Journal of Mechanical Science and Technology

, Volume 33, Issue 11, pp 5425–5435 | Cite as

Stochastic accelerated degradation model involving multiple accelerating variables by considering measurement error

  • Junxing Li
  • Zhihua WangEmail author
  • Chengrui Liu
  • Ming Qiu


In accelerated degradation tests, products are usually exposed to several environmental variables or operating conditions. This motivates the need for developing an accelerated degradation model involving multiple accelerating variables. Among the current literature, the conventional accelerated degradation models involving multiple accelerating variables have not considered the measurement error, which inevitably exists in practical degradation datasets. Therefore, a Wiener process-based accelerated degradation model that simultaneously considering multiple accelerating variables and measurement error is proposed. Then approximate closed-form expressions for the failure time distribution (FTD) and its percentiles are derived. The expectation maximization (EM) algorithm is adopted to estimate unknown parameters. Moreover, a multivariate normality testing method is developed to test the fitting goodness of the degradation model. Finally, a comprehensive simulation study and a real application are given to validate the proposed method. The result shows that the proposed model can provide precise estimates even for small sample size of approximately five, and the estimated mean square errors (MSEs) of the mean time to failure (MTTF) and the FTD percentile of the proposed model can be improved by at least 70 % compared with those of the reference methods when the sample size is same.


Accelerated degradation analysis Multiple accelerating variables Measurement error Wiener process EM algorithm 


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The authors are grateful to the anonymous reviewers, and the editor, for their critical and constructive review of the manuscript. This study was supported by the National Basic Research Program of China (Grant No. 2018YFB2000203), National Natural Science Foundation of China (Grant No. 11872085) and Key Scientific Research Projects of Henan Colleges and Universities of China (Grant No. 19A460002).


  1. [1]
    D. G. Chen, Y. Lio, H. K. T. Ng and T. R. Tsai, Statistical Modeling for Degradation Data, Springer, Germany (2017).CrossRefGoogle Scholar
  2. [2]
    A. F. Shahraki, O. P. Yadav and H. Liao, A review on degradation modelling and its engineering applications, International Journal of Performability Engineering, 13 (3) (2017) 299–314.Google Scholar
  3. [3]
    Z. Zhang, X. Si, C. Hu and Y. Lei, Degradation data analysis and remaining useful life estimation: A review on Wiener-process-based methods, European Journal of Operational Research, 271 (3) (2018) 775–796.MathSciNetCrossRefGoogle Scholar
  4. [4]
    W. Q. Meeker and L. Escobar, Statistical Methods for Reliability Data, John Wiley & Sons, USA (2014).zbMATHGoogle Scholar
  5. [5]
    W. Q. Meeker, L. Escobar and C. J. Lu, Accelerated degradation tests: Modeling and analysis, Technometrics, 40 (2) (1998) 89–99.CrossRefGoogle Scholar
  6. [6]
    G. A. Whitmore and F. Schenkelberg, Modelling accelerated degradation data using wiener diffusion with a time scale transformation, Lifetime Data Analysis, 3 (1) (1997) 27–45.CrossRefGoogle Scholar
  7. [7]
    C. M. Liao and S. T. Tseng, Optimal design for step-stress accelerated degradation tests, IEEE Transactions on Reliability, 55 (1) (2006) 59–66.CrossRefGoogle Scholar
  8. [8]
    H. Lim and B. J. Yum, Optimal design of accelerated degradation tests based on Wiener process models, Journal of Applied Statistics, 38 (2) (2011) 309–325.MathSciNetCrossRefGoogle Scholar
  9. [9]
    S. Tang, X. Guo, C. Yu, H. Xue and Z. Zhou, Accelerated degradation tests modeling based on the nonlinear Wiener process with random effects, Mathematical Problems in Engineering (2014) 1–11.Google Scholar
  10. [10]
    L. Sun, X. Gu and P. Song, Accelerated degradation process analysis based on the nonlinear Wiener process with covariates and random effects, Mathematical Problems in Engineering (2016) 1–13.Google Scholar
  11. [11]
    J. Li, Z. Wang, X. Liu, Y. Zhang, H. Fu and C. Liu, A Wiener process model for accelerated degradation analysis considering measurement errors, Microelectronics Reliability, 65 (2016) 8–15.CrossRefGoogle Scholar
  12. [12]
    S. Hao, J. Yang and C. Berenguer, Nonlinear step-stress accelerated degradation modelling considering three sources of variability, Reliability Engineering & System Safety, 172 (2018) 207–215.CrossRefGoogle Scholar
  13. [13]
    H. Wang, Y. Zhao and X. Ma, Mechanism equivalence in designing optimum step-stress accelerated degradation test plan under Wiener process, IEEE Access, 6 (2018) 4440–4451.CrossRefGoogle Scholar
  14. [14]
    X. Y. Li, J. P. Wu, H. G. Ma, X. Li and R. Kang, A random fuzzy accelerated degradation model and statistical analysis, IEEE Transactions on Fuzzy Systems, 26 (3) (2018) 1638–1650.CrossRefGoogle Scholar
  15. [15]
    D. He, Y. Wang and G. Chang, Objective bayesian analysis for the accelerated degradation model based on the inverse Gaussian process, Applied Mathematical Modelling, 61 (2018) 341–350.MathSciNetCrossRefGoogle Scholar
  16. [16]
    F. Duan and G. Wang, Optimal design for constant-stress accelerated degradation test based on gamma process, Communication in Statistics- Theory and Methods (2018) 1–25.Google Scholar
  17. [17]
    S. Limon, O. P. Yadav and H. Liao, A literature review on planning and analysis of accelerated testing for reliability assessment, Quality and Reliability Engineering International, 33 (8) (2017) 2361–2383.CrossRefGoogle Scholar
  18. [18]
    C. Park and W. J. Padgett, Stochastic degradation models with several accelerating variables, IEEE Transactions on Reliability, 55 (2) (2006) 379–390.CrossRefGoogle Scholar
  19. [19]
    L. Liu, X. Li, F. Sun and N. Wang, A general accelerated degradation model based on the Wiener process, Materials, 9 (12) (2016) 1–20.Google Scholar
  20. [20]
    G. A. Whitmore, Estimating degradation by a Wiener diffusion process subject to measurement error, Lifetime Data Analysis, 1 (3) (1995) 307–319.CrossRefGoogle Scholar
  21. [21]
    Z. S. Ye, Y. Wang, K. L. Tsui and M. Pecht, Degradation data analysis using Wiener processes with measurement errors, IEEE Transactions on Reliability, 62 (4) (2013) 772–780.CrossRefGoogle Scholar
  22. [22]
    J. Li, Z. Wang, Y. Zhang, H. Fu, C. Liu and S. Krishnas-wamy, Degradation data analysis based on a generalized Wiener process subject to measurement error, Mechanical Systems & Signal Processing, 94 (2017) 57–72.CrossRefGoogle Scholar
  23. [23]
    J. Li, Z. Wang, Y. Zhang, C. Liu and H. Fu, A nonlinear Wiener process degradation model with autoregressive errors, Reliability Engineering & System Safety, 173 (2018) 48–57.CrossRefGoogle Scholar
  24. [24]
    X. S. Si, C. H. Hu and Z. X. Zhang, Data-Driven Remaining Useful Life Prognosis Techniques, Springer, Germany (2017).CrossRefGoogle Scholar
  25. [25]
    T. K. Moon, The expectation-maximization algorithm, Signal Processing Magazine IEEE, 13 (6) (1996) 47–60.CrossRefGoogle Scholar
  26. [26]
    D. Pan, J. Liu, F. Huang, J. Cao and A. Alsaedi, A Wiener process model with truncated normal distribution for reliability analysis, Applied Mathematical Modelling, 50 (2017) 333–346.MathSciNetCrossRefGoogle Scholar
  27. [27]
    D. Pan, Y. Wei, H. Fang and W. Yang, A reliability estimation approach via Wiener degradation model with measurement errors, Applied Mathematics and Computation, 320 (2018) 131–141.MathSciNetCrossRefGoogle Scholar
  28. [28]
    X. S. Si, W. Wang, C. H. Hu, M. Y. Chen and D. H. Zhou, A Wiener-process-based degradation model with a recursive filter algorithm for remaining useful life estimation, Mechanical Systems & Signal Processing, 35 (2013) 219–237.CrossRefGoogle Scholar
  29. [29]
    Z. Huang, Z. Xu, W. Wang and Y. Sun, Remaining useful life prediction for a nonlinear heterogeneous wiener process model with an adaptive drift, IEEE Transactions on Reliability, 64 (2) (2015) 687–700.CrossRefGoogle Scholar
  30. [30]
    O. Aalen, O. Borgan and H. Gjessing, Survival and Event History Analysis: A Process Point of View, Springer Science & Business Media, Germany (2008).CrossRefGoogle Scholar
  31. [31]
    N. D. Singpurwalla, Survival in dynamic environments, Statistical Science, 10 (1) (1995) 86–103.CrossRefGoogle Scholar
  32. [32]
    X. Wang, N. Balakrishnan and B. Guo, Residual life estimation based on a generalized Wiener degradation process, Reliability Engineering & System Safety, 124 (2014) 13–23.CrossRefGoogle Scholar
  33. [33]
    X. S. Si, W. Wang, C. H. Hu, D. H. Zhou and M. G. Pecht, Remaining useful life estimation based on a nonlinear diffusion degradation process, IEEE Transactions on Reliability, 61 (1) (2012) 50–67.CrossRefGoogle Scholar
  34. [34]
    K. Xiao, Research on the accelerated degradation testing for the O type rubber sealing ring used by ammunition, M.S. Thesis, Nanjing University of Science and Technology, Nanjing, China (2014).Google Scholar
  35. [35]
    S. A. Murphy, Likelihood ratio-based confidence intervals in survival analysis, Journal of the American Statistical Association, 90 (432) (1995) 1399–1405.MathSciNetCrossRefGoogle Scholar

Copyright information

© KSME & Springer 2019

Authors and Affiliations

  • Junxing Li
    • 1
  • Zhihua Wang
    • 2
    Email author
  • Chengrui Liu
    • 3
  • Ming Qiu
    • 1
  1. 1.School of Mechatronical EngineeringHenan University of Science and TechnologyLuoyangChina
  2. 2.School of Aeronautic Science and EngineeringBeihang UniversityBeijingChina
  3. 3.Beijing Institute of Control EngineeringBeijingChina

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