab Eng Syst Safe, doi: 10.1016/j.ress.2004.03.002 (this issue).
39. Regan H, Ferson S, Berleant D. Equivalence of five methods for bounding uncertainty, International Journal of Approximate Reasoning (in revision).
40. Sandia National Laboratory. Epistemic Uncertainty Workshop, Albuquerque; August 6-7, 2002. http://www.sandia.gov/epistemic/eup_workshop1.htm .
41. Shafer G. A mathematical theory of evidence. 1976;Princeton University Press, Princeton, NJ.
42. Sobol I.M. A Primer for the Monte Carlo method. 1994;CRC Press, Boca Raton.
43. Statool. Software, http://class.ee.iastate.edu/berleanthome/Research/ Pdfs/versions/statool/distribution/index.htm .
44. Sunaga T. Theory of an interval algebra and its application to numerical analysis. 1958;Gaukutsu Bunken Fukeyu-kai, Tokyo.
45. Williamson RC. Probabilistic arithmetic. PhD Thesis. Australia: University of Queensland; 1989. http://axiom.anu.edu.au/~williams/cv/pubs- standalonepubs-standalone.html .
46. Williamson R.C., Downs T. Calculating convolutions and dependency bounds. Int J Approx Reason. 4:1990;89-158.
47. Young R.C. The algebra of many valued-quantities. Math Ann. 104:1931;260-290.