Bivariate degradation modeling and reliability analysis based on a shared frailty factor with truncated normal distribution
##plugins.themes.bootstrap3.article.main##
##plugins.themes.bootstrap3.article.sidebar##
Published
Jan 13, 2026
Lu Li
Zhihua Wang
Abstract
For a complex product, a single performance characteristic(PC) often fails to fully reflect its degradation process, making it essential to consider the joint degradation of multiple PCs. In this paper, we propose a bivariate degradation model based on a shared frailty factor with the truncated normal distribution, using Wiener processes to characterize the marginal distributions of the PCs. The assumption of the truncated normal distribution aligns better with the physical background where the degradation rates of PCs are non-negative during actual degradation processes. Furthermore, a method for inferring unknown parameters is developed by employing the expectation maximization algorithm. Under this modeling assumption, it became possible to obtain an analytical expression for the product's lifetime distribution on the basis of the concept of the first hitting time. Therefore, in this paper, we further extend the normal distribution integral lemmas to the case of the truncated normal distribution, and provide analytical expression for the cumulative distribution function of the product lifetime. Finally, the rational effectiveness of the proposed model and methods is validated through a numerical simulation example and a case study on wheel wear.
##plugins.themes.bootstrap3.article.details##
Keywords
Bivariate degradation, Shared frailty factor, Truncated normal distribution, First hitting time, Expectation maximization algorithm
References
Barui, S., Mitra, D., & Balakrishnan, N. (2024). Flexible modelling of a bivariate degradation process with a shared frailty and an application to fatigue crack data. Rel. Eng. Syst. Saf., 242(2), 1.1-1.13.
Freitas, M. A., de Toledo, M. L. G., Colosimo, E. A., & Pires, M. C. (2009). Using degradation data to assess reliability: a case study on train wheel degradation. John Wiley & Sons, Ltd.(5).
Hougaard, P. (2000). Analysis of multivariate survival data. Springer.
Kawakubo, Y., Miyazawa, S., Nagata, K., & Kobatake, S. (2003). Wear-life prediction of contact recording head. IEEE Trans. Magn., 39(2, 1), 888-892.
Lu, C. J., & Meeker, W. Q. (1993). Using degradation measures to estimate a time-to-failure distribution. Technometrics, 35(2), 161-174.
Meeker, W. Q., & Escobar, L. A. (1998). Statistical methods for reliability data. New York: John Wiley & Sons.
Michiels, S., Baujat, B., Mahé, C., Sargent, D. J., & Pignon, J. P. (2005). Random effects survival models gave a better understanding of heterogeneity in individual patient data meta-analyses. J. Clin. Epidemiol., 58(3), 238-245.
Owen, D. B. (1981). Corrections to: A table of normal integrals. Commun. Stat. Simul. Comput..
Pan, D., Liu, J. B., Huang, F., Cao, J., & Alsaedi, A. (2017). A wiener process model with truncated normal distribution for reliability analysis. Appl. Math. Model, 50(oct.), 333-346.
Peng, W., Li, Y.-F., Yang, Y.-J., Zhu, S.-P., & Huang, H.-Z. (2016). Bivariate analysis of incomplete degradation observations based on inverse gaussian processes and copulas. IEEE Trans. Rel., 65(2), 624-639.
Song, K., & Cui, L. (2022). A common random effect induced bivariate gamma degradation process with application to remaining useful life prediction. Rel. Eng. Syst. Saf.(3), 219.
Wang, F., & Li, H. (2017). System reliability under prescribed marginals and correlations: Are we correct about the effect of correlations? Rel. Eng. Syst. Saf., 173(MAY), 94-104.
Wang, X., Balakrishnan, N., & Guo, B. (2015). Residual life estimation based on nonlinear-multivariate wiener processes. J. Stat. Comput. Sim., 85(9), 1742-1764.
Wang, Z., Hu, C., Wang, W., Zhou, Z., & Si, X. (2014, DEC). A case study of remaining storage life prediction using stochastic filtering with the influence of condition monitoring. Rel. Eng. Syst. Saf., 132, 186-195.
Whitmore, G. A. (1995). Estimating degradation by a wiener diffusion process subject to measurement error. Lifetime Data Anal., 1(3), 307-319.
Xu, A., Shen, L., Wang, B., & Tang, Y. (2018). On modeling bivariate wiener degradation process. IEEE Trans. Rel., 67(3), 897-906.
Xu, A., Wang, B., Zhu, D., Pang, J., & Lian, X. (2024). Bayesian reliability assessment of permanent magnet brake under small sample size. IEEE Trans. Rel..
Yan, B., Wang, H., & Ma, X. (2023). Correlation-driven multivariate degradation modeling and rul prediction based on wiener process model. Qual. Reliab. Eng. Int., 39(8), 3203-3229.
Zhai, Q., & Ye, Z.-S. (2023). A multivariate stochastic degradation model for dependent performance characteristics. Technometrics, 65(3), 315-327.
Zhang, X., & Wilson, A. (2017). System reliability and component importance under dependence: a copula approach. Technometrics, 59(2), 215-224.
Zhang, Z., Si, X., Hu, C., & Lei, Y. (2018). Degradation data analysis and remaining useful life estimation: a review on wiener-process-based methods. Eur. J. Oper. Res., 271(3), 775-796.
Zheng, B., Chen, C., Lin, Y., Ye, X., & Zhai, G. (2023). Reliability analysis based on a bivariate degradation model considering random initial state and its correlation with degradation rate. IEEE Trans. Rel., 72(1), 37-48.
Zhu, J., Wang, Y., Huang, Y., Gopaluni, R. B., Cao, Y., Heere, M., . . . Ehrenberg, H. (2022). Data-driven capacity estimation of commercial lithium-ion batteries from voltage relaxation. Nat. Commun., 13(1).
Freitas, M. A., de Toledo, M. L. G., Colosimo, E. A., & Pires, M. C. (2009). Using degradation data to assess reliability: a case study on train wheel degradation. John Wiley & Sons, Ltd.(5).
Hougaard, P. (2000). Analysis of multivariate survival data. Springer.
Kawakubo, Y., Miyazawa, S., Nagata, K., & Kobatake, S. (2003). Wear-life prediction of contact recording head. IEEE Trans. Magn., 39(2, 1), 888-892.
Lu, C. J., & Meeker, W. Q. (1993). Using degradation measures to estimate a time-to-failure distribution. Technometrics, 35(2), 161-174.
Meeker, W. Q., & Escobar, L. A. (1998). Statistical methods for reliability data. New York: John Wiley & Sons.
Michiels, S., Baujat, B., Mahé, C., Sargent, D. J., & Pignon, J. P. (2005). Random effects survival models gave a better understanding of heterogeneity in individual patient data meta-analyses. J. Clin. Epidemiol., 58(3), 238-245.
Owen, D. B. (1981). Corrections to: A table of normal integrals. Commun. Stat. Simul. Comput..
Pan, D., Liu, J. B., Huang, F., Cao, J., & Alsaedi, A. (2017). A wiener process model with truncated normal distribution for reliability analysis. Appl. Math. Model, 50(oct.), 333-346.
Peng, W., Li, Y.-F., Yang, Y.-J., Zhu, S.-P., & Huang, H.-Z. (2016). Bivariate analysis of incomplete degradation observations based on inverse gaussian processes and copulas. IEEE Trans. Rel., 65(2), 624-639.
Song, K., & Cui, L. (2022). A common random effect induced bivariate gamma degradation process with application to remaining useful life prediction. Rel. Eng. Syst. Saf.(3), 219.
Wang, F., & Li, H. (2017). System reliability under prescribed marginals and correlations: Are we correct about the effect of correlations? Rel. Eng. Syst. Saf., 173(MAY), 94-104.
Wang, X., Balakrishnan, N., & Guo, B. (2015). Residual life estimation based on nonlinear-multivariate wiener processes. J. Stat. Comput. Sim., 85(9), 1742-1764.
Wang, Z., Hu, C., Wang, W., Zhou, Z., & Si, X. (2014, DEC). A case study of remaining storage life prediction using stochastic filtering with the influence of condition monitoring. Rel. Eng. Syst. Saf., 132, 186-195.
Whitmore, G. A. (1995). Estimating degradation by a wiener diffusion process subject to measurement error. Lifetime Data Anal., 1(3), 307-319.
Xu, A., Shen, L., Wang, B., & Tang, Y. (2018). On modeling bivariate wiener degradation process. IEEE Trans. Rel., 67(3), 897-906.
Xu, A., Wang, B., Zhu, D., Pang, J., & Lian, X. (2024). Bayesian reliability assessment of permanent magnet brake under small sample size. IEEE Trans. Rel..
Yan, B., Wang, H., & Ma, X. (2023). Correlation-driven multivariate degradation modeling and rul prediction based on wiener process model. Qual. Reliab. Eng. Int., 39(8), 3203-3229.
Zhai, Q., & Ye, Z.-S. (2023). A multivariate stochastic degradation model for dependent performance characteristics. Technometrics, 65(3), 315-327.
Zhang, X., & Wilson, A. (2017). System reliability and component importance under dependence: a copula approach. Technometrics, 59(2), 215-224.
Zhang, Z., Si, X., Hu, C., & Lei, Y. (2018). Degradation data analysis and remaining useful life estimation: a review on wiener-process-based methods. Eur. J. Oper. Res., 271(3), 775-796.
Zheng, B., Chen, C., Lin, Y., Ye, X., & Zhai, G. (2023). Reliability analysis based on a bivariate degradation model considering random initial state and its correlation with degradation rate. IEEE Trans. Rel., 72(1), 37-48.
Zhu, J., Wang, Y., Huang, Y., Gopaluni, R. B., Cao, Y., Heere, M., . . . Ehrenberg, H. (2022). Data-driven capacity estimation of commercial lithium-ion batteries from voltage relaxation. Nat. Commun., 13(1).
Section
Regular Session Papers
This work is licensed under a Creative Commons Attribution 3.0 Unported License.