IDEAS home Printed from https://ideas.repec.org/a/spr/metcap/v18y2016i1d10.1007_s11009-014-9411-x.html
   My bibliography  Save this article

Rare Event Probability Estimation in the Presence of Epistemic Uncertainty on Input Probability Distribution Parameters

Author

Listed:
  • Mathieu Balesdent

    (Onera - The French Aerospace Lab)

  • Jérôme Morio

    (Onera - The French Aerospace Lab)

  • Loïc Brevault

    (Onera - The French Aerospace Lab
    CNES - Launchers Directorate)

Abstract

The accurate estimation of rare event probabilities is a crucial problem in engineering to characterize the reliability of complex systems. Several methods such as Importance Sampling or Importance Splitting have been proposed to perform the estimation of such events more accurately (i.e., with a lower variance) than crude Monte Carlo method. However, these methods assume that the probability distributions of the input variables are exactly defined (e.g., mean and covariance matrix perfectly known if the input variables are defined through Gaussian laws) and are not able to determine the impact of a change in the input distribution parameters on the probability of interest. The problem considered in this paper is the propagation of the input distribution parameter uncertainty defined by intervals to the rare event probability. This problem induces intricate optimization and numerous probability estimations in order to determine the upper and lower bounds of the probability estimate. The calculation of these bounds is often numerically intractable for rare event probability (say 10−5), due to the high computational cost required. A new methodology is proposed to solve this problem with a reduced simulation budget, using the adaptive Importance Sampling. To this end, a method for estimating the Importance Sampling optimal auxiliary distribution is proposed, based on preceding Importance Sampling estimations. Furthermore, a Kriging-based adaptive Importance Sampling is used in order to minimize the number of evaluations of the computationally expensive simulation code. To determine the bounds of the probability estimate, an evolutionary algorithm is employed. This algorithm has been selected to deal with noisy problems since the Importance Sampling probability estimate is a random variable. The efficiency of the proposed approach, in terms of accuracy of the found results and computational cost, is assessed on academic and engineering test cases.

Suggested Citation

  • Mathieu Balesdent & Jérôme Morio & Loïc Brevault, 2016. "Rare Event Probability Estimation in the Presence of Epistemic Uncertainty on Input Probability Distribution Parameters," Methodology and Computing in Applied Probability, Springer, vol. 18(1), pages 197-216, March.
    • Handle: RePEc:spr:metcap:v:18:y:2016:i:1:d:10.1007_s11009-014-9411-x
    DOI: 10.1007/s11009-014-9411-x
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11009-014-9411-x
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s11009-014-9411-x?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Scott Ferson & William L. Oberkampf, 2009. "Validation of imprecise probability models," International Journal of Reliability and Safety, Inderscience Enterprises Ltd, vol. 3(1/2/3), pages 3-22.
    2. Helton, Jon C., 2011. "Quantification of margins and uncertainties: Conceptual and computational basis," Reliability Engineering and System Safety, Elsevier, vol. 96(9), pages 976-1013.
    3. Rubinstein, Reuven Y., 1997. "Optimization of computer simulation models with rare events," European Journal of Operational Research, Elsevier, vol. 99(1), pages 89-112, May.
    4. Pieter-Tjerk de Boer & Dirk Kroese & Shie Mannor & Reuven Rubinstein, 2005. "A Tutorial on the Cross-Entropy Method," Annals of Operations Research, Springer, vol. 134(1), pages 19-67, February.
    5. Limbourg, Philipp & de Rocquigny, Etienne & Andrianov, Guennadi, 2010. "Accelerated uncertainty propagation in two-level probabilistic studies under monotony," Reliability Engineering and System Safety, Elsevier, vol. 95(9), pages 998-1010.
    6. Morio, Jérôme, 2011. "Non-parametric adaptive importance sampling for the probability estimation of a launcher impact position," Reliability Engineering and System Safety, Elsevier, vol. 96(1), pages 178-183.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Youngjun Choe & Henry Lam & Eunshin Byon, 2018. "Uncertainty Quantification of Stochastic Simulation for Black-box Computer Experiments," Methodology and Computing in Applied Probability, Springer, vol. 20(4), pages 1155-1172, December.
    2. Chabridon, Vincent & Balesdent, Mathieu & Bourinet, Jean-Marc & Morio, Jérôme & Gayton, Nicolas, 2018. "Reliability-based sensitivity estimators of rare event probability in the presence of distribution parameter uncertainty," Reliability Engineering and System Safety, Elsevier, vol. 178(C), pages 164-178.
    3. Sarazin, Gabriel & Morio, Jérôme & Lagnoux, Agnès & Balesdent, Mathieu & Brevault, Loïc, 2021. "Reliability-oriented sensitivity analysis in presence of data-driven epistemic uncertainty," Reliability Engineering and System Safety, Elsevier, vol. 215(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Mattrand, C. & Bourinet, J.-M., 2014. "The cross-entropy method for reliability assessment of cracked structures subjected to random Markovian loads," Reliability Engineering and System Safety, Elsevier, vol. 123(C), pages 171-182.
    2. El Masri, Maxime & Morio, Jérôme & Simatos, Florian, 2021. "Improvement of the cross-entropy method in high dimension for failure probability estimation through a one-dimensional projection without gradient estimation," Reliability Engineering and System Safety, Elsevier, vol. 216(C).
    3. Balesdent, Mathieu & Morio, Jérôme & Marzat, Julien, 2015. "Recommendations for the tuning of rare event probability estimators," Reliability Engineering and System Safety, Elsevier, vol. 133(C), pages 68-78.
    4. J Morio & R Pastel, 2012. "Plug-in estimation of d-dimensional density minimum volume set of a rare event in a complex system," Journal of Risk and Reliability, , vol. 226(3), pages 337-345, June.
    5. Heath, Jeffrey W. & Fu, Michael C. & Jank, Wolfgang, 2009. "New global optimization algorithms for model-based clustering," Computational Statistics & Data Analysis, Elsevier, vol. 53(12), pages 3999-4017, October.
    6. Erik Hintz & Marius Hofert & Christiane Lemieux & Yoshihiro Taniguchi, 2022. "Single-Index Importance Sampling with Stratification," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 3049-3073, December.
    7. Kin-Ping Hui, 2011. "Cooperative Cross-Entropy method for generating entangled networks," Annals of Operations Research, Springer, vol. 189(1), pages 205-214, September.
    8. Caballero, Rafael & Hernández-Díaz, Alfredo G. & Laguna, Manuel & Molina, Julián, 2015. "Cross entropy for multiobjective combinatorial optimization problems with linear relaxations," European Journal of Operational Research, Elsevier, vol. 243(2), pages 362-368.
    9. A. Shibu & M. Reddy, 2014. "Optimal Design of Water Distribution Networks Considering Fuzzy Randomness of Demands Using Cross Entropy Optimization," Water Resources Management: An International Journal, Published for the European Water Resources Association (EWRA), Springer;European Water Resources Association (EWRA), vol. 28(12), pages 4075-4094, September.
    10. Nguyen, Hoa T.M. & Chow, Andy H.F. & Ying, Cheng-shuo, 2021. "Pareto routing and scheduling of dynamic urban rail transit services with multi-objective cross entropy method," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 156(C).
    11. Hao Su & Qun Niu & Zhile Yang, 2023. "Optimal Power Flow Using Improved Cross-Entropy Method," Energies, MDPI, vol. 16(14), pages 1-33, July.
    12. Papaioannou, Iason & Geyer, Sebastian & Straub, Daniel, 2019. "Improved cross entropy-based importance sampling with a flexible mixture model," Reliability Engineering and System Safety, Elsevier, vol. 191(C).
    13. Ferdinand Bollwein & Stephan Westphal, 2022. "Oblique decision tree induction by cross-entropy optimization based on the von Mises–Fisher distribution," Computational Statistics, Springer, vol. 37(5), pages 2203-2229, November.
    14. Chabridon, Vincent & Balesdent, Mathieu & Bourinet, Jean-Marc & Morio, Jérôme & Gayton, Nicolas, 2018. "Reliability-based sensitivity estimators of rare event probability in the presence of distribution parameter uncertainty," Reliability Engineering and System Safety, Elsevier, vol. 178(C), pages 164-178.
    15. Dirk P. Kroese & Sergey Porotsky & Reuven Y. Rubinstein, 2006. "The Cross-Entropy Method for Continuous Multi-Extremal Optimization," Methodology and Computing in Applied Probability, Springer, vol. 8(3), pages 383-407, September.
    16. Jiaqiao Hu & Michael C. Fu & Steven I. Marcus, 2007. "A Model Reference Adaptive Search Method for Global Optimization," Operations Research, INFORMS, vol. 55(3), pages 549-568, June.
    17. Chen, Jun-Yu & Feng, Yun-Wen & Teng, Da & Lu, Cheng & Fei, Cheng-Wei, 2022. "Support vector machine-based similarity selection method for structural transient reliability analysis," Reliability Engineering and System Safety, Elsevier, vol. 223(C).
    18. Certa, Antonella & Hopps, Fabrizio & Inghilleri, Roberta & La Fata, Concetta Manuela, 2017. "A Dempster-Shafer Theory-based approach to the Failure Mode, Effects and Criticality Analysis (FMECA) under epistemic uncertainty: application to the propulsion system of a fishing vessel," Reliability Engineering and System Safety, Elsevier, vol. 159(C), pages 69-79.
    19. Xi Chen & Enlu Zhou, 2015. "Population model-based optimization," Journal of Global Optimization, Springer, vol. 63(1), pages 125-148, September.
    20. Benoumechiara Nazih & Bousquet Nicolas & Michel Bertrand & Saint-Pierre Philippe, 2020. "Detecting and modeling critical dependence structures between random inputs of computer models," Dependence Modeling, De Gruyter, vol. 8(1), pages 263-297, January.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:metcap:v:18:y:2016:i:1:d:10.1007_s11009-014-9411-x. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.