Reliability, Risk, and Uncertainty Analysis Using Generic Expectation Functions
Publication: Journal of Environmental Engineering
Volume 127, Issue 10
Abstract
In engineering design and analysis, mathematical models that generally involve a number of uncertain parameters are frequently employed for decision making. Over the years, a number of techniques have been developed to quantify model output uncertainty contributed by uncertain input parameters. Typically, the methods that are easy to apply may give inaccurate estimates of model output uncertainty. Other methods that reliably produce very accurate results are either difficult to apply or require intensive computational effort. This paper describes the development of generic expectation functions as a function of means and coefficients of variation of input random variables. The generic expectation functions are straightforward to develop, and apply to problems related to reliability, risk, and uncertainty analysis. Several expectation functions based on commonly used probability distributions have been developed. Using them, any order of moment can be estimated exactly. It is found that if exact moments of the model output are available, one can find a good estimate of reliability, risk, and uncertainty of a system without knowing its model output distribution exactly. This technique is applicable when an output variable is a function of several independent random variables in multiplicative, additive, or combined (multiplicative and additive) forms. A practical example is presented to demonstrate the application of generic expectation functions.
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
Abramowitz, M., and Stegun, I. A. ( 1972). Handbook of mathematical functions with formulas, graphs, and mathematical tables, 9th Ed., Dover, New York.
2.
Ang, A. H. S., and Tang, W. H. ( 1984). Probability concepts in engineering planning and design. Volume II: Decision, risk, and reliability, Wiley, New York.
3.
Bates, B. C. ( 1988). “Nonlinear, discrete flood event models. 2: Assessment of statistical nonlinearity.” J. Hydro., Amsterdam, 99, 77–89.
4.
Bates, B. C., and Townley, L. R. ( 1988). “Nonlinear discrete flood event models. 3: Analysis of prediction uncertainty.” J. Hydro., Amsterdam, 99, 91–101.
5.
Benjamin, J. R., and Cornell, C. A. ( 1970). Probability, statistics, and decisions for civil engineers. McGraw-Hill, New York.
6.
Burn, D. H., and McBean, E. A. ( 1985). “Optimization modeling of water quality in an uncertain environment.” Water Resour. Res., 21(7), 934–940.
7.
Cawlfield, J. D., and Wu, M. C. ( 1993). “Probabilistic sensitivity analysis for one dimensional reactive transport in porous media.” Water Resour. Res., 29(3), 661–672.
8.
Cesare, M. A. (1991). “First-order analysis of open-channel flow.”J. Hydr. Engrg., ASCE, 117(2), 242–247.
9.
Chang, C. H., Tung, Y. K., and Yang, J. C. ( 1995). “Evaluation of probabilistic point estimate methods.” Appl. Math. Modeling, 19(2), 95–105.
10.
Fisher, R. A., and Cornish, E. A. ( 1960). “The percentile points of distributions having known cumulants.” Technometrics, 2(2), 209–225.
11.
Haan, C. T. ( 1977). Statistical methods in hydrology, The Iowa State University Press, Ames, Iowa.
12.
Haan, C. T., Barfield, B. J., and Hayes, J. C. ( 1994). Design hydrology and sedimentology for small catchments, Academic, San Diego.
13.
Harr, M. E. ( 1989). “Probability estimates for multivariate analyses.” Appl. Math. Modeling, 13, 313–318.
14.
Hasofer, A. M., and Lind, N. C. (1974). “Exact and invariant second-moment code format.”J. Engrg. Mech. Div., ASCE, 100(1), 111–121.
15.
Kendall, M., Stuart, A., and Ord, J. K. ( 1987). Kendall's advanced theory of statistics. Volume 1: Distribution theory, 5th Ed., Oxford University Press, New York.
16.
Krenkel, P. A., and Ruane, R. J. ( 1979). “Chapter 18: Basic approach to water quality modeling.” Modeling of rivers, H. W. Shen, ed., Wiley, New York.
17.
Lee, H. L., and Mays, L. W. (1986). “Hydraulic uncertainties in flood levee capacity.”J. Hydr. Engrg., ASCE, 112(10), 928–934.
18.
Mays, L. W., and Tung, Y. K. ( 1992). Hydrosystems engineering and management, McGraw-Hill, New York.
19.
Melching, C. S. ( 1995). “Reliability estimation.” Computer models of watershed hydrology, V. P. Singh, ed., Water Resources Publications, Littleton, Colo., 69–118.
20.
Melching, C. S., and Anmangandla, S. (1992). “Improved first-order uncertainty method for water-quality modeling.”J. Envir. Engrg., ASCE, 118(5), 791–805.
21.
Melching, C. S., and Yen, B. C. ( 1986). “Slope influence on storm sewer risk.” Stochastic and risk analyses in hydraulic engineering, B. C. Yen, ed., Water Resources Publications, Littleton, Colo., 79–89.
22.
Melching, C. S., Yen, B. C., and Wenzel, H. G. Jr. (1991). “Output reliability as guide for selection of rainfall-runoff models.”J. Water Resour. Plng. and Mgmt., ASCE, 117(3), 383–398.
23.
Mishra, S. ( 1998). “A methodology for probabilistic risk-based performance assessments using the first-order reliability method.” Proc., Risk-Based Perf. Assessment and Decision Making Topical Meeting, American Nuclear Society, Pasco, Wash., 1–6.
24.
Novotny, V., and Olem, H. ( 1994). Water quality: Prevention, identification, and management of diffuse pollution, Van Nostrand Reinhold, New York.
25.
Rackwitz, R. ( 1976). “Practical probabilistic approach to design.” Bull. No. 112, Comite European du Beton, Paris.
26.
Rosenblueth, E. ( 1975). “Point estimates for probability moments.” Proc., Nat. Acad. of Sci. U.S.A., 72(10), 3812–3814.
27.
Rosenblueth, E. ( 1981). “Point estimates in probabilities.” Appl. Math. Modeling, 72(10), 3812–3814.
28.
Sitar, N., Cawlfield, J. D., and Kiureghian, A. D. ( 1987). “First-order reliability approach to stochastic analysis of subsurface flow and contaminant transport.” Water Resour. Res., 23(5), 794–804.
29.
Stevens, E. ( 1993). “A method to predict the accuracy of a first order approximation of model output variance.” PhD thesis, Department of Biosystems and Agricultural Engineering, Oklahoma State University, Stillwater.
30.
Tung, Y. K. (1990). “Mellin transform applied to uncertainty analysis in hydrology/hydraulics.”J. Hydr. Engrg., ASCE, 116(5), 659–674.
31.
Tung, Y. K. ( 1996). “Uncertainty analysis in water resources engineering.” Proc., Stochastic hydraulics '96, K. S. Tickle, I. C. Goulter, C. Xu, S. A. Wasimi, and F. Bouchart, eds., Balkema, Rotterdam, The Netherlands, 29–46.
32.
Tung, Y. K., and Mays, L. W. ( 1980). “Optimal risk-based design of water resources engineering projects.” Tech. Rep. No. 171, Center for Research in Water Resources, University of Texas, Austin.
33.
Tyagi, A. ( 2000). “A simple approach to reliability, risk, and uncertainty analysis of hydrologic, hydraulic, and environmental engineering systems.” PhD thesis, Oklahoma State University, Stillwater.
34.
U.S. Environmental Protection Agency (USEPA). ( 1989). “Risk assessment guidance for superfund. Volume 1: Human health evaluation manual (Part A). Interim final.” Rep. No. EPA/540/1-89/002, U.S. Office of Emergency and Remedial Response, Washington, D.C.
35.
Yen, B. C., Cheng, S. T., and Melching, C. S. ( 1986). “First order reliability analysis.” Stochastic and risk analyses in hydraulic engineering, B. C. Yen, ed., Water Resources Publications, Littleton, Colo., 79–89.
Information & Authors
Information
Published In
Journal of Environmental Engineering
Volume 127 • Issue 10 • October 2001
Pages: 938 - 945
History
Received: Aug 29, 2000
Published online: Oct 1, 2001
Published in print: Oct 2001
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.