An expectation maximization algorithm to model failure times by continuous-time Markov chains. (English) Zbl 1203.90055

Summary: In many applications, the failure rate function may present a bathtub shape curve. In this paper, an expectation maximization algorithm is proposed to construct a suitable continuous-time Markov chain which models the failure time data by the first time reaching the absorbing state. Assume that a system is described by methods of supplementary variables, the device of stage, and so on. Given a data set, the maximum likelihood estimators of the initial distribution and the infinitesimal transition rates of the Markov chain can be obtained by our novel algorithm. Suppose that there are \(m\) transient states in the system and that there are \(n\) failure time data. The devised algorithm only needs to compute the exponential of \(m\times m\) upper triangular matrices for \(O(nm^{2})\) times in each iteration. Finally, the algorithm is applied to two real data sets, which indicates the practicality and efficiency of our algorithm.


90B25 Reliability, availability, maintenance, inspection in operations research
65C60 Computational problems in statistics (MSC2010)
60J22 Computational methods in Markov chains
Full Text: DOI EuDML


[1] C. K. Pil, M. Rausand, and J. Vatn, “Reliability assessment of reliquefaction systems on LNG carriers,” Reliability Engineering and System Safety, vol. 93, pp. 1345-1353, 2008.
[2] S. Y. Chen, C. Y. Yao, G. Xiao, Y. S. Ying, and W. L. Wang, “Fault detection and prediction of clocks and timers based on computer audition and probabilistic neural networks,” in Proceedings of the 8th International Workshop on Artificial Neural Networks (IWANN ’05), vol. 3512 of Lecture Notes on Computer Science, pp. 952-959, 2005.
[3] S. Y. Chen, Y. F. Li, and J. W. Zhang, “Vision processing for realtime 3-D data acquisition based on coded structured light,” IEEE Transactions on Image Processing, vol. 17, no. 2, pp. 167-176, 2008. · Zbl 05516617
[4] X. Zhao and L. Cui, “On the accelerated scan finite Markov chain imbedding approach,” IEEE Transactions on Reliability, vol. 58, no. 2, pp. 383-388, 2009.
[5] S. Y. Chen, Y. F. Li, Q. Guan, and G. Xiao, “Real-time three-dimensional surface measurement by color encoded light projection,” Applied Physics Letters, vol. 89, no. 11, article no. 111108, 2006.
[6] “IEC61511. Functional safety-safety instrumented systems for the process industry sector,” Geneva, Switzerland, IEC, 2004.
[7] M. Xie, Y. Tang, and T. N. Goh, “A modified Weibull extension with bathtubshaped failure rate function,” Reliability Engineering and System Safety, vol. 76, pp. 279-285, 2002.
[8] A. Pievatolo, E. Tironi, and I. Valadé, “Semi-Markov processes for power system reliability assessment with application to uninterruptible power supply,” IEEE Transactions on Power Systems, vol. 19, no. 3, pp. 1326-1333, 2004.
[9] H. R. Guo, H. Liao, W. Zhao, and A. Mettas, “A new stochastic model for systems under general repairs,” IEEE Transactions on Reliability, vol. 56, no. 1, pp. 40-49, 2007.
[10] M. Li, “Fractal time series-a tutorial review,” Mathematical Problems in Engineering, vol. 2010, Article ID 157264, 26 pages, 2010. · Zbl 1191.37002
[11] J. Endrenyi, G. J. Anders, and A. M. Leite Da Suva, “Probabilistic evaluation of the effect of maintenance on reliability-an application,” IEEE Transactions on Power Systems, vol. 13, no. 2, pp. 576-583, 1998.
[12] M. Li and W. Zhao, “Representation of a stochastic traffic bound,” IEEE Transactions on Parallel and Distributed Systems, vol. 21, no. 9, pp. 1368-1372, 2010.
[13] M. Li and S. C. Lim, “Modeling network traffic using generalized Cauchy process,” Physica A, vol. 387, no. 11, pp. 2584-2594, 2008.
[14] D. R. Cox, “The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 51, pp. 433-441, 1955. · Zbl 0067.10902
[15] C. Singh, R. Billinton, and S. Y. Lee, “The method of stages for non-Markov models,” IEEE Transactions on Reliability, vol. 26, no. 2, pp. 135-137, 1977. · Zbl 0368.60105
[16] Y. Lam, “Calculating the rate of occurrence of failures for continuous-time markov chains with application to a two-component parallel system,” Journal of the Operational Research Society, vol. 46, pp. 528-536, 1995. · Zbl 0838.90057
[17] E. A. Oliveira, A. C.M. Alvim, and P. Melo, “Unavailability analysis of safety systems under aging by supplementary variables with imperfect repair,” Annals of Nuclear Energy, vol. 32, no. 2, pp. 241-252, 2005.
[18] A. Bobbio, A. Horváth, M. Scarpa, and M. Telek, “Acyclic discrete phase type distributions: properties and a parameter estimation algorithm,” Performance Evaluation, vol. 54, no. 1, pp. 1-32, 2003.
[19] R. J. Elliott, L. Aggoun, and J. B. Moore, Hidden Markov Models, vol. 29 of Applications of Mathematics, Springer, New York, NY, USA, 1995. · Zbl 0819.60045
[20] R. J. Elliott, Z. P. Chen, and Q. H. Duan, “Insurance claims modulated by a hidden Brownian marked point process,” Insurance: Mathematics & Economics, vol. 45, no. 2, pp. 163-172, 2009. · Zbl 1231.91182
[21] R. N. Bhattacharya and E. C. Waymire, Stochastic Processes with Applications, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, New York, NY, USA, 1990. · Zbl 0744.60032
[22] C. F. Van Loan, “Computing integrals involving the matrix exponential,” IEEE Transactions on Automatic Control, vol. 23, no. 3, pp. 395-404, 1978. · Zbl 0387.65013
[23] F. Carbonell, J. C. Jímenez, and L. M. Pedroso, “Computing multiple integrals involving matrix exponentials,” Journal of Computational and Applied Mathematics, vol. 213, no. 1, pp. 300-305, 2008. · Zbl 1133.65012
[24] R. J. Elliott, “New finite-dimensional filters and smoothers for noisily observed Markov chains,” IEEE Transactions on Information Theory, vol. 39, no. 1, pp. 265-271, 1993. · Zbl 0779.93093
[25] R. J. Elliott and W. P. Malcolm, “Discrete-time expectation maximization algorithms for Markov-modulated Poisson processes,” IEEE Transactions on Automatic Control, vol. 53, no. 1, pp. 247-256, 2008. · Zbl 1367.62246
[26] M. V. Aarset, “How to identify bathtub hazard rate,” IEEE Transactions on Reliability, vol. 36, no. 1, pp. 106-108, 1987. · Zbl 0625.62092
[27] M. Xie and C. D. Lai, “Reliability analysis using an additive Weibull model with bathtub-shaped failure rate function,” Reliability Engineering and System Safety, vol. 52, no. 1, pp. 87-93, 1996.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.