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A probabilistic approach for representation of interval uncertainty

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  • Zaman, Kais
  • Rangavajhala, Sirisha
  • McDonald, Mark P.
  • Mahadevan, Sankaran

Abstract

In this paper, we propose a probabilistic approach to represent interval data for input variables in reliability and uncertainty analysis problems, using flexible families of continuous Johnson distributions. Such a probabilistic representation of interval data facilitates a unified framework for handling aleatory and epistemic uncertainty. For fitting probability distributions, methods such as moment matching are commonly used in the literature. However, unlike point data where single estimates for the moments of data can be calculated, moments of interval data can only be computed in terms of upper and lower bounds. Finding bounds on the moments of interval data has been generally considered an NP-hard problem because it includes a search among the combinations of multiple values of the variables, including interval endpoints. In this paper, we present efficient algorithms based on continuous optimization to find the bounds on second and higher moments of interval data. With numerical examples, we show that the proposed bounding algorithms are scalable in polynomial time with respect to increasing number of intervals. Using the bounds on moments computed using the proposed approach, we fit a family of Johnson distributions to interval data. Furthermore, using an optimization approach based on percentiles, we find the bounding envelopes of the family of distributions, termed as a Johnson p-box. The idea of bounding envelopes for the family of Johnson distributions is analogous to the notion of empirical p-box in the literature. Several sets of interval data with different numbers of intervals and type of overlap are presented to demonstrate the proposed methods. As against the computationally expensive nested analysis that is typically required in the presence of interval variables, the proposed probabilistic representation enables inexpensive optimization-based strategies to estimate bounds on an output quantity of interest.

Suggested Citation

  • Zaman, Kais & Rangavajhala, Sirisha & McDonald, Mark P. & Mahadevan, Sankaran, 2011. "A probabilistic approach for representation of interval uncertainty," Reliability Engineering and System Safety, Elsevier, vol. 96(1), pages 117-130.
    • Handle: RePEc:eee:reensy:v:96:y:2011:i:1:p:117-130
    DOI: 10.1016/j.ress.2010.07.012
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    References listed on IDEAS

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    1. Baudrit, C. & Dubois, D., 2006. "Practical representations of incomplete probabilistic knowledge," Computational Statistics & Data Analysis, Elsevier, vol. 51(1), pages 86-108, November.
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    Cited by:

    1. Carlo Drago, 2017. "Interval Based Composite Indicators," Working Papers 2017.42, Fondazione Eni Enrico Mattei.
    2. Sankararaman, Shankar & Mahadevan, Sankaran, 2011. "Likelihood-based representation of epistemic uncertainty due to sparse point data and/or interval data," Reliability Engineering and System Safety, Elsevier, vol. 96(7), pages 814-824.
    3. Lv, Y. & Yan, X.D. & Sun, W. & Gao, Z.Y., 2015. "A risk-based method for planning of bus–subway corridor evacuation under hybrid uncertainties," Reliability Engineering and System Safety, Elsevier, vol. 139(C), pages 188-199.
    4. Kais Zaman & Saraf Anika Kritee, 2014. "An Optimization-Based Approach to Calculate Confidence Interval on Mean Value with Interval Data," Journal of Optimization, Hindawi, vol. 2014, pages 1-8, July.
    5. Liu, H.B. & Jiang, C. & Jia, X.Y. & Long, X.Y. & Zhang, Z. & Guan, F.J., 2018. "A new uncertainty propagation method for problems with parameterized probability-boxes," Reliability Engineering and System Safety, Elsevier, vol. 172(C), pages 64-73.
    6. Yao, Wen & Chen, Xiaoqian & Huang, Yiyong & van Tooren, Michel, 2013. "An enhanced unified uncertainty analysis approach based on first order reliability method with single-level optimization," Reliability Engineering and System Safety, Elsevier, vol. 116(C), pages 28-37.

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