Elsevier

Computers & Structures

Volume 244, February 2021, 106427
Computers & Structures

A likelihood-free approach towards Bayesian modeling of degradation growths using mixed-effects regression

https://doi.org/10.1016/j.compstruc.2020.106427Get rights and content

Highlights

  • Degradation modeling using a Bayesian mixed-effects regression model with flexible error distribution.

  • Solving the intractable likelihood problem using the approximate Bayesian computation (ABC) method.

  • Case study on the degradation of the nuclear piping system.

Abstract

Mixed-effects regression models are widely applicable for predicting degradation growths in structural components. The Bayesian inference method is used to estimate the regression parameters when the degradation data are confounded by measurement and parameter uncertainties. The Gibbs sampler (GS), commonly used for this purpose, works when the regression errors are assumed as normally distributed that allows for the analytical formulation of the likelihood function. In case of a more general regression error distribution (e.g., mixture models), the likelihood becomes analytically intractable and computationally expensive to a degree that any likelihood-based Bayesian inference scheme (e.g., GS, Metropolis-Hastings sampler) can no longer be used for solving a practical problem.

This paper proposes a practical likelihood-free approach for parameter estimation based on the approximate Bayesian computation (ABC) method. The ABC method implements forward simulation coupled with a rejection mechanism to sample from a target posterior distribution thereby eliminating the need to evaluate the likelihood function. The advantages of the proposed method are illustrated by analyzing degradation data obtained from a Canadian nuclear power plant.

Introduction

The structural integrity and system performance of large engineering systems are impacted by various forms of degradation mechanisms. Modeling and forecasting such mechanisms are accomplished by collecting degradation data from periodic in-service inspections of various components. Subsequently, the degradation prediction is transformed into system and component lifetimes that are crucial inputs for risk-based life-cycle management of engineering structures.

In practice, various data-driven models are used for degradation assessment, such as regression models that are commonly used in corrosion assessment. The statistical estimation of such models is often challenged by both aleatoric and epistemic uncertainties. The aleatoric uncertainty comes from the inherent randomness of a degradation process. Whereas, the sources of epistemic/parameter uncertainty are noise in the measurements, limited resolution of the inspection probe, and small sample size.

The uncertainties in model parameters are suitably handled by a Bayesian formulation of the underlying degradation model. Primarily, there are two main approaches available for Bayesian inference, one is likelihood-based and another is likelihood-free. Given that the model likelihood is either analytically tractable or computationally inexpensive, the likelihood-based approach is preferred since it produces accurate results. By contrast, if the model likelihood turns out to be analytically intractable or computationally expensive, the likelihood-free approach is most suitable for efficient sampling from the target posterior distribution. For example, the mixed-effects regression model with a normally distributed error term always produces an analytically tractable likelihood function. However, in the context of degradation modeling, the error term in a regression model represents a combination of many factors such as inspection error, human error (for manually operated tools), model error, and error due to the change of inspection tools at different inspection campaigns. Thus, the normality assumption for the regression error may not always be a good choice. Generally speaking, a non-normal/flexible error model (e.g., the mixture of distributions model) may act as a better choice; however, this choice makes the model likelihood intractable and computationally expensive due to the presence of high-dimensional integrals.

The most common Bayesian inference method for estimating regression model parameters is the Gibbs sampler (GS) – a likelihood-based Markov chain Monte Carlo (MCMC) method [1]. GS directly generates samples from the joint posterior distribution of the model parameters with the help of individual conditional posterior distributions. However, even if the model likelihood is analytically tractable, implementing the GS algorithm is not always feasible, because: (1) from satisfying the prior conjugacy to the derivation of the individual conditional posterior distributions is a complicated and tedious assignment (especially when a correlation structure is present between the parameters), and (2) in some cases, sampling from the conditional posterior distributions is challenging. Hence, even though the standard GS algorithm produces accurate results for the normal regression models, it can not be considered a practical approach.

The approximate Bayesian computation (ABC) method [2] is a popular likelihood-free Bayesian inference scheme. Based on a predefined distance function, ABC directly generates samples from the target posterior distribution by comparing the observed data set with numerous simulated data sets. The algorithm employs an “accept-reject” mechanism through a tolerance threshold on the distance values, and retains the relevant parameter samples that satisfy the acceptance criterion. ABC is particularly useful when the model likelihood is intractable or computationally expensive to evaluate – making it a perfect choice for handling flexible non-normal error distributions in a regression model. In addition, if data simulation from a forward model is computationally cheap, ABC can achieve reasonable efficiency during the sampling procedure [3].

ABC was primarily proposed to solve intractable likelihood-related problems in biostatistics [2], [4], [5], [6]. However, recently ABC has grown in popularity in several other fields such as engineering (e.g., [3], [7], [8], [9], [10]), astronomy (e.g., [11]), archeology (e.g., [12]), psychology (e.g., [13]), geology (e.g., [14]) and hydrology (e.g., [15]). To solve the parameter estimation problem in complex systems, several well-known sampling algorithms, such as MCMC [16], population Monte Carlo [17], sequential Monte Carlo [18], subset simulation [19], and ellipsoidal nested sampling [20], have been used within the ABC algorithm.

This study introduces the application of ABC, a likelihood-free novel computational method for Bayesian parameter inference of mixed-effects regression models. Whether a normal or flexible regression model is chosen, the proposed method can be applied to solve all types (tractable or intractable likelihoods) of parameter estimation problems. ABC is integrated with subset simulation [19] – a highly efficient rare event simulation technique that greatly reduces the rejection rate in ABC. Although the current study focuses only on the linear mixed-effects regression (LMER) model, the proposed approach can be extended to other mixed-effects models (e.g., nonlinear regression, stochastic processes) in a straight-forward manner.

The LMER model is utilized in modeling and prediction of flow-accelerated corrosion (FAC) in the primary piping system of a nuclear reactor. FAC is a major form of degradation seen in the feeder piping system of a Canada deuterium uranium (CANDU®) reactor [21], [22]. The LMER model systematically handles the inspection data by considering both the system-level fixed effects, as well as component-level random effects. The model is well suited for pooling unbalanced data from component-specific measurements across the pipe population to obtain robust estimates of the model parameters [23]. The analysis is extended by further utilizing the posterior parameter samples to obtain the lifetime distribution and survival function of the components under study.

The paper is organized as follows. Section 2 presents the degradation modeling problem in general, and introduces the LMER model and its assumptions related to a degradation process. Section 3 presents the likelihood-free Bayesian inference methodology and the details of its implementation for the LMER model. Section 4 contains a practical application of the proposed methodology applied to wall thickness loss data of a nuclear piping system. Finally, Section 5 presents the summary and conclusions of the study.

Section snippets

Degradation model

Suppose the data at hand consist of degradation measurements of N components at various inspection times. The ith component is measured at times ti1,ti2,,timi producing noisy degradation measurements represented as yi1,yi2,,yimi; where mi is the number of inspections conducted on the ith component. Assuming the true degradation growths to be xi1,xi2,,ximi, the noisy degradation growths are represented as yij=xij+zij,i=1,,N,j=1,,mi, where zij represents measurement noise. The properties of

Bayesian inference

Let us assume that the set of parameters of the underlying degradation model is represented as Θ. Given the observed data Dobs={y1,y2,,yN} and a joint prior distribution f(Θ), the Bayesian inference gives the posterior distribution of Θ asf(Θ|Dobs)=L(Θ|Dobs)f(Θ)ΘL(Θ|Dobs)f(Θ)dΘL(Θ|Dobs)f(Θ)where L(Θ|Dobs) is the likelihood function and the integration in the denominator is called the normalizing constant. This is also known as the conventional likelihood-based framework of Bayesian inference.

System description

Nuclear power plant components are affected by various degradation processes, such as flow-accelerated corrosion (FAC) (e.g., [10]), pitting corrosion (e.g., [25]), creep (e.g., [44]), and fatigue (e.g., [45]). For instance, FAC is a major form of degradation seen in the feeder piping system of a CANDU® reactor assembly [21], [22]. The feeder pipes contain pressurized heavy-water coolant which carries the heat generated by a CANDU® reactor core [46]. Fig. 1 shows the typical layout of the

Summary and conclusions

Degradation assessment is a critical step to maintain a safe and reliable operation of engineering systems such as nuclear power plants. Degradation data collected from multiple inspection campaigns are utilized for modeling and predicting the extent of future degradation and component lifetime. The predicted lifetime distribution of the system of components is further utilized for component maintenance and replacement planning. Thus, the lifetime distribution is an important input for

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

The authors are grateful to the University Network of Excellence in Nuclear Engineering (UNENE) and the National Sciences and Engineering Research Council of Canada (NSERC) for providing financial support to this study. The authors appreciate the constructive comments received from the anonymous reviewers, which have lead to a significant improvement to an earlier draft of the paper.

References (48)

  • Yan Wang et al.

    Evaluation of Bayesian source estimation methods with prairie grass observations and gaussian plume model: A comparison of likelihood functions and distance measures

    Atmosph Environ

    (2017)
  • Majid K. Vakilzadeh et al.

    Approximate Bayesian computation by subset simulation using hierarchical state-space models

    Mech Syst Signal Process

    (2017)
  • Lin Jeen-Shang et al.

    Nonlinear structural identification using extended kalman filter

    Comput Struct

    (1994)
  • Y.Q. Ni et al.

    Identification of non-linear hysteretic isolators from periodic vibration tests

    J Sound Vib

    (1998)
  • M.D. Pandey et al.

    Understanding the mechanics of creep deformation to develop a surrogate model for contact assessment in CANDU fuel channels

    Nucl Eng Des

    (2018)
  • Jon Wakefield

    Bayesian and frequentist regression methods

    (2013)
  • Jean-Michel Marin et al.

    Approximate Bayesian computational methods

    Statist Comput

    (2012)
  • Mikael Sunnåker et al.

    Approximate Bayesian computation

    PLoS Comput Biol

    (2013)
  • Mark A Beaumont

    Approximate Bayesian computation in evolution and ecology

    Ann Rev Ecol Evol Systemat

    (2010)
  • Scott A. Sisson et al.

    Handbook of approximate Bayesian computation

    (2018)
  • Anis Ben Abdessalem et al.

    Model selection and parameter estimation in structural dynamics using approximate Bayesian computation

    Mech Syst Signal Process

    (2018)
  • Piao Chen et al.

    Parametric analysis of time-censored aggregate lifetime data

    IISE Trans

    (2020)
  • Indranil Hazra et al.

    Estimation of flow-accelerated corrosion rate in nuclear piping system

    J Nucl Eng Radiat Sci

    (2020)
  • Pacchiardi Lorenzo, Kunzli Pierre, Schoengens Marcel, Chopard Bastien, Dutta Ritabrata. Distance-learning for...
  • Cited by (3)

    • Likelihood-free Hamiltonian Monte Carlo for modeling piping degradation and remaining useful life prediction using the mixed gamma process

      2022, International Journal of Pressure Vessels and Piping
      Citation Excerpt :

      The approximate Bayesian computation (ABC) method, on the other hand, generates sample from a target posterior utilizing forward simulation and an accept–reject rule to circumvent the likelihood evaluation step. The convenience of bypassing the computation of the likelihood function has made the ABC methods highly popular for solving real-life problems across several other fields of study; such as in biostatistics (e.g., [37,38]), engineering (e.g., [39–44]), archaeology (e.g., [45]), psychology (e.g., [46]), astronomy (e.g., [47]), geology (e.g., [48]), and hydrology (e.g., [49]). In a recent study by Hazra et al. [30], the authors used a Bayesian gamma process to model noisy degradation data; they found that the currently available ABC with Markov chain Monte Carlo (ABC-MCMC) scheme shows promising results, although it requires significant thinning of samples due to its poor mixing properties resulting in high sample repetitions.

    • A simulation-based Bayesian approach to predict the distribution of maximum pit depth in steam generator tubes

      2022, Nuclear Engineering and Design
      Citation Excerpt :

      ABC methods are highly popular in several fields of study for solving practical problems that suffer from intractable likelihoods. Various applications of the method can be found in biostatistics (e.g., Marin et al., 2012), psychology (e.g., Turner and Van Zandt, 2012), geology (e.g., Lorenzo Pacchiardi et al., 2019), hydrology (e.g., Sadegh and Vrugt, 2014), archeology (e.g., Crema et al., 2014), astronomy (e.g., Jennings and Madigan, 2017), and engineering (e.g., Abdessalem et al., 2018; Hazra and Pandey, 2021). Given that the basic ABC schemes suffer from convergence issues in cases where diffuse priors are used or the parameter space is high-dimensional, the sequential ABC samplers have been developed that suitably handle the convergence issue.

    View full text