Sains Malaysiana 53(9)(2024): 3215-3227

http://doi.org/10.17576/jsm-2024-5309-24

Bayesian Approach for Estimating the Parameters of the Time-to-Failure Distribution and Its Percentiles under Power Degradation Model

(Pendekatan Bayesian untuk Menganggar Parameter Taburan Masa Kegagalan dan Peratusannya di bawah Model Penurunan Kuasa)

LAILA NAJI BA DAKHN^1,2, MOHD AFTAR ABU BAKAR¹, RAZIK RIDZUAN MOHD TAJUDDIN^1,* & KAMARULZAMAN IBRAHIM¹

¹Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia

²Department of Mathematics, Faculty of Science, Hadhramout University, Mukalla, Hadhramout, Yemen

Diserahkan: 8 Januari 2024/Diterima: 9 Julai 2024

Abstract

The degradation models are often applied on the degradation data for studying time-to-failure distribution. In this study, the Bayesian approach is applied on the power degradation model for estimating the parameters of the time-to-failure distribution and its percentiles. Two different distributions are assumed for the degradation parameter of the model. The degradation parameter is firstly assumed to follow the skew-normal distribution with three jointly independently distributed parameters such that the gamma prior is assumed for the shape parameter, while the scale and the location parameters are assumed uniform. The second distribution assumed for the degradation parameter is the log-logistic distribution with two jointly independent random parameters where the shape parameter is assumed gamma, while the scale parameter is assumed uniform. Based on the Gibbs sampling method carried out under the JAGS platform, the models considered are applied on the simulated data and the NASA turbofan Jet engine dataset and the results found are compared. In modeling the time-to-failure distribution, it is shown that based on the simulated data and real data, the Bayesian approach for the power degradation model with the skew-normal degradation parameter outperformed the Bayesian approach for the power degradation model with the log-logistic degradation parameter.

Keywords: Bayesian approach; log-logistic distribution; power degradation model; skew-normal distribution; time-to-failure distribution

Abstrak

Model degradasi sering digunakan pada data degradasi untuk mempelajari taburan masa kegagalan. Dalam kajian ini, pendekatan Bayes digunakan pada model degradasi kuasa untuk menganggarkan parameter taburan masa kegagalan dan persentilnya. Dua taburan yang berbeza diandaikan untuk parameter degradasi model. Parameter degradasi pertama diandaikan mengikut taburan normal terpencong dengan tiga parameter yang tertabur secara tak bersandar dengan diandaikan prior gamma untuk parameter bentuk, sementara parameter skala dan lokasi diandaikan tertabur secara seragam. Taburan kedua yang diandaikan untuk parameter degradasi adalah taburan log-logistik dengan dua parameter rawak bercantum yang tak bersandar dengan andaian parameter bentuk tertabur secara gamma, sementara parameter skala tertabur secara seragam. Berdasarkan kaedah pensampelan Gibbs yang dilaksanakan di platform JAGS, model yang dipertimbangkan digunakan pada data simulasi dan set data enjin jet kipas turbo NASA dan hasil yang ditemui dibandingkan. Dalam pemodelan taburan masa kegagalan, didapati bahawa berdasarkan data yang disimulasikan dan data sebenar, prestasi pendekatan Bayes untuk model degradasi kuasa dengan parameter degradasi normal terpencong mengatasi prestasi pendekatan Bayes untuk model degradasi kuasa dengan parameter degradasi log-logistik.

Kata kunci: Masa kegagalan; model degradasi kuasa; pendekatan Bayes; taburan log-logistik; taburan normal terpencong

RUJUKAN

Alhamidie, A.A., Kamarulzaman Ibrahim, Alodat, M.T. & Wan Zawiah Wan Zin. 2019. Bayesian inference for linear regression under alpha-skew-normal prior. Sains Malaysiana 48(1): 227-235.

Azzalini, A. 1985. A class of distributions which includes the normal ones. Scandinavian Journal of Statistics 12(2): 171-178.

Dakhn, L.N.B., Mohd Aftar Abu Bakar & Kamarulzaman Ibrahim. 2023. Bayesian estimation of time to failure distributions based on skew normal degradation model: An application to GaAs laser degradation data. Sains Malaysiana 52(2): 641-653. https://doi.org/http://doi.org/10.17576/jsm-2023-5202-24

Dakhn, L.N.B., Ebrahem, M.A-Haj. & Eidous, O. 2017. Semi-parametric method to estimate the time-to- failure distribution and its percentiles for simple linear degradation model.Journal of Modern Applied Statistical Methods 16(2): 322-346. https://doi.org/10.22237/jmasm/1509495420

Brooks, S.P. & Gelman, A. 1998. General methods for monitoring convergence of iterative simulations. Journal of Computational and Graphical Statistics 7(4): 434-455. https://doi.org/10.1080/10618600.1998.10474787

Chen, S., Wang, M., Huang, D., Wen, P., Wang, S. & Zhao, S. 2020. Remaining useful life prediction for complex systems with multiple indicators based on particle filter and parameter correlation. IEEE Access 8: 215145-215156. https://doi.org/10.1109/ACCESS.2020.3041682

Eidous, O., Ebrahem, M.A-Haj. & Dakhn, L.N.B. 2017. Estimating the time-to-failure distribution and its percentiles for simple linear degradation model using double kernel method. Journal of Probability and Statistical Science 15(1): 121-134.

Gelman, A. & Hill, J. 2007. Data Analysis Using Regression and Multilevel/Hierarchical Models. New York: Cambridge University Press.

Gelman, A. & Rubin, D.B. 1992. Inference from iterative simulation using multiple sequences. Statistical Science 7(4): 457-511.

Geweke, J. 1991. Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments. The Fourth Valencia International Meeting on Bayesian Statistics, Peniscola, Spain, April 15-20.

Ghaderinezhad, F., Christophe, L. & Nicola, L. 2020. Bayesian inference for skew-symmetric distributions. Symmetry 12(4): 491. https://doi.org/10.3390/sym12040491

Gonzalez, J.R., Vazquez, M., Algora, C. & Nunez, N. 2010. Real-time reliability test for a CPV module based on a power degradation model. Progress in Photovoltaics: Research and Applications 20(1): 6-11. https://doi.org/10.1002/pip

Guure, C.B. & Noor Akma Ibrahim. 2014. Approximate Bayesian estimates of Weibull parameters with Lindley’s method. Sains Malaysiana 43(9): 1433-1437.

Kundu, D. 2008. Bayesian inference and life testing plan for the Weibull distribution in presence of progressive censoring. Technometrics 50(2): 144-154. https://doi.org/10.1198/004017008000000217

Meeker, W.Q. & Escobar, L.A. 1998. Statistical Method for Reliability Data. New York: John Wiley & Sons, Inc.

Oliveira, R.P.B., Loschi, R.H. & Freitas, M.A. 2018. Skew-heavy-tailed degradation models: An application to train wheel degradation. IEEE Transactions on Reliability 67(1): 129-141. https://doi.org/10.1109/TR.2017.2765485

Pan, D., Liu, J-B. & Yang, W. 2018. A new result on lifetime estimation based on Skew-Wiener degradation model. Statistics and Probability Letters 138: 157-164. https://doi.org/10.1016/j.spl.2018.03.009

Puggard, W., Niwitpong, S-A. & Niwitpong, S. 2022. Comparison analysis on the coefficients of variation of two independent Birnbaum-Saunders distributions by constructing confidence intervals for the ratio of coefficients of variation. Sains Malaysiana 51(7): 2265-2281.

Saxena, A., Goebel, K., Simon, D. & Eklund, N. 2008. Damage propagation modeling for aircraft engine run-to-failure simulation. International Conference on Prognostics and Health Management, PHM 2008. https://doi.org/10.1109/PHM.2008.4711414

Sen, S., Maiti, S.S. & Chandra, N. 2016. The Xgamma distribution: Statistical properties and application. Journal of Modern Applied Statistical Methods 15(1): 774-788. https://doi.org/10.22237/jmasm/1462077420

Shafiq, M., Alamgir & Muhammad Atif. 2016. On the estimation of three parameters lognormal distribution based on fuzzy life time data. Sains Malaysiana 45(11): 1773-1777.

Siju, K.C. & Kumar, M. 2018. Bayesian estimation of reliability using time-to- failure distribution of parametric degradation models. Journal of Statistical Computation and Simulation 88(9): 1717-1748. https://doi.org/10.1080/00949655.2018.1445743

Spiegelhalter, D.J., Best, N.G., Carlin, B.P. & Van Der Linde, A. 2002. Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society 64(4): 583-639.

Wu, J., Wu, C., Cao, S., Or, S.W., Deng, C. & Shao, X. 2019. Degradation data-driven time-to-failure prognostics approach for rolling element bearings in electrical machines. IEEE Transactions on Industrial Electronics 66(1): 529-539. https://doi.org/10.1109/TIE.2018.2811366

Yadav, A.S., Saha, M., Tripathi, H. & Kumar, S. 2021. The exponentiated Xgamma distribution: A new monotone failure rate model and its applications to lifetime data. Statistica 81(3): 303-334. https://doi.org/10.6092/issn.1973-2201/10578

*Pengarang untuk surat-menyurat; email: rrmt@ukm.edu.my