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The actual failure times of individual components are usually unavailable in many applications. Instead, only aggregate failure-time data are collected by actual users, due to technical and/or economic reasons. When dealing with such data for reliability estimation, practitioners often face the challenges of selecting the underlying failure-time distributions and the corresponding statistical inference methods. So far, only the exponential, normal, gamma and inverse Gaussian distributions have been used in analyzing aggregate failure-time data, due to these distributions having closed-form expressions for such data. However, the limited choices of probability distributions cannot satisfy extensive needs in a variety of engineering applications. PHase-type (PH) distributions are robust and flexible in modeling failure-time data, as they can mimic a large collection of probability distributions of non-negative random variables arbitrarily closely by adjusting the model structures. In this article, PH distributions are utilized, for the first time, in reliability estimation based on aggregate failure-time data. A Maximum Likelihood Estimation (MLE) method and a Bayesian alternative are developed. For the MLE method, an Expectation-Maximization algorithm is developed for parameter estimation, and the corresponding Fisher information is used to construct the confidence intervals for the quantities of interest. For the Bayesian method, a procedure for performing point and interval estimation is also introduced. Numerical examples show that the proposed PH-based reliability estimation methods are quite flexible and alleviate the burden of selecting a probability distribution when the underlying failure-time distribution is general or even unknown.
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REIN: Reliability Estimation via Importance sampling with Normalizing flows
We introduce a novel framework called REIN: Reliability Estimation by learning an Importance sampling (IS) distribution with Normalizing flows (NFs). The NFs learn probability space maps that transform the probability distribution of the input random variables into a quasi-optimal IS distribution. NFs stack together invertible neural networks to construct differentiable bijections with efficiently computed Jacobian determinants. The NF 'pushes forward' a realization from the input probability distribution into a realization from the IS distribution, with importance weights calculated using the change of variables formula. We also propose a loss function to learn a NF map that minimizes the reverse Kullback-Leibler divergence between the 'pushforward' distribution and a sequentially updated target distribution obtained by modifying the optimal IS distribution. We demonstrate REIN's efficacy on a set of benchmark problems that feature very low failure rates, multiple failure modes and high dimensionality, while comparing against other variance reduction methods. We also consider two simple applications, the reliability analyses of a thirty-four story building and a cantilever tube, to demonstrate the applicability of REIN to practical problems of interest. As compared to other methods, REIN is shown to be useful for high-dimensional reliability estimation problems with very small failure probabilities.
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- Award ID(s):
- 1663667
- PAR ID:
- 10653777
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Reliability Engineering & System Safety
- Volume:
- 242
- Issue:
- C
- ISSN:
- 0951-8320
- Page Range / eLocation ID:
- 109729
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1016/j.ress.2023.109729
