An artificial neural network supported stochastic process for degradation modeling and prediction
Introduction
Modern manufactured products, such as bipolar transistors [1], seals [2,3], and LED-based lights [4], have seen a growing demand for high reliability. Generally, to ensure the reliability of the using products, degradation modeling is performed based on run-to-failure degradation data before or after use, and online predictions are performed based on real-time degradation data when the predicting individual is being used [5,6]. Multiple pieces of research have been published in degradation modeling and online prediction. Several methods have been applied in degradation modeling, such as the degradation-path-based modeling method [7] and stochastic process-based modeling method [8]. Several methods have been presented in online degradation prediction, such as stochastic process [9] and machine learning [10]. The presented research is focused on studying stochastic processes due to their flexibility in both degradation modeling and online prediction [11,12].
Three factors are needed to apply the stochastic process in degradation modeling and prediction, these are degradation process, degradation path, and model parameters. Inverse Gaussian, Gamma, and Wiener processes are the three most widely used in degradation modeling and prediction [13], [14], [15]. Generally, the determination (selected or combined) of the degradation is based on the degradation dataset and corresponding prior knowledge. Selection of the degradation process for specific products is based on the degradation dataset by some criterion, such as Akaike Information Criteria (AIC) and Bayesian Information Criteria (BIC) [16,17]. Furthermore, considering that there may be more than one stochastic process can be used to fit a specific degradation dataset, the selected processes can be combined by some model fusion method, such as adjustment factor method, model averaging method, and Dempster-Shafer (D-S) evidence theory [18], [19], [20]. Inverse Gaussian process and Gamma process are only suitable to model monotonic degradation processes, while Wiener process can be used to model both monotonic and non-monotonic degradation processes. Hence, in the presented research, the Wiener process is presented as an example due to its advantage in degradation modeling for monotonic and non-monotonic degradation processes.
Published degradation paths include the linear path, the phase-based path, and the path with a time-varying degradation rate. The linear degradation path is the most widely used because most nonlinear degradation processes can transform into linear degradation processes [21], [22], [23], [24], [25]. The above transformation is practical and effective only when the degradation path is determined exactly. Hence, the accuracy of the determined degradation path is important for degradation modeling and prediction.
The multi-phase degradation paths, called phase based path here, are also widely used. For the phase based paths, the degradation path is divided into several stages with different degradation rates. The change points and degradation rates are important factors that need to be evaluated. For instance, several methods have been proposed in the two phase degradation modeling to evaluate the degradation rates and change point. Bae et al. proposed hierarchical Bayesian models for two-phase degradation phenomena where a change point regression approach models the two-phase degradation paths [26]. The model parameters, including degradation rates and change point, are evaluated by the Gibbs sampling algorithms. Altun and Comert [27] have presented a change point detection method using forward and backward data analysis. Furthermore, the measurement errors are considered in Wiener process based degradation modeling in Ref. [28]. To consider the abrupt jump at change point, a Wiener process with an abrupt jump at change point is proposed in Ref. [29]. The expectation-maximization algorithm is applied to evaluate the change points. Furthermore, similar researches on phase based degradation paths can be found in Refs. [30], [31], [32], [33].
Besides the described phase based paths, several degradation paths with time-varying degradation rates have also been proposed. In Ref. [34], a new class of Wiener processes has been presented to consider the dependence between degradation mean and variance. Several common degradation paths with time-varying degradation rates are considered candidate paths, and the paths are compared using AIC. Wang and Wu [35] have studied a class of inverse Gaussian process models based on paths with different time-varying degradation rates and presented the corresponding maximum likelihood estimation. Three kinds of degradation rates have been discussed. Additional researches on degradation paths with changing degradation rates can also found in Refs. [36], [37], [38], [39], [40]. Furthermore, to meet the stochastic process to engineering practices, Ma et al. [41,42] have proposed a mechanism analysis based degradation path obtaining method in stochastic process based degradation modeling. The degradation path is determined by analyzing the failure mechanism of the hydraulic piston pump. The authors argue that it is a practical way to determine the degradation path based on failure mechanism analysis. Simultaneously, this kind of method is only suitable when the failure mechanism can be analyzed clearly.
As discussed above, there are multiple degradation paths, and the goodness-of-fit test is usually used to select the best degradation path among the candidate paths. However, almost all previously published research needs to determine path or candidate paths based on prior experience and knowledge before using the stochastic process to model the degradation process. Hence, the absence of previous information about the degradation path or candidate paths makes the application of the stochastic process in degradation modeling and prediction difficult. Unfortunately, the degradation path uncertainty issue is inevitable. The artificial neural network (ANN) has a strong ability in data fitting and relationship describing. ANN has been introduced in tribology to fit and describe the relationship between the wear rate and its affecting factors, including sliding speed and lubrication characteristics [43,44]. According to experimental results, the ANN supported models can predict the wear precisely. ANN has been applied in chemical engineering showing superiority in data fitting and relationship description [45,46]. Furthermore, in reliability engineering, the ANN has been used in fault diagnosis [47], reliability estimation [48], remaining useful life prediction [49], and health prognostics [50].
The ANN has strong capability in data fitting and path describing. However, it is hard to find ANN in degradation modeling, particularly in stochastic process based degradation modeling. Hence, to handle the degradation path uncertainty, ANN is used to fit the degradation path, and an ANN supported stochastic process is proposed in this paper. The main distinguishing features of the proposed method compared to the previously published studies are as follows. (1) The stochastic process is combined with ANN to handle degradation path uncertainty. (2) The degradation path is fitted based on ANN by training the ANN supported stochastic process to minimize the minus log-likelihood offline, based on a run-to-failure degradation dataset. (3) The process parameters are inferred by online Bayesian inference combined with offline moment estimation, based on real-time degradation data of predicting individual. It should be noted that, in this paper, ‘online’ means the state when the predicting individual is being used, and ‘offline’ means the state before or after use. Two improvements have been achieved in the presented research. First, the proposed method can model the degradation process even without prior information about the degradation path. Second, in previously published papers, the initial degradation is assumed to be zero, while this assumption is freed in the proposed ANN supported Wiener process.
This paper is organized as follows. In Section 2, the motivation of applying ANN in the stochastic process is declared, and the main methodology of the presented research is also described. Section 3 presents the proposed ANN supported stochastic process and corresponding inference method. Section 4 verifies the proposed ANN supported stochastic process and corresponding inference method by a simulation study. In Section 5, an actual degradation dataset is used to illustrate the effectiveness of the proposed ANN supported stochastic process and corresponding inference method in actual engineering practices. Section 6 concludes the presented research, and a possible topic for future work is also discussed.
Section snippets
Motivation
A degradation process, {Y(t)| t≥0}, which can be described as a stochastic process, has the following properties:
(a) Y(t) has independent increments, e.g. Y(t4) - Y(t3) and Y(t2) - Y(t1) are independent only if t4> t3>t2>t1.
(b) The degradation increments ΔY(t)=Y(t+Δt) - Y(t) can be given by Eq.(1).where f is a probability density function (PDF) with parameter vector θs. Λ(t) is a monotonic increasing function and ΔΛ(t)=Λ(t+Δt)-Λ(t). Furthermore, in the presented research, the
Weiner process with random effects
In the presented research, the Wiener process is presented as an example due to its advantage in degradation modeling for both monotonic and non-monotonic degradation processes. The Wiener degradation process has the following properties [12].
(a) Y(t) has independent increments, e.g. Y(t4) - Y(t3) and Y(t2) - Y(t1) are independent only if t4> t3>t2>t1.
(b) The degradation increments ΔY(t)=Y(t+Δt) - Y(t) are given by Eq.(2).where N(•) means Gaussian distribution, and θS=(μ,λ
Simulation experiment
In this section, a simulation experiment is conducted to validate the proposed ANN supported degradation path and parameter estimation algorithm.
Case study on spindle systems
An actual degradation dataset of machining accuracy of spindle systems is used to illustrate the proposed ANN-supported Wiener process and its corresponding method in degradation modeling and prediction [38]. There are five samples in the degradation dataset, as presented in Fig. 11 and Table 7.
Conclusions
An ANN supported stochastic process, corresponding degradation modeling, and prediction methods are proposed. The stochastic process is combined with ANN to handle degradation path uncertainty. Furthermore, the assumption that the initial degradation is zero is freed. The degradation path is fitted based on ANN by training the proposed ANN supported stochastic process to minimize the minus log-likelihood offline. Three previously used distributions describe the process parameters, including
CRediT authorship contribution statement
Di Liu: Conceptualization, Methodology, Software, Investigation, Visualization, Writing – original draft. Shaoping Wang: Resources, Project administration, Writing – review & editing.
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
This study was co-supported by the National Natural Science Foundation of China (51620105010, 51575019), Natural Science Foundation of Beijing Municipality (17L1003) and China Postdoctoral Science Foundation (2020M680289).
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