Convergence properties of an interval probabilistic approach to system reliability estimation

Based on a black box model of a complex system, and on intervals and probabilities describing the known information about the inputs, we want to estimate the system's reliability. This problem is motivated by a number of problem areas, most specifically in engineering reliability analysis under conditions of poor measurement and high complexity of system models. Using the results of tests performed on the system's computer model, we can estimate the lower and upper bounds of the probability that the system is in a desirable state. This is equivalent to using Monte-Carlo sampling to estimate cumulative belief and plausibility values of functionally propagated finite random intervals. In this paper, we prove that these estimates are correct in the sense that under reasonable assumptions, these estimates converge to the actual probability bounds.

[1]  A. Dempster.  Upper and Lower Probabilities Generated by a Random Closed Interval , 1968 .

[2]  J. Kacprzyk,et al.  Advances in the Dempster-Shafer theory of evidence , 1994 .

[3]  D. Bell,et al.  Evidence Theory and Its Applications , 1991 .

[4]  R. Stephenson.  A and V , 1962, The British journal of ophthalmology.

[5]  J. C. Helton,et al.  Bounds on belief and plausibility of functionally propagated random sets , 2002, 2002 Annual Meeting of the North American Fuzzy Information Processing Society Proceedings. NAFIPS-FLINT 2002 (Cat. No. 02TH8622).

[6]  Jon C. Helton,et al.  Challenge Problems : Uncertainty in System Response Given Uncertain Parameters ( DRAFT : November 29 , 2001 ) , 2001 .

[7]  Olaf Wolkenhauer,et al.  Random-sets: theory and applications , 2001 .

[8]  Jon C. Helton,et al.  An exploration of alternative approaches to the representation of uncertainty in model predictions , 2003, Reliab. Eng. Syst. Saf..

[9]  Ronald R. Yager,et al.  Arithmetic and Other Operations on Dempster-Shafer Structures , 1986, Int. J. Man Mach. Stud..

[10]  Jon C. Helton,et al.  Alternative representations of epistemic uncertainty , 2004, Reliab. Eng. Syst. Saf..

[11]  Jon C. Helton,et al.  Sensitivity analysis in conjunction with evidence theory representations of epistemic uncertainty , 2006, Reliab. Eng. Syst. Saf..

[12]  C. Joslyn.  Approximate representations of random intervals for hybrid uncertainty quantification in engineering modeling , 2004 .

[13]  V. Kreinovich,et al.  Limit Theorems and Applications of Set-Valued and Fuzzy Set-Valued Random Variables , 2002 .

[14]  Jon C. Helton,et al.  Summary from the epistemic uncertainty workshop: consensus amid diversity , 2004, Reliab. Eng. Syst. Saf..