An integrated degradation modeling framework considering model uncertainty and calibration

Highlights

• An integrated degradation modeling framework based on the wavelet density estimation is proposed.

• Assumption of zero mean normally distributed errors of general path models is released.

• Model uncertainty for stochastic process models is dealt with.

• The effectiveness and feasibility of the proposed method are illustrated through a numerical example and a case study.

Abstract

General path models and stochastic process models are two widely applied categories of probabilistic degradation models. The former explains the randomness of degradation data as normal distributed errors with zero mean. The latter describes degradation with stochastic processes such as Wiener process, Gamma process and Inverse Gaussian process. For general path models, a limitation is the assumption of normally distributed errors. For stochastic process models, model uncertainty with respect to the available stochastic processes should be considered, but the widely-applied model selection methods in consideration of model uncertainty are unable to warn when all the candidate models fit data poorly. Therefore, an integrated degradation modeling framework based on wavelet density estimation is proposed, which can calibrate the distribution of errors for general path models and deal with model uncertainty for stochastic process models. The proposed framework can select the best stochastic process if certain stochastic processes fit the degradation data well. Otherwise, all the candidate stochastic processes can be calibrated, which overcomes the drawback of model selection methods. The effectiveness and feasibility of the proposed framework are illustrated through a case study and a numerical example.

Introduction

For the reliability analysis of the high-quality products, the sparsity of failure data leads to the wide use of degradation data. In degradation data analysis, accurate degradation models are critical [1], for which probability theory is often applied to deal with the uncertainty in data. Because the observations are always imprecise and disturbed by measurement noise, the probabilistic degradation models are developed to describe the randomness of the degradation path, which contain deterministic part and stochastic part. The deterministic part describes the trend of a degradation process which is obtained from the physics-based method and data-driven method. The stochastic part describes the uncertainty in the data. As described in [2] and summarized in [3], two widely applied categories of probabilistic degradation models are general path models and stochastic process models. The general path models describe the degradation path with a real value function called general path and explain the randomness of obtained data as errors caused by imperfect measurements [4]. In general path models, the general path can be determined from mechanism analysis, engineering experience or the data-driven method. Lu and Meeker [4] proposed a general path model for fatigue crack growth data based on the Paris law. The point estimation and pointwise confidence intervals of the product reliability are calculated by Monte Carlo simulation and bootstrap method. Bae and Kvam [5] proposed a nonlinear general path model for the degradation data of vacuum fluorescent displays, in which the statistical inference was given based on adaptive importance sampling approximation. Zhou et al. [6] proposed a non-parametric general path model based on the cubic spline bases. However, all the above researches described the errors between degradation observations and general paths with a normal distribution, which seems arbitrary and inappropriate in practice. On one hand, the complexity and irregularity of working condition of monitoring equipment may cause the measurement errors non-Gaussian [7]. On the other hand, even though degradation measurement errors follow normal distribution, the errors between the measurements and the values of general path functions may not be the same because the general path function is only a simplified mathematical approximation of the true degradation path. Inappropriate assumption will lead to inaccurate general path model, therefore, it is expected that the distribution of the errors can be calibrated according to the degradation data when the normal distribution is inappropriate. In addition, note that another type of probabilistic models similar to the general path models is Gaussian process regression model. Gaussian process regression is a non-parametric Bayesian machine learning method which can provide not only a regression curves (with the posterior mean of the Gaussian process) but also the corresponding uncertainty estimation (with the posterior variance of the Gaussian process). However, in contrast to the general path models, Gaussian process regression fit the degradation data by mapping the input into a feature space with dimension equal to the number of data points, which may lead to overfitting with limited data. On the other hand, the Gaussian process regression requires Gaussian assumption of measurement error to obtain the closed-form posterior probability density function (PDF).

In stochastic process model, stochastic process such as Wiener process, Gamma process and Inverse Gaussian (IG) process are applied to make prediction based on the statistical inference of the degradation independent increments. The unknown parameters of the stochastic process are estimated based on the degradation data, and the trend and randomness of the degradation processes can be both captured in the models. Among the above three kinds of stochastic models, the Wiener process has been used intensively [8]. Whitmore and Schenkelberg [9] proposed a Wiener process to fit the accelerated degradation data of self-regulating heating cable, in which the effects of time transformation were considered. Si et al. [10] proposed a Wiener-process-based degradation model to describe degradation of the gyros in an inertial navigation system, and the unknown parameters are updated with the recursive filter algorithm and expectation maximization algorithm. The Wiener process was also applied in the analysis of the head wear data of hard disk drives by Wang et al. [11]. In contrast to the Wiener process, the Gamma process and the IG process are mainly utilized for the monotonic degradation process. Lawless and Crowder [12] proposed a Gamma process with covariates and random effects for the crack growth data fitting. Ye and Chen [13] explained the physical meanings of IG process and applied it to fit the GaAs laser degradation dataset. Wang and Xu [14] proposed the IG process with covariates and random effects for degradation data analysis, in which the Expectation Maximization algorithm was proposed to calculate the maximum likelihood estimates of the unknown parameters. Note that the problem of model uncertainty arises in the application of stochastic process models [15], which refers to the uncertainty involved in the determination of the best stochastic process among several candidates [16], especially when their statistical inferences are different from each other. The widely applied method to deal with model uncertainty is to establish a model set consisting of all the candidates such as Wiener process, Gamma process and IG process. Then, the model selection methods such as AIC [17] and Bayes model selection [18] are utilized to assess all the candidate models and guide their selection. For example, the AIC method was applied to compare Wiener process with Gamma process in [19], [20]. However, an inherent drawback of the model selection method is its disability of warning when all the candidate models fit the degradation data poorly. To overcome this issue, the idea of model averaging is considered such as the Bayesian model averaging method applied in [21]. But as illustrated in [22], the essence of the Bayesian model averaging method is still model selection which cannot overcome the inherent drawback. Therefore, a new method to deal with model uncertainty is expected which can select the best model when certain candidate models can well fit the degradation data, or calibrate the candidate models when all of them fit the degradation data poorly.

Based on the above motivations, an integrated degradation modeling framework considering model calibration and model uncertainty is proposed in this paper. For general path models, the proposed framework can relax the assumption of normally distributed errors and calibrate their distribution according to the data. For stochastic process models, the proposed framework can deal with model uncertainty by selecting the best stochastic process from several candidates. Furthermore, the candidate models can be calibrated in this framework if all of them fit the data poorly, which overcomes the above-mentioned drawback of model selection methods such as AIC and Bayesian model selection. The proposed framework is based on wavelet density estimation (WDE), which is a non-parametric density estimation method approximating the unknown data-generating distribution by a series of wavelet basis functions. WDE can approximate all the PDFs belonging to $L^2\left(R\right)$ but the risk of overfitting has to be reduced simultaneously. Therefore, in the proposed model calibration module, the parametric models are compared with the wavelet density estimator and the statistical inference is obtained by the calibrated model rather than WDE. Based on the proposed model calibration module, the zero-mean normal distribution of errors in general path model can be calibrated as follows: First, the PDF of the original distribution is represented as wavelet series expansion based on the discrete wavelet transformation. Second, a wavelet density estimator is obtained from the corresponding degradation data based on WDE. Third, the wavelet series expansion and the wavelet density estimator are compared with each other. Note that the wavelet series expansion and the wavelet density estimator are both vectors in the subspace of the square-integrable function space, $L^2\left(R\right)$, so that the comparison of two PDFs can be simplified into the comparison of two corresponding vectors. Then, because of the standard orthogonality of the wavelet basis, this comparison can be further simplified into the comparison of scaling coefficients and wavelet coefficients. Finally, the scaling coefficient or wavelet coefficient in the wavelet series expansion which has the maximum absolute difference from that in the wavelet density estimator is replaced by the latter for model calibration. Finally, the calibrated wavelet series expansion is transformed back into the calibrated PDF. To deal with the model uncertainty with respect to three candidate stochastic process models (Wiener process, Gamma process, and IG process), the above-mentioned first step and second step are carried out to obtain three wavelet series expansions of the candidate stochastic processes and a wavelet density estimator of the degradation increments. Then, the norms between the wavelet series expansions and the wavelet density estimator are calculated to measure the similarities of PDFs. And an index is proposed, if certain indexes are smaller than a predefined threshold, the stochastic process with the smallest norm will be selected. Otherwise, all the stochastic processes will be calibrated according to the above-mentioned third and final steps.

In the proposed framework, a new degradation model will be developed with better goodness of fitting owing to the consideration of model calibration and model uncertainty. The reasonability of model calibration based on WDE can be concluded as follows: (1) as a non-parametric density estimator, the wavelet density estimator can approximate all the PDFs belonging to $L^2\left(R\right)$ rather than the PDFs belonging to a certain parametric family, which enables it to fit the data better and calibrate the parametric models by comparison; (2) based on the standard orthogonality of the wavelet basis, the comparison of two PDFs can be simplified into the comparison of scaling coefficients and wavelet coefficients, which reduces the complexity in the application; (3) compared with other orthogonal basis such as sine and cosine functions, wavelet bases have local properties in space as well as in frequency. The location properties enable WDE to do better in describing local oscillations, and enable the wavelet series to represent a function in a sparse way [23], which reduces the computing costs. Besides, the selected compactly supported wavelet makes the multiple integral in discrete wavelet transformation practicable; (4) for different degradation datasets, the variability of available wavelet functions guarantees that their accurate density estimation can be achieved. In reliability engineering, Antoniadis et al. [24] proposed the WDE method in hazard rate estimation for the right-censored data. To the best knowledge of the authors, the WDE has not been applied in the degradation data analysis. Besides, The proposed integrated framework can reduce the error caused by model misspecification Model misspecification refers to the difference between the actual data generating process and the best model in hypothesis space [25]. In degradation data analysis, the actual data generating process is unknown and the hypothesis space is the parametric family of the employed degradation models, of which multiple candidates are available. Under this condition, the proposed model calibration module and the module dealing with model uncertainty can reduce the error caused by model misspecification. Because the module dealing with model uncertainty can select the best candidate hypothesis space according to the data rather than choosing one arbitrarily, and the model calibration module can extend the original hypothesis spaces to make them more likely to contain the actual data generation process.

The contribution of the proposed framework is two-folds. First, it is the first-time application of the WDE method in degradation modeling, which can calibrate the developed general path model and increase their accuracy. Second, an innovative method considering model uncertainty is proposed to select the best model when certain candidate models well fit the degradation data, or calibrate the candidate models when all of them fit the degradation data poorly, which accomplishes the model selection and overcomes the drawback of other model selection methods. The contributions are illustrated with a case study and a numerical example.

The rest of this paper is organized as follows. In Section 2, the general path models and stochastic process models are introduced. In Section 3, the WDE is introduced. Based on WDE, the proposed integrated framework considering model calibration and model uncertainty is presented. In Section 4, the contributions of the proposed framework are illustrated through a case study and a numerical example. Section 5 concludes the work and future directions.