Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 5246108, 13 pages

http://dx.doi.org/10.1155/2016/5246108

## Accelerated Degradation Process Analysis Based on the Nonlinear Wiener Process with Covariates and Random Effects

^{1}Nanjing University of Science and Technology, Nanjing, Jiangsu 210094, China^{2}Science and Technology on Combustion and Explosion Laboratory, Xi’an, Shanxi 710065, China

Received 13 September 2016; Accepted 27 November 2016

Academic Editor: Eusebio Valero

Copyright © 2016 Li Sun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

It is assumed that the drift parameter is dependent on the acceleration variables and the diffusion coefficient remains the same across the whole accelerated degradation test (ADT) in most of the literature based on Wiener process. However, the diffusion coefficient variation would also become obvious in some applications with the stress increasing. Aiming at the phenomenon, the paper concludes that both the drift parameter and the diffusion parameter depend on stress variables based on the invariance principle of failure mechanism and Nelson assumption. Accordingly, constant stress accelerated degradation process (CSADP) and step stress accelerated degradation process (SSADP) with random effects are modeled. The unknown parameters in the established model are estimated based on the property of degradation and degradation increment, separately for CASDT and SSADT, by the maximum likelihood estimation approach with measurement error. In addition, the simulation steps of accelerated degradation data are provided and simulated step stress accelerated degradation data is designed to validate the proposed model compared to other models. Finally, a case study of CSADT is conducted to demonstrate the benefits of our model in the practical engineering.

#### 1. Introduction

For many highly reliable products, it is not an easy task to obtain their life information by using traditional life test because failures are not likely to occur in a certain period of time, even by censoring life test and accelerated life test. In such a case, degradation data which is related to life is used due to the following reasons: ease of obtaining, low cost, short test period, and informative data. And it has been widely used in classification [1], residual life estimation [2], reliability assessment [3], and so on. To model the degradation data, two classes of models have been well exploited, general path model and stochastic process model [4, 5]. The general path model is first introduced by Lu and Meeker in 1993 [6] whose failure time is determined with known random parameters. But it may not be good at describing the inherent randomness of each product and the unexplained randomness and dynamics due to unobserved environmental factors. There are various stochastic process models, including Wiener process [7], Gamma process [8], Geometric Brownian Motion Process [9], and Inverse Gaussian Process [10, 11]. Beyond all of the stochastic process models, Wiener process has been used intensively for its flexible and meaningful characteristic. In addition, Wiener process has more advantages than other stochastic process models for nonmonotonic degradation data.

Most of the degradation data mentioned above is degradation data under normal stress or field degradation data. However, the life information should be obtained in a shorter period of time for some products, especially for newly developed products and highly reliable components. Instead, it is a lengthy and drawn-out process to collect field degradation data. Under the circumstances, ADT is a suitable choice to gather the life information quickly and efficiently.

In general, with more accelerated degradation data and higher measuring precision, we can achieve higher accuracy for forecasting parameters, but the experiment cost would increase correspondingly. So we can deal with the optimal accelerated degradation plan (including the optimal settings for the sample size, accelerated stresses, measurement frequency, and termination time) for a Wiener degradation process by minimizing the approximate variance of the estimated mean time to failure under the constraint that the total experimental cost does not exceed a prespecified budget or minimizing the testing cost under the condition of a maximum acceptable approximate standard error. Some well-known references on the optimization of CSADT based on Wiener process are Lim and Yum [5] and Tsai et al. [12]. The optimization of SSADT based on Wiener process can be referred to in the research of Liao and Tseng [13], Tang et al. [14], and Hu et al. [15]. In addition, accelerated degradation model is another hot area which has attracted much attention of the researchers. Liao and Tseng [13] modeled the step stress accelerated degradation data of LED lamps. Considering unit-to-unit variability, a step stress accelerated degradation model based on the basic Wiener process was proposed by Tang [16]. However, it is often found that the degradation path is not always linear. So an accelerated degradation process modeling method with random effects for the nonlinear Wiener process was established by Tang et al. [17] later. Wang et al. [18] proposed a Bayesian evaluation method to integrate the ADT data from laboratory with the failure data from field.

The above literature all assumed that the drift parameter is dependent on the acceleration variables and the diffusion coefficient remains the same across the whole ADT. But when the degradation rate increases, the degradation variation would also become larger in some applications [19]. WHITMORE [20] fitted the degradation data of each product separately with a time scale transformed Wiener process and then the parameter transformation was tentatively identified based on the plots against the reciprocal of the absolute temperature. The plots revealed that both the drift parameter and the diffusion parameter of self-regulating heating cable are increasing with the increment of temperature. Doksum and Hoyland [21] introduced the conception of the multiplicative factor, and it was assumed that the drift parameter and diffusion parameter are multiples of the multiplicative factor whose expression with accelerated stress level provides a good model fit of one of the four empirical models. Liao and Elsayed [22] extended an accelerated degradation model to predict field reliability by considering the stress variations where the drift parameter and the diffusion parameter are expressed by the different function of the constant stress vector. Ye [19] proposed a new random effects Wiener process model such that the drift parameter is a particular multiple of the diffusion parameter and the unknown parameters were calculated by EM algorithm. The relationship between the drift parameter and diffusion parameter was either an assumption or just the fitting according to specific test data.

The paper is motivated by the latest paper of Wang et al. [23] in which they deduced that the ratio of drift parameters under two different stresses is equal to the acceleration factor, as well as the ratio of diffusion parameters. Based on this conclusion, we model the constant stress accelerated degradation process (CSADP) and the step stress accelerated degradation process (SSADP) in consideration of random effects. Moreover, the unknown parameters in the model, including measurement error, are obtained by using the maximum likelihood estimation (MLE) method. Besides, a numerical example and a case study are presented to verify the superiority of the model proposed in this paper compared with other two models.

The remainder of this paper is organized as follows: Section 2 develops the nonlinear Wiener process and deduces the relationships of parameters in ADT and the probability density function (PDF) and cumulative distribution function (CDF) under a certain stress with random effects. Section 3 models the degradation process in CSADT and SSADT. Section 4 describes the procedure for parameter estimation for two cases. Two numerical examples and a practical example are presented to verify the proposed model in Sections 5 and 6, separately. Section 7 concludes the paper with a discussion.

#### 2. Nonlinear Wiener Process with Covariates and Random Effects

##### 2.1. The Wiener Process with Time Scale Transformation

The time-transformed Wiener process is commonly used to model the nonlinear accelerated degradation data [17]. Let denote the degradation value at time ; then the Wiener degradation process with time scale transformation can be represented as follows [20]:where is the drift parameter, is the diffusion parameter, denotes the clock or calendar time, and is the transformed time whose selection can be referred to in Section 6 of literature [24]. is the standard Brownian motion which represents the stochastic dynamics of the degradation process at transformed time scale. If , the nonlinear Wiener process becomes the traditional Wiener process [17]. Generally, if reaches a specific value which is related to the failure mechanism in most cases for the first time, the product is announced to be failed and the time is thus called the first hitting time (FHT). Given , , and , it is known that the transformed FHT in such a case follows an inverse Gaussian distribution [11], with corresponding PDF and CDF aswhere denotes a standard normal distribution function.

##### 2.2. Deducing the Relationship of Parameters in ADT Based on Nonlinear Wiener Process

ADT is a method to accelerate the degradation of products by elevating stress, and the obtained degradation data are then used to extrapolate the information through accelerating model to obtain the estimates of life or performance of products at normal use condition. To ensure the accuracy of the extrapolation, the failure mechanism under the accelerated stress and the normal stress must keep the same which is also the premise of the ADT. One of the most common methods for consistency inspection of the failure mechanism is based on statistical method [25]. The principle of this method is that the acceleration factor is a constant and independent of testing time if the failure mechanism remains unchanged. The definition of the acceleration factor is given below according to the Nelson assumption [26].

Specify and suppose represents the predetermined cumulative failure probability. is defined as the testing time when the accumulated failure probability comes to under normal stress , as well as under accelerated stress .

Then the acceleration factor of stress relative to stress can be defined as

The expression can be obtained from (5) and plug it into (4). Then take the first-order derivative with respect to and we have the following equation for any :

The expression of and can be deducted according to (2); then

The acceleration factor is a constant that does not change with if and only if the relationship of the parameters is satisfied with Instead of the hypothesis that the diffusion parameter is a constant and does not change with the stress, the conclusion that both the drift parameter and the diffusion parameter depend on stress variables could be drawn based on the previous derivation. At the same time, it was testified that a unit with high realization of the drift parameter would possess a high degradation rate and a high variation in the degradation path in theory which is in conformity with the viewpoint of [19]. The relationship between parameters and accelerated stress variables can be set up by acceleration models based on engineering background. The frequently used acceleration model includes the Arrhenius model, the inverse power model, and the Eyring model whose expressions and acceleration factors are listed in Table 1. Specify , , and to the three models, separately. The accelerated model of drift parameter , the diffusion parameter , and the accelerated factor can be uniformly written aswhere is the abbreviated form of for simplification of the expressions.