A random-effects Wiener degradation model based on accelerated failure time

doi:10.1016/j.ress.2018.07.003

Reliability Engineering & System Safety

Volume 180, December 2018, Pages 94-103

https://doi.org/10.1016/j.ress.2018.07.003 Get rights and content

Highlights

•
A Wiener process model based on random acceleration effect is proposed.
•
The IG distribution is used to model the heterogeneous degradation rate.
•
The model is extended to constant-stress ADT analysis.
•
EM algorithm is developed for model parameter estimation.

Abstract

Due to the variability of raw materials and the fluctuation in the manufacturing process, degradation of products may exhibit unit-to-unit variability in a population. The heterogeneous degradation rates can be viewed as random effects, which are often modeled by a normal distribution. Despite of its mathematical convenience, the normal distribution has certain limitations in modeling the random effects. In this study, we propose a novel random-effects Wiener process model based on ideas from accelerated failure time principle. An inverse Gaussian (IG) distribution can be used to characterize the unit-specific heterogeneity in degradation paths, which overcomes the disadvantages of the traditional models and provides more flexibility in the degradation modeling using Wiener processes. Properties of the model are investigated, and statistical inference based on the maximum likelihood estimation and the EM algorithm is established. An extension of the model to the constant-stress accelerated degradation test (ADT) is developed. The effectiveness and applicability of the proposed model are validated using a laser degradation dataset and an LED ADT dataset.

Introduction

The Wiener process is one of the most popular degradation models in recent decades [5], [8], [20], [26], [30]. Denote the degradation of a quality characteristic by X(t), e.g., the capacity of a battery cell, the following Wiener process is often used to model its evolution with time: $X (t) = v t + σ B (t) .$ In this basic Wiener process model, $B (t)$ is the standard Brownian motion, the drift rate v > 0 corresponds to the mean rate of the degradation process, and the diffusion coefficient σ > 0 quantifies the magnitude of the process fluctuation.

The basic Wiener process model assumes all the units in a population have the same drift rate v and diffusion coefficient σ. However, this assumption has certain limitations in many practical applications. Particularly, units from a same batch, due to the variability of raw materials and the fluctuation in the manufacturing process, probably exhibit heterogeneities in their degradation paths [2], [17]. When using the basic Wiener process to analyze such degradation data, the heterogeneities among individuals are neglected, which may lead to inaccurate statistical inference. To capture the heterogeneities among units, random-effects Wiener process models, which allow some parameters of the degradation process to be unit-specific and follow certain distributions across the population, have been proved to be useful [3], [21], [24], [28], [29]. In this context, most studies assume that the diffusion coefficient σ is invariant while the drift rate v is normally distributed for the units in a population [3], [21], [29]. Some studies also take the heterogeneity of the diffusion coefficient into account. Wang [24] allowed both drift and diffusion to be random, and assumed the diffusion σ² follows an inverse Gamma distribution and v follows a Gaussian distribution conditional on σ. Ye et al. [28] assumed that σ is a linear function of v and assigned a Gaussian distribution for the reciprocal of the drift rate.

The Gaussian distribution for the heterogeneous drift, although widely adopted, is restrictive in the following aspects. First, the Gaussian distribution has a support over the whole real line, which is often conflicted with the practical applications where the drift rate is believed to be one-sided, i.e., positive or negative. To address this incompatibility, many studies assume that the negative part of the Gaussian distribution is negligible, e.g., [28]. However, when the mean of v is not significantly larger than the standard deviation of v, the probability of a negative realization of v may not be neglected. Some studies truncated the Gaussian distribution [22], but it makes the model much more complicated. Second, the Gaussian distribution has a symmetric, bell-shaped probability density function (PDF), which is restrictive in modeling unit-to-unit heterogeneity. For example, we examine the degradation data of 15 laser devices from Meeker and Escobar [11], and fit each degradation path individually by a basic Wiener process in Eq. (1). As shown in Fig. 1, a skewed distribution appears to be more suitable to the unit-specific drift rates of the laser devices. Therefore, the performance of the Wiener process model with Gaussian drift may not be satisfactory due to these restrictions.

To overcome the deficiency, this paper proposes a novel random-effects Wiener process model exploiting the ideas in accelerated test modeling. In accelerated life/degradation tests (ALT/ADT), two or more groups of units are tested in harsh operation stresses to accelerate the failure/degradation process. Thus, sufficient reliability information can be observed which otherwise may be difficult to obtain under normal use conditions within practical test duration [4], [15], [23]. Since the reliability in normal stress is concerned, it is of vital importance to extrapolate the model in higher stress levels to the use stress level. The accelerated failure time (AFT) is a widely accepted concept, where the degradation path/lifetime under accelerated conditions is equivalent to the degradation path/lifetime that the same unit would have had under the normal use condition after properly scaling the time axis by the acceleration factor [12], [13]. Therefore, the degradation paths/lifetimes under different stress levels are comparable after transformed to a same baseline stress level.

Inspired by this idea, the heterogeneity in the degradation rate can be viewed as a random acceleration phenomenon caused by unobserved random effects, which can be modeled by random scaling of the time axis. In this context, an inverse Gaussian (IG) distribution can be used to characterize the random effects, and a novel random-effects Wiener process model is developed. As a model for random effects, the IG assumption naturally accommodates the practical requirement that the drift rate should be positive. Moreover, it promises a wide applicability of the Wiener process model with IG random effects as the shape of the IG distribution is quite flexible [19]. Because the proposed model inherits the ideas in accelerated testing, it is also convenient to extend the proposed model to analyze the ADT data.

The remainder of the paper is organized as follows. Section 2 presents the detailed model formulation and the relevant properties. Section 3 develops an EM algorithm for the maximum likelihood (ML) estimation of the proposed model. In Section 4, we extend the proposed model to constant-stress ADT modeling. The application of the proposed model to real degradation data is illustrated in Section 5. Conclusions are given in Section 6.

Section snippets

Model formulation

In this study, we consider the following random-effects Wiener process model $X (t) = v Λ (t) + κ B (v Λ (t)),$ where v > 0 is the unit-specific drift rate, κ > 0 is a diffusion parameter, and $B (\cdot)$ is the standard Brownian motion. $Λ (t) = Λ (t; θ)$ is a determined function with parameter θ to capture possible non-linear degradation patterns [18], [26]. For example, Λ(t) can be linear $Λ (t) = t$ or of the power law form $Λ (t) = t^{θ}$ . Generally, Λ(t) is specified according to degradation physics or empirical observations [29]

Point estimation

We first assume the parameters θ in Λ( · ) are determined and focus on the estimation of Θ. For the proposed model, we consider the unobservable drift parameter v_i for each unit as missing data. Denote $V = {(v_{1}, \dots, v_{n})}^{T}$ . Then, we can obtain the following log-likelihood function for the complete data {V, X} $ℓ_{c} (Θ | V, X) = ℓ_{v} + ℓ_{X},$ where $ℓ_{v} = - \frac{n}{2} \ln (2 π) + \frac{n}{2} \ln ζ - \frac{3}{2} \sum_{i = 1}^{n} \ln v_{i} - \sum_{i = 1}^{n} \frac{ζ {(v_{i} - μ)}^{2}}{2 μ^{2} v_{i}},$ and $ℓ_{X} = \sum_{i = 1}^{n} {- \frac{m_{i}}{2} \ln (2 π) - \frac{m_{i}}{2} \ln κ^{2} v_{i} - \frac{1}{2} \sum_{j = 1}^{m_{i}} \ln Δ Λ_{i, j} - \frac{1}{2 κ^{2} v_{i}} \sum_{j = 1}^{m_{i}} \frac{{(Δ X_{i, j} - v_{i} Δ Λ_{i, j})}^{2}}{Δ Λ_{i, j}}} .$

With the complete-data log-likelihood, the

Extension to ADT data

Consider a constant-stress ADT under K stress levels $ξ_{1}, \dots, ξ_{K}$ . Suppose n_k units are allocated in stress ξ_k for degradation testing. The degradation in each stress level can be modeled by the proposed Wiener process model with IG drift. From the AFT point of view, we assume the degradation of an unit in the kth stress level can be modeled as $X_{k, i} (t) = μ \cdot r_{k, i} h (ξ_{k}) Λ (t) + σ B (r_{k, i} h (ξ_{k}) Λ (t)) .$ Here, h(ξ_k) is a determined acceleration factor that depends on the acceleration variables. In such a formulation,

Laser degradation data

To illustrate the application of the proposed model, we apply it to analyze the GaAs laser degradation data from Meeker and Escobar [11]. The operating current of the laser device increases over time, and the laser device would fail when the operating current exceeds a specified threshold. The dataset consists of the degradation data of 15 testing samples, where each unit is measured at times ${250, 500, \dots, 4000}$ . The increase of the operating current of the 15 units are illustrated in Fig. 2.

From

Conclusions

In this study, we proposed a novel random-effects Wiener process model for degradation data based on linear accelerated failure time principle. The new Wiener process model exploits an IG distribution to describe the heterogeneous degradation rates in the population and has the diffusion coefficient dependent on the drift rate. Compared with the existing models, the proposed Wiener process model with IG drift has the following features:

•
The IG distribution has a support on $(0, + \infty)$ and a flexible

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View full text

A random-effects Wiener degradation model based on accelerated failure time

Highlights

Abstract

Introduction

Section snippets

Model formulation

Point estimation

Extension to ADT data

Laser degradation data

Conclusions

Reliab Eng Syst Saf

Eur J Oper Res

IEEE Trans Reliab

IEEE Trans Reliab

IEEE Trans Reliab

Technometrics

IEEE Trans Reliab

Handbook of mathematical functions with formulas, graphs, and mathematical tables, chap. 9.6 Modified BesselFunctions I and K

A degradation-based model for joint optimization of burn-in, quality inspection, and maintenance: a light display device application

Int J Adv ManufTechnol

Remaining useful life prediction for a nonlinear heterogeneous wiener process model with an adaptive drift

IEEE Trans Reliab

Statistical properties of the generalized inverse gaussian distribution

A cumulative-exposure-based algorithm for failure data from a load-sharing system

IEEE Trans Reliab

Reliability inference for field conditions from accelerated degradation testing

Nav Res Logist

Optimal design of accelerated degradation tests based on wiener process models

J Appl Stat

Finding the observed information matrix when using the em algorithm

J R Stat Soc Ser B (Methodological)