Probability Approximations by Log Likelihood Maximization
Publication: Journal of Engineering Mechanics
Volume 117, Issue 3
Abstract
The computation of multivariate integrals is an important mathematical problem in reliability theory. Various approximation methods have been developed for this task. In the so‐called FORM and SORM methods, it is assumed that all variables have been transformed into independent standard normal ones. Only then can the methods be applied. Here, it is shown that such transformations are not necessary. It is sufficient to maximize the log likelihood function of the probability distribution in the original space, then to approximate the function and limit‐state function near the maximum points by second‐order Taylor expansions to obtain asymptotic approximations. In the same way, asymptotic sensitivity factors for the parameter dependence of the failure probability can be found. The advantages of this method are that the often‐complicated numerical transformation into the normal space is avoided and that the results have a natural interpretation in terms of the parameters of the original random variables and limit‐state function.
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References
1.
Abramowitz, M., and Stegun, I. (1965). Handbook of mathematical functions. Dover, New York, N.Y.
2.
Ang, A. H., and Tang, W. H. (1984). Probability concepts in engineering planning and design, John Wiley & Sons, New York, N.Y.
3.
Bjerager, P. (1988a). “On computational methods for structural reliability analysis.” New directions in structural system reliability, D. Frangopol, ed., Univ. of Colorado, Boulder, Colo., 52–67.
4.
Bjerager, P. (1988b). “Probability integration by directional simulation.” J. Engrg. Mech., ASCE, 114(8), 1–17.
5.
Bleistein, N., and Handelsman, R. (1986). Asymptotic expansions of integrals. Dover, New York, N.Y.
6.
Breitung, K. (1984). “Asymptotic approximations for multinomial integrals.” J. Engrg. Mech., ASCE, 110(3), 357–366.
7.
Breitung, K. (1989). “Asymptotic approximations for probability integrals.” J. Probabilistic Engrg. Mech., 4(4), 187–190.
8.
Breitung, K., and Hohenbichler, M. (1989). “Asymptotic approximations for multivariate integrals with an application to multinomial probabilities.” J. Multivariate Anal., 30, 80–97.
9.
Bucher, C. (1988). “Adaptive sampling—an iterative fast Monte Carlo method.” Struct. Safety, 5, 119–126.
10.
Copson, E. (1965). Asymptotic expansions. Cambridge University Press, Cambridge, U.K.
11.
Der Kiureghian, A., Lin, H., and Hwang, S. (1987). “Second‐order reliability approximations.” J. Engrg. Mech., ASCE, 113(8), 1208–1225.
12.
Ditlevsen, O. (1981). “Principle of normal tail approximation.” J. Engrg. Mech., ASCE, 107(6), 1191–1209.
13.
Dolinski, K. (1983). “First‐order second‐moment approximation in reliability of systems: critical review and alternative approach.” Struct. Safety, 1, 211–213.
14.
Fleming, W. (1977). Functions of several variables. Springer‐Verlag, New York, N.Y.
15.
Freudenthal, A. M. (1956). “Safety and the probability of structural failure.” Trans. ASCE, 121, 1337–1397.
16.
Hasofer, A., and Lind, N. (1974). “An exact and invariant first‐order reliability format.” J. Engrg. Mech., ASCE, 100(1), 111–121.
17.
Hohenbichler, M. (1984). “Mathematics Grundlagen der Zuverlässigkeitstheorie erster Ordnung und einige Erweiterungen,” thesis presented to the Technical University of Munich, at Munich, Germany, in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
18.
Hohenbichler, M., and Rackwitz, R. (1981). “Non‐normal dependent vectors in structural safety.” J. Engrg. Mech., ASCE, 107(6), 1227–1241.
19.
Hohenbichler, M., Gollwitzer, S., Kruse, W., and Rackwitz, R. (1987). “New light on first‐ and second‐order reliability methods.” Struct. Safety, 4, 267–284.
20.
Kendall, M., and Stuart, A. (1979). The advanced theory of statistics, Vol. 2, Fourth Ed., Charles Griffin and Company Ltd., London, U.K.
21.
Madsen, H., Krenk, S., and Lind, N. (1986). Methods of structural safety. Prentice‐Hall Inc., Englewood Cliffs, N.J.
22.
Miller, K. (1980). An introduction to vector stochastic processes. Krieger, Huntington, N.Y.
23.
Murdoch, D. (1957). Linear algebra for undergraduates. John Wiley & Sons, New York, N.Y.
24.
Schuëller, G., and Stix, R. (1987). “A critical appraisal of methods to determine failure probabilities.” Struct. Safety, 4, 293–309.
25.
Shinozuka, M. (1983). “Basic analysis of structural safety.” J. Struct. Engrg., ASCE, 109(3), 721–740.
26.
Wu, Y., and Wirsching, P. (1987). “New algorithm for structural reliability estimation.” J. Engrg. Mech., ASCE, 113(9), 1319–1336.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 117 • Issue 3 • March 1991
Pages: 457 - 477
Copyright
Copyright © 1991 ASCE.
History
Published online: Mar 1, 1991
Published in print: Mar 1991
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