A likelihood-free approach towards Bayesian modeling of degradation growths using mixed-effects regression
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 producing noisy degradation measurements represented as ; where is the number of inspections conducted on the ith component. Assuming the true degradation growths to be , the noisy degradation growths are represented as , where represents measurement noise. The properties of
Bayesian inference
Let us assume that the set of parameters of the underlying degradation model is represented as . Given the observed data and a joint prior distribution , the Bayesian inference gives the posterior distribution of aswhere 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.
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