# A General Accelerated Degradation Model Based on the Wiener Process

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The General ADT Model

#### 2.1. Models

**S**;

**η**), it is assumed to be dependent with the acceleration variable

**S**as the acceleration model. The acceleration model for both single and multiple acceleration variables is denoted as:

**S**= [s

_{1}, s

_{2}, …, s

_{p}] and

**η**= [η

_{0}, η

_{1}, …, η

_{p}], where s

_{v}is the vth acceleration stress type and η

_{v}is the vth constant coefficient. While φ(.) is the continuous function of acceleration variables s

_{i}, i = 1, 2, …, p. For instance, if φ(s) = exp(1/s), Equation (2) presents the Arrhenius relationship (exponential type); while if φ(s) = s, Equation (2) presents the Eyring relationship (power rule type).

**S**;

**η**) are varied from product to product. Hence, we consider it as a random variable to present this kind of variation. Similar methods can be found in Peng and Tseng [22], Wang [23], and Si, et al. [24]. Here, for simplicity, the coefficient η

_{0}in Equation (2) is assumed to follow a normal distribution with mean value a and variance b, i.e., η

_{0}~ N(a, b). The parameter values should be such that Pr(μ(

**S**;

**η**) < 0) is nearly zero to avoid negative values existing in the drift coefficient of Equation (1). Thus, if such a random effect is not considered (b = 0), Equation (2) will become the traditional acceleration model in Park and Padgett [25].

**S**;

**η**)z + σB(z).

**S**;

**η**) ~ N(μ

_{0}, ${\sigma}_{0}^{2}$), Equation (1) and its limiting cases (i.e., Equations (3) and (4)) are the degradation models used in traditional degradation analysis [19,20,28].

_{0}, while Equations (3) and (4) are M

_{1}and M

_{2}. Since model M

_{1}and M

_{2}are widely used in the ADT field, in this paper, we concentrate on the comparison of M

_{1}and M

_{2}with M

_{0}to verify the effectiveness of the proposed model for both linear and nonlinear ADT analyses.

#### 2.2. Derivation of the Failure Time Distribution under the Given Stress Level

**S**

_{0}= [${s}_{1}^{\left(0\right)}$, ${s}_{2}^{\left(0\right)}$,⋯, ${s}_{p}^{\left(0\right)}$], where ${s}_{v}^{\left(0\right)}$ is the vth acceleration stress type under normal level. Thus, the drift coefficient μ(

**S**

_{0};

**η**) follows a normal distribution. Therefore, we simplify the notation of μ(

**S**

_{0};

**η**) to μ in the derivation of the failure time distribution. From Equation (2), it is known that μ ~ N(μ

_{0}, ${\sigma}_{0}^{2}$), where ${\mu}_{0}=a\cdot {\prod}_{v=1}^{p}{\left[\varphi \right({s}_{v}^{\left(0\right)}\left)\right]}^{{\eta}_{v}}$ and ${\sigma}_{0}^{2}=b\cdot {\prod}_{v=1}^{p}{\left[\varphi \right({s}_{v}^{\left(0\right)}\left)\right]}^{2{\eta}_{v}}$.

^{−1}(z;γ). We define ρ(z;θ) = Λ(τ

^{−1}(z;γ);θ). So, Equation (1) becomes [20]:

**Theorem**

**1.**

_{0}, ${\sigma}_{0}^{2}$), and ω, A, B, C∈

**R**, then:

^{2}(z;θ)${\sigma}_{0}^{2}$ + σ

^{2}z.

^{2}(t;θ)${\mathsf{\sigma}}_{0}^{2}$+ σ

^{2}τ(t;γ); G = Λ(t;θ) − h(τ(t;γ);θ)τ(t;γ).

^{θ}and τ(t;γ) = t

^{γ}, Equation (12) becomes:

_{T}(t)dt = 1 should be satisfied. Therefore, the PDF and cumulative distribution function (CDF) of the FPT for model M

_{0}are modified as:

_{1}, i.e., Λ(t;θ) = t and τ(t;γ) = t, the PDF of the FPT for M

_{1}is known as an inverse Gaussian distribution [29]. Considering the random effect, the PDF and CDF of the FPT are [22,28]:

_{T}(t)dt = 1.

_{2}, i.e., Λ(t;θ) = τ(t;γ), the PDF of the FPT for M

_{2}is in accordance with Equation (15) through a time-scale transformation by replacing t into Λ(t;θ) or τ(t;γ) [18,30], which is also a limiting case of Equation (14).

## 3. Statistical Inference

**S**

_{1},

**S**

_{2}and

**S**

_{3}, while, on the right, all samples are in the same group and tested from the lowest stress level to the highest in a step-by-step manner, i.e.,

**S**

_{1}→

**S**

_{2}→

**S**

_{3}. Thus, the advantage of SSADT is that it can save the number of samples with a shorter test time [31].

_{v}], v = 1, 2, …, p. The analytic expressions of those parameters are hard to obtain directly. Hence, a two-stage maximum likelihood estimation (MLE) method is proposed to address this issue. In the first stage, the parameters related to the degradation process model are estimated, i.e., Ω

_{1}= [θ, γ, σ] in Equation (1). In the second stage, the rest of the parameters related to the acceleration model, i.e., Ω

_{2}= [a, b, η

_{v}] in Equation (2), are given accordingly.

#### 3.1. Estimation of Ω_{1} for CSADT

_{ijk}is the kth degradation value of unit j under the ith stress level and t

_{ijk}is the corresponding measurement time, i = 1, 2, …, K, j = 1, 2, …, n

_{i}, k = 1, 2, …, m

_{ij}, where K is the number of stress levels, n

_{i}is the number of test samples under the ith stress level, and m

_{ij}is the number of measurements for unit j under the ith stress level. Let

**X**

_{ij}= (x

_{ij}

_{1}, x

_{ij}

_{2}, …, x

_{ijm}

_{ij})

^{′}and

**t**

_{ij}= (Λ(t

_{ij}

_{1};θ), Λ(t

_{ij}

_{2};θ), …, Λ(t

_{ijm}

_{ij};θ))

^{′}. According to the properties of the Wiener process, the degradation value

**X**

_{ij}follows a multivariate normal distribution:

_{ij}is the drift coefficient of unit j under the ith stress level:

#### 3.2. Estimation of Ω_{2} for CSADT

_{2}is related to the acceleration model in Equation (2). From Section 3.1, the estimates of the drift coefficients ${\mu}_{ij}$ for unit j under the ith stress level are given and the corresponding stresses are ${s}_{v}^{\left(i\right)}$, i = 1, 2, …, K, j = 1, 2, …, n

_{i}. With the consideration of unit-to-unit variation in Equation (2), the relationship among them is denoted as:

_{2}is:

_{1}, we compute the estimation of $\widehat{a}$ and $\widehat{b}$ relying upon $\left({s}_{v}^{\left(i\right)}|{\eta}_{v}\right)$, which are:

#### 3.3. Estimation of Ω_{1} and Ω_{2} for SSADT

_{1}during the time interval $[0,{t}_{1}]$ is obviously a subset of CSADT data. For the time intervals $({t}_{1},{t}_{2}]$ and $({t}_{2},{t}_{3}]$, a transformation will be given since the initial values of both time and degradation value are not zero, as they are in CSADT.

_{1}~ N(μ

_{2}(Λ(t;θ) − Λ(t

_{1};θ)), σ

^{2}(τ(t;γ) − τ(t

_{1};γ))) where μ

_{2}is the drift coefficient when $t\in ({t}_{1},{t}_{2}]$. Given that x

^{′}= x − x

_{1}, Λ

^{′}(t;θ) = Λ(t;θ) − Λ(t

_{1};θ) and τ

^{′}(t;γ) = τ(t;γ) − τ(t

_{1};γ). The SSADT data in the time interval $({t}_{1},{t}_{2}]$ can be interpreted as a subset of CSADT data where the transferred degradation value is from 0 to x

^{′}during the time interval $[0,{t}^{*}]$, where ${t}^{*}$ is Λ

^{′}(t;θ) or τ

^{′}(t;γ) accordingly. This procedure is the same for the data in the time interval (t

_{2}, t

_{3}].

_{ijk}, which is the kth degradation value of unit j at the stress level i and t

_{ijk}is the corresponding measurement time, i = 1, 2, …, K, j = 1, 2, …, n, k = 1, 2, …, m

_{ij}. The transformations are given in Equations (26)–(28):

_{m}

_{×n}is the m × n all ones matrix; t

_{0jk}= 0 and x

_{0jk}= 0.

## 4. Case Study

^{θ}and τ(t;γ) = t

^{γ}will be used in this paper. In order to compare the s-fits of model M

_{0}with M

_{1}and M

_{2}, the Akaike information criterion (AIC) is introduced:

_{max}is the maximum value of the log-likelihood function in Equation (17), and n

_{p}is the number of unknown parameters. The lower the AIC value is, the better the model fits.

_{min}is the minimum of AIC

_{i}values. Then, Δ = 0 for the best model, Δ ≤ 2 for models having substantial support, 4 ≤ Δ ≤ 7 for models having considerably less support and Δ > 10 for models having essentially no support compared to the best models.

#### 4.1. Simulation Example

^{2}= 0.01, a = 20, b = 5, and η

_{1}= −1500. The Arrhenius model is selected as the acceleration model, which is:

_{0}at the normal stress level. Given by Equation (32) and the parameter settings, Pr(μ

_{0}< 0) = 1.8720 × 10

^{−19}, which is approximately near zero. Thus, the non-negative assumption can be satisfied to compute the PDF of the FPT. In the following, both the model comparison and sensitivity analysis are conducted with the abovementioned parameter values.

#### 4.1.1. Model Comparison

_{0}, M

_{1}and M

_{2}have 6, 4 and 5 parameters, respectively. Meanwhile, the absolute error (AE) of the candidate model M

_{i}(i = 0, 1, 2) to the real model M

_{real}is given by Equation (35) to quantitatively analyze the reliability evaluation results:

_{T}(t) is the CDF of the FPT at the normal stress level given by Equation (13), t

_{j}= 0.1, 1.1, …, 699.1, for hundreds of hours in this study, N

_{t}= 700. Herein, if AE > 0, it means that model M

_{i}overestimates the reliability evaluation results comparing with the true values, otherwise it underestimates.

_{max}and Δ values at each sample size, M

_{0}is the most suitable model, then is model M

_{2}, and the worst is model M

_{1}. The reason is that the time-scale transformation model M

_{2}can capture the nonlinear property of the degradation process to some extent, but still perform worse than M

_{0}. As to M

_{1}, it tries to linearize the nonlinear degradation process, which leads to dreadful fitting results and poor parameter estimation compared with the true results.

_{1}and M

_{2}overestimate the reliability evaluation results at normal stress levels. With the sample size increasing, the AEs for M

_{1}and M

_{2}become larger since that they are not the right models for ADT analysis and the level of error will be amplified when more data are available for model validation. Meanwhile, model M

_{0}slightly underestimates the reliability evaluation results from AE indexes, and the accuracies are improved by several orders of magnitude when the sample size goes from five to 30. The results demonstrate that model M

_{0}is the most applicable model for nonlinear ADT analysis and can provide accurate reliability and lifetime evaluation results.

_{0}and M

_{2}with the real values when the sample size n = 10. It is clear that model M

_{0}is closer to the real values than M

_{2}. The mean time to failure (MTTF) for M

_{0}and M

_{2}are 8340 and 8720 h, while the true value is 8430 h. The results verify the effectiveness of model M

_{0}than M

_{2}with respect to nonlinear ADT analysis.

#### 4.1.2. Sensitivity Analysis

_{0}with different values of model parameters [θ, γ, σ

^{2}, η

_{0}(i.e., a, b), η

_{1}] for the simulation example.

_{0}is robust, its relative error of reliability evaluation results at normal stress levels should be as small as possible when compared with the real model M

_{real}. Herein, we repeated the simulation procedure of SSADT data for N

_{s}= 100 times and n = 10 samples will be generated at each time point. Then, the relative error for model M

_{0}(RE of M

_{0}) is given through Equation (36):

^{6}= 15,625 and N

_{s}= 100 repeated simulations, it will lead to heavy computational effort to compute the results. Thus, the orthogonal design of experiments is introduced to reduce the number of combinations, but still be able to find the sensitive parameters from the response (RE of M

_{0}) at each factor level [35]. The orthogonal array L

_{25}(5

^{6}) is selected with 25 overall tests rather than 5

^{6}.

_{0}. From Table 2, it is known that the sensitivity of the parameters is ranked as θ > b > η

_{1}> γ > σ

^{2}> a for the simulation study. Hence, special attention should be given to those parameters when using them for reliability and lifetime evaluation at normal stress levels. Furthermore, the absolute bias δ ranged from 0.01806 to 0.05734, indicating that model M

_{0}is quite robust for lifetime and reliability evaluation.

_{0}, we may be interested in the performances of M

_{0}on a wide range of the coefficients of variations (CVs). Five new factor levels are given to a and b with other parameters fixed, i.e., 20%, 60%, 100%, 140% and 180%. Twenty five tests were conducted with N

_{s}= 100 repeated simulations according to the values at the first three columns in Table 2. The results are shown in Figure 4. As we can see, the relative errors of M

_{0}remain as significantly lower values than 0.05 with the CVs ranging 0~0.25. While the relative errors rise to the values around 0.3 with the CVs ranging 0.25~0.75, the robustness of M

_{0}is still shown.

#### 4.2. Real Applications

_{0}over M

_{1}and M

_{2}for both linear and nonlinear ADT analyses.

#### 4.2.1. LED Application

_{1}in our approach. As we can see from Figure 5, the tested LEDs experience nonlinear degradation paths. Therefore, a linear model like M

_{1}may not be appropriate for ADT analysis in this case. Thus, we use model M

_{0}to fit the data and compare the results with model M

_{1}and M

_{2}to verify its effectiveness on nonlinear ADT analysis. The inverse power model is used as the acceleration model, i.e.,:

_{0}< 0) = 6.1870 × 10

^{−11}, which verifies that there is no danger of there being any negative drift coefficient. Figure 6 presents the fitting results. It is obvious that model M

_{0}is the most applicable model with the maximum log-likelihood and lowest AIC value, which is the same as the fitting results. As with the linear model M

_{1}, the fitting results are worse than the other two models. Thus, the linear model in [36] is not appropriate for the LED application, while the performance of model M

_{2}lies between model M

_{0}and M

_{1}.

_{0}should be selected for the LED ADT data analysis with the lowest AIC value.

_{0}, M

_{1}, and M

_{2}are 1167.2, 1345.0, and 13,169.2 h, while the 95% confidence intervals are [202.1, 3459.1], [364.1, 3744.1], and [1660.1, 52,787.1] h, respectively. However, the estimated lifetime using the degradation-path model in Chaluvadi [36] (p. 103) is 1346 h. It is interesting to see that the intervals of M

_{0}and M

_{1}can capture this value to show the consistency of evaluation results under different models, while M

_{2}computes significantly larger values, meaning that model M

_{2}is unreliable. A reasonable explanation may be that the time-scale transformation is inapplicable for the nonlinear ADT analysis for LEDs. With respect to model M

_{1}, its lifetime evaluation results are closer to model M

_{0}, although it fits worse on the ADT data than model M

_{2}. In Tang, et al. [18], the time-scale transformation model M

_{2}is used to analyze the same set of data and their 95% confidence interval is [1672, 53,466] h, which, however, is also not valid in the LED case, as discussed. For the proposed model M

_{0}, its evaluation results are reliable with the best nonlinear ADT data fitting.

#### 4.2.2. Resistor Application

_{1}) and applied voltage (s

_{2}). Hence, in order to evaluate their lifetimes, the CSADT is conducted under nine constant stress levels with two acceleration variables, i.e., s

_{1}= 3.5, 4.0 and 5.0 in hundred Kelvin, s

_{2}= 10, 15 and 20 volts. The normal stress level is s

_{1}= 3.2315 and s

_{2}= 5. Ten resistors are tested at each stress level. For more details about the description of resistor data, readers are referred to Park and Padgett [25]. The original data is modified to ensure that the initial degradation values are equal to zero and the threshold ω is, therefore, set to be 0.2. The CSADT data at the first stress level is presented in Figure 8. It is clear that the degradation processes follow linear paths which are the same as the data in other stress levels.

_{0}< 0) = 6.0094 × 10

^{−6}, which verifies that there is no danger of there being any negative drift coefficient. Intuitively, compared with M

_{1}and M

_{2}, model M

_{0}displays the best fit with the maximum log-likelihood value and minimum AIC value. However, according to the Δ values, there is substantial support for model M

_{2}and considerably less support for model M

_{1}. The fitting results for the different models are shown in Figure 9, which imply that the performances of those models are similar in the resistor case since that the degradation paths are approximately linear.

_{1}is the best model with a deterministic drift coefficient, while supports are given to the other two models.

_{0}can be simplified into M

_{2}with the assumption (θ = γ) since M

_{2}has substantial support. Hence, the likelihood ratio (LR) test is implemented with the log-likelihood values in Table 5. The resulting LR statistic is 3.57 (<${\chi}_{1,0.05}^{2}$ = 3.84). Thus, we accept the assumption and choose M

_{2}.

_{0}is slightly sharper than that of M

_{1}and M

_{2}.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Elsayed, E.A.; Chen, A.C.K. Recent research and current issues in accelerated testing. In Proceedings of the 1998 IEEE International Conference on Systems, Man, and Cybernetics, San Diego, CA, USA, 11–14 October 1998; pp. 4704–4709.
- Meeker, W.Q.; Escobar, L.A. Statistical Methods for Reliability Data; John Wiley & Sons: New York, NY, USA, 1998. [Google Scholar]
- Nelson, W.B. Accelerated Testing: Statistical Models, Test Plans, and Data Analysis; John Wiley & Sons: New York, NY, USA, 1990. [Google Scholar]
- Thomas, E.V.; Bloom, I.; Christophersen, J.P.; Battaglia, V.S. Statistical methodology for predicting the life of lithium-ion cells via accelerated degradation testing. J. Power Sources
**2008**, 184, 312–317. [Google Scholar] [CrossRef] - Bae, S.J.; Kim, S.-J.; Park, J.I.; Lee, J.-H.; Cho, H.; Park, J.-Y. Lifetime prediction through accelerated degradation testing of membrane electrode assemblies in direct methanol fuel cells. Int. J. Hydrogen Energy
**2010**, 35, 9166–9176. [Google Scholar] [CrossRef] - Wang, F.-K.; Lu, Y.-C. Useful lifetime analysis for high-power white LEDs. Microelectron. Reliab.
**2014**, 54, 1307–1315. [Google Scholar] [CrossRef] - Wang, F.-K.; Chu, T.-P. Lifetime predictions of LED-based light bars by accelerated degradation test. Microelectron. Reliab.
**2012**, 52, 1332–1336. [Google Scholar] [CrossRef] - Meeker, W.Q.; Escobar, L.A.; Lu, C.J. Accelerated degradation tests: Modeling and analysis. Technometrics
**1998**, 40, 89–99. [Google Scholar] [CrossRef] - Park, J.I.; Bae, S.J. Direct prediction methods on lifetime distribution of organic light-emitting diodes from accelerated degradation tests. IEEE Trans. Reliab.
**2010**, 59, 74–90. [Google Scholar] [CrossRef] - Ling, M.H.; Tsui, K.L.; Balakrishnan, N. Accelerated degradation analysis for the quality of a system Based on the gamma process. IEEE Trans. Reliab.
**2015**, 64, 463–472. [Google Scholar] [CrossRef] - Whitmore, G.A.; Schenkelberg, F. Modelling accelerated degradation data using wiener diffusion with A time scale transformation. Lifetime Data Anal.
**1997**, 3, 27–45. [Google Scholar] [CrossRef] [PubMed] - Ye, Z.S.; Chen, L.P.; Tang, L.C.; Xie, M. Accelerated degradation test planning using the inverse gaussian process. IEEE Trans. Reliab.
**2014**, 63, 750–763. [Google Scholar] [CrossRef] - Escobar, L.A.; Meeker, W.Q. A review of accelerated test models. Stat. Sci.
**2006**, 21, 552–577. [Google Scholar] [CrossRef] - Park, C.; Padgett, W.J. Accelerated degradation models for failure based on geometric Brownian motion and gamma processes. Lifetime Data Anal.
**2005**, 11, 511–527. [Google Scholar] [CrossRef] [PubMed] - Pan, Z.Q.; Balakrishnan, N. Multiple-steps step-stress accelerated degradation modeling based on wiener and gamma processes. Commun. Stat. Simul. Comput.
**2010**, 39, 1384–1402. [Google Scholar] [CrossRef] - Wang, L.Z.; Pan, R.; Li, X.Y.; Jiang, T.M. A Bayesian reliability evaluation method with integrated accelerated degradation testing and field information. Reliab. Eng. Syst. Saf.
**2013**, 112, 38–47. [Google Scholar] [CrossRef] - Liao, H.T.; Elsayed, E.A. Reliability inference for field conditions from accelerated degradation testing. Nav. Res. Logist.
**2006**, 53, 576–587. [Google Scholar] [CrossRef] - Tang, S.J.; Guo, X.S.; Yu, C.Q.; Xue, H.J.; Zhou, Z.J. Accelerated degradation tests modeling based on the nonlinear wiener process with random effects. Math. Probl. Eng.
**2014**, 2014, 1–11. [Google Scholar] [CrossRef] - Wang, X.L.; Jiang, P.; Guo, B.; Cheng, Z.J. Real-time reliability evaluation with a general wiener process-based degradation model. Qual. Reliab. Eng. Int.
**2014**, 30, 205–220. [Google Scholar] [CrossRef] - Wang, X.L.; Balakrishnan, N.; Guo, B. Residual life estimation based on a generalized Wiener degradation process. Reliab. Eng. Syst. Saf.
**2014**, 124, 13–23. [Google Scholar] [CrossRef] - Liu, L.; Li, X.-Y.; Jiang, T.-M. Nonlinear accelerated degradation analysis based on the general wiener process. In Proceedings of the 25th European Safety and Reliability Conference (ESREL 2015), Zurich, Switzerland, 7–10 September 2015; pp. 2083–2088.
- Peng, C.Y.; Tseng, S.T. Mis-specification analysis of linear degradation models. IEEE Trans. Reliab.
**2009**, 58, 444–455. [Google Scholar] [CrossRef] - Wang, X. Wiener processes with random effects for degradation data. J. Multivar. Anal.
**2010**, 101, 340–351. [Google Scholar] [CrossRef] - Si, X.S.; Wang, W.B.; Hu, C.H.; Zhou, D.H.; Pecht, M.G. Remaining useful life estimation based on a nonlinear diffusion degradation process. IEEE Trans. Reliab.
**2012**, 61, 50–67. [Google Scholar] [CrossRef] - Park, C.; Padgett, W.J. Stochastic degradation models with several accelerating variables. IEEE Trans. Reliab.
**2006**, 55, 379–390. [Google Scholar] [CrossRef] - Li, X.; Jiang, T.; Sun, F.; Ma, J. Constant stress ADT for superluminescent diode and parameter sensitivity analysis. In Proceedings of the 8th International Conference on Reliability, Maintainability and Safety, Chengdu, China, 20–24 July 2009.
- Lim, H.; Yum, B.-J. Optimal design of accelerated degradation tests based on Wiener process models. J. Appl. Stat.
**2011**, 38, 309–325. [Google Scholar] [CrossRef] - Si, X.-S.; Wang, W.; Hu, C.-H.; Chen, M.-Y.; Zhou, D.-H. A Wiener-process-based degradation model with a recursive filter algorithm for remaining useful life estimation. Mech. Syst. Signal Process.
**2013**, 35, 219–237. [Google Scholar] [CrossRef] - Chhikara, R.S.; Folks, J.L. The Inverse Gaussian Distribution: Theory, Methodology, and Applications; CRC Press: New York, NY, USA, 1988; Volume 95, p. 232. [Google Scholar]
- Ye, Z.-S.; Xie, M. Stochastic modelling and analysis of degradation for highly reliable products. Appl. Stoch. Model. Bus.
**2015**, 31, 16–32. [Google Scholar] [CrossRef] - Tang, L.C.; Yang, G.; Xie, M. Planning of step-stress accelerated degradation test. In Proceedings of the 2004 Annual Reliability and Maintainability Symposium, Los Angeles, CA, USA, 26–29 January 2004; pp. 287–292.
- Lagarias, J.C.; Reeds, J.A.; Wright, M.H.; Wright, P.E. Convergence properties of the Nelder-Mead simplex method in low dimensions. SIAM J. Optim.
**1998**, 9, 112–147. [Google Scholar] [CrossRef] - Van Noortwijk, J.M. A survey of the application of gamma processes in maintenance. Reliab. Eng. Syst. Saf.
**2009**, 94, 2–21. [Google Scholar] [CrossRef] - Burnham, K.P.; Anderson, D.R. Multimodel inference understanding AIC and BIC in model selection. Sociol. Method Res.
**2004**, 33, 261–304. [Google Scholar] [CrossRef] - Taguchi, G.; Yokoyama, Y. Taguchi Methods: Design of Experiments; American Supplier Institute: Nasr City Cairo, Egypt, 1993; Volume 4. [Google Scholar]
- Chaluvadi, V. Accelerated Life Testing of Electronic Revenue Meters. Master Thesis, Clemson University, Clemson, SC, USA, 2008. [Google Scholar]

**Figure 3.**The (

**a**) PDFs and (

**b**) CDFs of the FPT for models M

_{0}and M

_{2}with the real values when n = 10.

**Figure 5.**The degradation paths for twenty four LEDs under two electric current levels: (

**a**) 35 mA and (

**b**) 40 mA.

**Figure 7.**The PDFs of the FPT for different models in the LED case: (

**a**) for M

_{0}and M

_{1}, (

**b**) for M

_{2}.

**Table 1.**Simulation example: parameter estimates with REs and RSEs in percentage (in parentheses), and AEs of reliability estimation for three candidate models under three sample sizes.

M_{i} | n | θ | γ | σ^{2} | η_{0} | η_{1} | l_{max} | n_{p} | AIC | Δ | AE | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

a | b | |||||||||||

M_{0} | 5 | 1.491 | 0.273 | 0.0177 | 21.62 | 4.33 | –1502 | 317 | 6 | –621 | 0 | –2.9×10^{−3} |

(−0.57, 3.3 × 10^{−3}) | (−32, 10) | (77, 59) | (8.1, 0.65) | (−13, 1.8) | (0.13, 1.7 × 10^{−4}) | |||||||

10 | 1.502 | 0.381 | 0.0097 | 18.65 | 4.42 | –1477 | 635 | –1259 | 0 | –1.2 × 10^{−3} | ||

(0.14, 2.0 × 10^{−4}) | (−4.7, 0.22) | (−3.1, 0.096) | (−6.7, 0.45) | (−12, 1.3) | (−1.5, 0.023) | |||||||

30 | 1.501 | 0.352 | 0.0114 | 20.90 | 5.18 | –1515 | 1898 | –3784 | 0 | 2.7 × 10^{−4} | ||

(0.055, 3.0 × 10^{−5}) | (−12, 1.4) | (14, 2.1) | (4.5, 0.20) | (3.6, 0.13) | (1.0, 0.011) | |||||||

M_{1} | 5 | 1 (fixed) | 1 (fixed) | 0.0607 | 8.5 × 10^{3} | 7.9 × 10^{5} | –3045 | −2.7 | 4 | 13.4 | 635 | 0.343 |

(−33, 11) | (150, 225) | (507, 2.6 × 10^{3}) | (4.2 × 10^{4}, 1.8 × 10^{7}) | (1.6 × 10^{7}, 2.5 × 10^{12}) | (103, 106) | |||||||

10 | 1 (fixed) | 1 (fixed) | 0.0579 | 8.2 × 10^{3} | 8.3 × 10^{5} | –3049 | 1.7 | 4.6 | 1263 | 0.365 | ||

(−33, 11) | (150, 225) | (479, 2.3 × 10^{3}) | (4.1 × 10^{4}, 1.7 × 10^{7}) | (1.7 × 10^{7}, 2.8 × 10^{12}) | (103, 107) | |||||||

30 | 1 (fixed) | 1(fixed) | 0.0578 | 9.2 × 10^{3} | 1.1 × 10^{6} | –3087 | 5.5 | –3.0 | 3781 | 0.374 | ||

(−33, 11) | (150, 225) | (478, 2.3 × 10^{3}) | (4.6 × 10^{4}, 2.1 × 10^{7}) | (2.2 × 10^{7}, 4.6 × 10^{12}) | (106, 112) | |||||||

M_{2} | 5 | 1.461 | =θ | 7.29 × 10^{−4} | 29.55 | 8.27 | –1577 | 217 | 5 | –425 | 197 | 3.2 × 10^{−3} |

(−2.6, 0.068) | (265, 704) | (−93, 86) | (48, 23) | (65, 43) | (5.1, 0.26) | |||||||

10 | 1.476 | =θ | 4.79 × 10^{−4} | 24.54 | 7.64 | –1544 | 490 | –971 | 288 | 4.2 × 10^{−3} | ||

(−1.6, 0.026) | (269, 724) | (−95, 91) | (23, 5.2) | (53, 28) | (2.9, 0.087) | |||||||

30 | 1.477 | =θ | 5.87 × 10^{−4} | 26.96 | 8.58 | –1577 | 1380 | –2750 | 1034 | 5.3 × 10^{−3} | ||

(−1.6, 0.025) | (269, 724) | (−94, 89) | (35, 12) | (72, 51) | (5.2, 0.27) |

**Table 2.**Sensitivity analysis of M

_{0}with five levels of parameters through the orthogonal array L

_{25}(5

^{6}) and Taguchi analysis.

Test No. | θ | γ | σ^{2} | η_{0} | η_{1} | RE of M_{0} | |
---|---|---|---|---|---|---|---|

a | b | ||||||

1 | 1 | 1 | 1 | 1 | 1 | 1 | 0.026812 |

2 | 1 | 2 | 2 | 2 | 2 | 2 | 0.049430 |

3 | 1 | 3 | 3 | 3 | 3 | 3 | 0.076352 |

4 | 1 | 4 | 4 | 4 | 4 | 4 | 0.107904 |

5 | 1 | 5 | 5 | 5 | 5 | 5 | 0.149784 |

6 | 2 | 1 | 2 | 3 | 4 | 5 | 0.101002 |

7 | 2 | 2 | 3 | 4 | 5 | 1 | 0.017453 |

8 | 2 | 3 | 4 | 5 | 1 | 2 | 0.005818 |

9 | 2 | 4 | 5 | 1 | 2 | 3 | 0.042759 |

10 | 2 | 5 | 1 | 2 | 3 | 4 | 0.068059 |

11 | 3 | 1 | 3 | 5 | 2 | 4 | 0.014726 |

12 | 3 | 2 | 4 | 1 | 3 | 5 | 0.060681 |

13 | 3 | 3 | 5 | 2 | 4 | 1 | 0.032238 |

14 | 3 | 4 | 1 | 3 | 5 | 2 | 0.018415 |

15 | 3 | 5 | 2 | 4 | 1 | 3 | 0.006743 |

16 | 4 | 1 | 4 | 2 | 5 | 3 | 0.019738 |

17 | 4 | 2 | 5 | 3 | 1 | 4 | 0.008050 |

18 | 4 | 3 | 1 | 4 | 2 | 5 | 0.010650 |

19 | 4 | 4 | 2 | 5 | 3 | 1 | 0.053845 |

20 | 4 | 5 | 3 | 1 | 4 | 2 | 0.031300 |

21 | 5 | 1 | 5 | 4 | 3 | 2 | 0.053440 |

22 | 5 | 2 | 1 | 5 | 4 | 3 | 0.044617 |

23 | 5 | 3 | 2 | 1 | 5 | 4 | 0.021416 |

24 | 5 | 4 | 3 | 2 | 1 | 5 | 0.009040 |

25 | 5 | 5 | 4 | 3 | 2 | 1 | 0.061280 |

MR1 | 0.08206 | 0.04314 | 0.03371 | 0.03659 | 0.01129 | 0.03833 | T = 1.09155 |

MR2 | 0.04702 | 0.03605 | 0.04649 | 0.03570 | 0.03577 | 0.06138 | |

MR3 | 0.02656 | 0.02929 | 0.02977 | 0.05302 | 0.06248 | 0.03804 | |

MR4 | 0.02472 | 0.04639 | 0.05108 | 0.03924 | 0.06341 | 0.04403 | |

MR5 | 0.03796 | 0.06343 | 0.05725 | 0.05376 | 0.04536 | 0.06623 | |

δ | 0.05734 | 0.03414 | 0.02748 | 0.01806 | 0.05212 | 0.03455 | |

Rank | 1 | 4 | 5 | 6 | 2 | 3 |

**Table 3.**LED application: parameter estimates for three candidate models with random drift coefficients (b ≠ 0).

Model | θ | γ | σ^{2} | η_{0} | η_{1} | l_{max} | n_{p} | AIC | Δ | |
---|---|---|---|---|---|---|---|---|---|---|

a | b | |||||||||

M_{0} | 0.442 | 0.117 | 73.784 | 0.273 | 0.0018 | 0.677 | –310 | 6 | 633 | 0 |

M_{1} | 1 (fixed) | 1 (fixed) | 0.761 | 1.69 × 10^{−6} | 5.62 × 10^{−14} | 3.112 | –389 | 4 | 785 | 152 |

M_{2} | 0.450 | =θ | 5.840 | 3.52 × 10^{−5} | 2.43 × 10^{−11} | 3.112 | –317 | 5 | 644 | 11 |

**Table 4.**LED application: parameter estimates for three candidate models with deterministic drift coefficients (b = 0).

Model | θ | γ | σ^{2} | η_{0} = a | η_{1} | l_{max} | n_{p} | AIC | Δ |
---|---|---|---|---|---|---|---|---|---|

M_{0} | 0.448 | 0.171 | 45.429 | 0.012 | 1.179 | –314.6634 | 5 | 639 | 0 |

M_{1} | 1 (fixed) | 1 (fixed) | 0.776 | 8.67 × 10^{−7} | 3.297 | –389.7358 | 3 | 785 | 146 |

M_{2} | 0.4477 | =θ | 6.238 | 1.83 × 10^{−5} | 3.297 | –319.9119 | 4 | 648 | 9 |

**Table 5.**Resistor application: parameter estimates for three candidate models with random drift coefficients (b ≠ 0).

Model | θ | γ | σ^{2} | η_{0} | η_{1} | η_{2} | l_{max} | n_{p} | AIC | Δ | |
---|---|---|---|---|---|---|---|---|---|---|---|

a | b | ||||||||||

M_{0} | 1.076 | 0.918 | 3.02 × 10^{−4} | 8.96 × 10^{−4} | 4.19 × 10^{−8} | 0.462 | 0.108 | 1698 | 7 | –3383 | 0 |

M_{1} | 1 (fixed) | 1 (fixed) | 2.59 × 10^{−4} | 0.0012 | 7.02 × 10^{−8} | 0.440 | 0.102 | 1694 | 5 | –3378 | 4.8 |

M_{2} | 1.046 | =θ | 2.35 × 10^{−4} | 0.0010 | 5.12 × 10^{−8} | 0.451 | 0.106 | 1697 | 6 | –3381 | 1.6 |

**Table 6.**Resistor application: parameter estimates for three candidate models with deterministic drift coefficients (b = 0).

Model | θ | γ | σ^{2} | η_{0} = a | η_{1} | η_{2} | l_{max} | n_{p} | AIC | Δ |
---|---|---|---|---|---|---|---|---|---|---|

M_{0} | 1.020 | 0.9498 | 3.33 × 10^{−4} | 0.0011 | 0.445 | 0.104 | 1647 | 6 | –3282 | 2.7 |

M_{1} | 1 (fixed) | 1 (fixed) | 3.02 × 10^{−4} | 0.0012 | 0.439 | 0.103 | 1646 | 4 | –3284 | 0 |

M_{2} | 1.008 | =θ | 2.97 × 10^{−4} | 0.0011 | 0.442 | 0.103 | 1646 | 5 | –3283 | 1.8 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Liu, L.; Li, X.; Sun, F.; Wang, N. A General Accelerated Degradation Model Based on the Wiener Process. *Materials* **2016**, *9*, 981.
https://doi.org/10.3390/ma9120981

**AMA Style**

Liu L, Li X, Sun F, Wang N. A General Accelerated Degradation Model Based on the Wiener Process. *Materials*. 2016; 9(12):981.
https://doi.org/10.3390/ma9120981

**Chicago/Turabian Style**

Liu, Le, Xiaoyang Li, Fuqiang Sun, and Ning Wang. 2016. "A General Accelerated Degradation Model Based on the Wiener Process" *Materials* 9, no. 12: 981.
https://doi.org/10.3390/ma9120981