Multi-state stochastic processes: a statistical flowgraph perspective. (English. French summary) Zbl 1416.62567
Summary: Two-state models (working/failed or alive/dead) are widely used in reliability and survival analysis. In contrast, multi-state stochastic processes provide a richer framework for modeling and analyzing the progression of a process from an initial to a terminal state, allowing incorporation of more details of the process mechanism. We review multi-state models, focusing on time-homogeneous semi-Markov processes (SMPs), and then describe the statistical flowgraph framework, which comprises analysis methods and algorithms for computing quantities of interest such as the distribution of first passage times to a terminal state. These algorithms algebraically combine integral transforms of the waiting time distributions in each state and invert them to get the required results. The estimated transforms may be based on parametric distributions or on empirical distributions of sample transition data, which may be censored. The methods are illustrated with several applications.
MSC:
| 62N05 | Reliability and life testing |
| 62N01 | Censored data models |
| 60K15 | Markov renewal processes, semi-Markov processes |
| 62Pxx | Applications of statistics |
Keywords:
censored data; empirical transform; Laplace transform; saddlepoint approximation; semi-Markov process; transform inversionReferences:
| [1] | Aalen, O., Borgan, Ø. & Gjessing, H.K. (2008). Survival and Event History Analysis: A Process Point of View. New York : Springer. · Zbl 1204.62165 |
| [2] | Abate, J. & Whitt, W. (1992). The Fourier‐series method for inverting transforms of probability distributions. Queueing Syst. , 10(1), 5-88. · Zbl 0749.60013 |
| [3] | Andersen, P.K., Borgan, Ø., Gill, R.D. & Keiding, N. (1993). Statistical Models Based on Counting Processes. New York : Springer. · Zbl 0769.62061 |
| [4] | Andersen, P.K. & Keiding, N. (2002). Multi‐state models for event history analysis. Stat. Methods Med. Res. , 11, 91-115. · Zbl 1121.62568 |
| [5] | Bartlett, M.S. (1955). An Introduction to Stochastic Processes, with Special Reference to Methods and Applications. Cambridge , UK : Cambridge University Press. · Zbl 0068.11801 |
| [6] | Billingsley, P. (1961). Statistical Inference for Markov Processes. Chicago , IL : University of Chicago Press. · Zbl 0106.34201 |
| [7] | Billingsley, P. (1979). Probability and Measure. New York : John Wiley & Sons. · Zbl 0411.60001 |
| [8] | Bogdanoff, J.L. & Kozin, F. (1985). Probabilistic Models of Cumulative Damage. New York : Wiley Interscience. |
| [9] | Broyles, J.R. & Montgomery, D.C. (2010). A statistical Markov chain approximation of transient hospital inpatient inventory. Eur. J. Oper. Res , 207(3), 1645-1657. · Zbl 1206.90077 |
| [10] | Butler, R.W. (2001). First passage distributions in semi‐Markov processes and their saddlepoint approximation. In Data Analysis from Statistical Foundations, Ed. A.K.Saleh (ed.), pp. 347-368. Hauppauge, NY: Nova Science Publishers. |
| [11] | Butler, R.W. & Bronson, D.A. (2002). Bootstrapping survival times in stochastic systems by using saddlepoint approximations. J. R. Stat. Soc. Ser. B Stat. Methodol. , 64(1), 31-49. · Zbl 1015.62045 |
| [12] | Casella, G. & Berger, R.L. (2002). Statistical Inference, 2nd ed. Pacific Grove , CA : Duxbury. |
| [13] | Çinlar, E. (1975). Introduction to Stochastic Processes. Englewood Cliffs , NJ : Prentice‐Hall. · Zbl 0341.60019 |
| [14] | Cheng, R.C.H. (1995). Bootstrap methods in computer simulation experiments. In Proceedings of the 1995 Winter Simulation Conference, Eds. C.Alexopoulos (ed.), K.Wang (ed.), W.R.Lilegdon (ed.) & D.Goldsman (ed.), pp. 171-177. Washington , DC : IEEE Computer Society. |
| [15] | Collins, D.H. (2009). Nonparametric Estimation of First Passage Time Distributions in Flowgraph Models, PhD dissertation. Albuquerque , NM : University of New Mexico. |
| [16] | Collins, D.H. & Huzurbazar, A.V. (2008). System reliability and safety assessment using nonparametric flowgraph models. J. Risk Reliab. , 222, 667-674. |
| [17] | Cox, D.R. & Miller, H.D. (1968). The Theory of Stochastic Processes. London : Methuen & Co. |
| [18] | Crowder, M. (2001). Classical Competing Risks. Boca Raton , FL : Chapman & Hall/CRC. · Zbl 0979.62078 |
| [19] | Csenki, A. (2009). Towards a unified view of finite automata and semi‐Markov flowgraph models. Int. J. Gen. Syst. , 38(6), 643-657. · Zbl 1191.68379 |
| [20] | Csörgő, S. (1982). The empirical moment generating function. In Nonparametric Statistical Inference, Eds. B.V.Gnedenko (ed.), M.L.Puri (ed.) & I.Vincze (ed.). Amsterdam: North‐Holland Publishing Company. |
| [21] | Csörgő, S. & Teugels, J.L. (1990). Empirical Laplace transform and approximation of compound distributions. J. Appl. Probab. , 27, 88-110. · Zbl 0703.62033 |
| [22] | Daniels, H.E. (1954). Saddlepoint approximations in statistics. Ann. Math. Stat. , 25, 631-650. · Zbl 0058.35404 |
| [23] | Daniels, H.E. (1980). Exact saddlepoint approximations. Biometrika , 67(1), 59-63. · Zbl 0423.62018 |
| [24] | Daniels, H.E. (1982). The saddlepoint approximation for a general birth process. J. Appl. Probab. , 19, 20-28. · Zbl 0482.60080 |
| [25] | David, H.A. & Moeschberger, M.L. (1978). The Theory of Competing Risks. New York : Macmillan. · Zbl 0434.62076 |
| [26] | Davies, B. & Martin, B. (1979). Numerical inversion of the Laplace transform: a survey and comparison of methods. J. Comput. Phys. , 33, 1-32. · Zbl 0416.65077 |
| [27] | Davison, A.C. & Hinkley, D.V. (1988). Saddlepoint approximations in resampling methods. Biometrika 73(3), 417-431. · Zbl 0651.62018 |
| [28] | Davison, A.C. & Hinkley, D.V. (1997). Bootstrap Methods and Their Application. Cambridge : Cambridge University Press. · Zbl 0886.62001 |
| [29] | Efron, B. (1967). The two sample problem with censored data. In Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Eds. L.M.Le Cam (ed.) & J.Neyman (ed.), pp. 831-853. Berkeley , CA : University of California Press. |
| [30] | Elkins, D.A. & Wortman, M.A. (2001). On numerical solution of the Markov renewal equation: tight upper and lower kernel bounds. Method. Comput. Appl. Probab. , 3, 239-253. · Zbl 0997.60098 |
| [31] | Epstein, C.L. & Schotland, J. (2008). The bad truth about Laplace’s transform. SIAM Rev. , 50(3), 504-520. · Zbl 1154.65094 |
| [32] | Feuerverger, A. (1989). On the empirical saddlepoint approximation. Biometrika , 76(3), 457-464. · Zbl 0674.62019 |
| [33] | Feuerverger, A. & Mureika, R.A. (1977). The empirical characteristic function and its applications. Ann. Stat. , 5(1), 88-97. · Zbl 0364.62051 |
| [34] | Field, C. & Ronchetti, E. (1990). Small Sample Asymptotics, IMS Lecture Notes‐Monograph Series No. 13, Hayward , CA : Institute of Mathematical Statistics. · Zbl 0742.62016 |
| [35] | Fienberg, S.E. & Holland, P.W. (1973). Estimation of multinomial cell probabilities. J. Amer. Statist. Assoc. , 68(343), 683-691. · Zbl 0267.62030 |
| [36] | Fleming, K.N. (2004). Markov models for evaluating risk‐informed in‐service inspection strategies for nuclear power plant piping systems. Reliab. Eng. Syst. Safety , 83, 27-45. |
| [37] | Gijbels, I., Pope, A. & Wand, M.P. (1999). Understanding exponential smoothing via kernel regression. J. R. Stat. Soc. Ser. B Stat. Methodol. , 61, 9-50. · Zbl 0913.62040 |
| [38] | Gilat, A. (2010). MATLAB: an Introduction with Applications, 4th ed. Hoboken , NJ : John Wiley & Sons. |
| [39] | Gill, R.D. (1980a). Censoring and Stochastic Integrals. Mathematisch Centrum , Amsterdam : Mathematical Centre Tracts 124. · Zbl 0456.62003 |
| [40] | Gill, R.D. (1980b). Nonparametric estimation based on censored observations of a Markov renewal process. Zeitschrift für Wahrscheinlickeitstheorie und Verwandte Gebeite , 53, 97-116. · Zbl 0417.60086 |
| [41] | Guihenneuc‐Jouyaux, C., Richardson, S. & Longini, I.M. (2000). Modeling markers of disease progression by a hidden Markov process: application to characterizing CD4 cell decline. Biometrics , 56, 733-741. · Zbl 1060.62619 |
| [42] | Gusella, V. (1998). Safety estimation method for structures with cumulative damage. J. Eng. Mech. , 124, 1200-1209. |
| [43] | Hamada, M.S., Wilson, A.G., Reese, C.S. & Martz, H.F. (2008). Bayesian Reliability. New York : Springer. · Zbl 1165.62074 |
| [44] | Hougaard, P. (1999). Multi‐state models: a review. Lifetime Data Anal. , 5, 239-264. · Zbl 0934.62112 |
| [45] | Huzurbazar, A.V. (2000). Modeling and analysis of engineering systems data using flowgraph models. Technometrics , 42, 300-306. |
| [46] | Huzurbazar, A.V. (2005). Flowgraph Models for Multistate Time‐to‐Event Data. Hoboken , NJ : Wiley Interscience. · Zbl 1055.62123 |
| [47] | Huzurbazar, A.V. & Williams, B.J. (2005). Flowgraph models for complex multistate system reliability. In Modern Statistical and Mathematical Methods in Reliability, Eds. A.Wilson (ed.), N.Limnios (ed.), S.Keller‐McNulty (ed.) & Y.Armijo (ed.). Singapore , Singapore : World Scientific. |
| [48] | Janssen, J. & Manca, R. (2006). Applied Semi‐Markov Processes. New York : Springer. · Zbl 1096.60002 |
| [49] | Jarrow, R.A., Lando, D. & Turnbull, S.M. (1997). A Markov model for the term structure of credit risk spreads. The Review of Financial Studies , 10(2), 481-523. |
| [50] | Kaplan, E.L. & Meier, P. (1958). Nonparametric estimation from incomplete observations. J. Amer. Statist. Assoc. , 53(282), 457-481. · Zbl 0089.14801 |
| [51] | Karlin, S. & Taylor, H.M. (1975). A First Course in Stochastic Processes, 2nd ed. London : Academic Press. · Zbl 0315.60016 |
| [52] | Karlin, S. & Taylor, H.M. (1981). A Second Course in Stochastic Processes, 2nd ed. London : Academic Press. · Zbl 0469.60001 |
| [53] | Kattan, M.W., Cowen, M.E. & Miles, B.J. (1997). A decision analysis for treatment of clinically localized prostate cancer. J. Gen. Intern. Med. , 12(5), 299-305. |
| [54] | Kemeny, J.G. & Snell, J.L. (1976). Finite Markov Chains. New York : Springer‐Verlag. · Zbl 0328.60035 |
| [55] | Klein, J.P. & Moeschberger, M.L. (2003). Survival Analysis: Techniques for Censored and Truncated Data, 2nd ed. New York : Springer‐Verlag. · Zbl 1011.62106 |
| [56] | Kleinrock, L. (1975). Queueing Systems Volume I: Theory. New York : John Wiley & Sons. · Zbl 0334.60045 |
| [57] | Kulkarni, V.G. (1996). Modeling and Analysis of Stochastic Systems. Boca Raton , FL : Chapman & Hall/CRC. |
| [58] | Law, A.M. & Kelton, D.W. (2000). Simulation Modeling and Analysis, 3rd ed. Boston : McGraw Hill. |
| [59] | Lawless, J.F. (1982). Statistical Models and Methods for Lifetime Data. New York : John Wiley & Sons. · Zbl 0541.62081 |
| [60] | Limnios, N. & Oprişan, G. (2001). Semi‐Markov Processes and Reliability. Boston : Birkhäuser. · Zbl 0990.60004 |
| [61] | Lorens, C.S. (1964). Flowgraphs for the Modeling and Analysis of Linear Systems. New York : McGraw‐Hill. |
| [62] | Mann, N.R., Schafer, R.E. & Singpurwalla, N.D. (1974). Methods for Statistical Analysis of Reliability and Life Data. New York : John Wiley & Sons. · Zbl 0339.62070 |
| [63] | Mason, S.J. (1953). Feedback theory—some properties of signal flow graphs. Proc. Inst. Radio Eng. , 41, 1144-1156. |
| [64] | Mason, S.J. (1956). Feedback theory—further properties of signal flow graphs. Proc. Inst. Radio Eng. , 44, 920-926. |
| [65] | Mazumdar, M. (1980). Component testing procedure for a series system with redundant subsystems. Technometrics , 22, 23-27. · Zbl 0437.62090 |
| [66] | McCabe, T.J. & Butler, C.W. (1989). Design complexity measurement and testing. Commun. ACM , 32(12), 1415-1425. |
| [67] | Meyer, D.D. (2000). Matrix Analysis and Applied Linear Algebra. Philadelphia , PA : SIAM. · Zbl 0962.15001 |
| [68] | Moeschberger, M.L. & Klein, J.P. (1985). A comparison of several methods of estimating the survival function when there is extreme right censoring. Biometrics , 41(1), 253-259. · Zbl 0618.62098 |
| [69] | Moore, E.H. & Pyke, R. (1968). Estimation of the transition distributions of a Markov renewal process. Ann. Inst. Statist. Math. , 20, 411-424. · Zbl 0196.19103 |
| [70] | Natvig, B. (2011). Multistate Systems Reliability Theory with Applications. Chichester , UK : John Wiley & Sons. · Zbl 1397.60008 |
| [71] | Nicola, V.F., Shahabuddin, P. & Nakayama, M.K. (2001). Techniques for fast simulaion of models of highly dependable systems. IEEE Trans. Reliab. , 50, 246-264. |
| [72] | Ouhbi, B. & Limnios, N. (2003). Nonparametric reliability estimation of semi‐Markov proccesses. J. Stat. Infer. Plan. , 109, 155-165. · Zbl 1038.62096 |
| [73] | Parzen, E. (1962). Stochastic Processes. San Francisco : Holden Day. · Zbl 0107.12301 |
| [74] | Pyke, R. (1961a). Markov renewal processes: definitions and preliminary results. Ann. Math. Stat. , 32, 1231-1242. · Zbl 0267.60089 |
| [75] | Pyke, R. (1961b). Markov renewal processes with finitely many states. Ann. Math. Stat. , 32, 1243-1259. · Zbl 0201.49901 |
| [76] | R Development Core Team (2008). R: A Language and Environment for Statistical Computing. Vienna , Austria : R Foundation for Statistical Computing. |
| [77] | Ross, S.M. (1992). Applied Probability Models with Optimization Applications. New York : Dover Publications. · Zbl 1191.60001 |
| [78] | Ross, S.M. (1996). Stochastic Processes, 2nd ed. New York : John Wiley & Sons. · Zbl 0888.60002 |
| [79] | Silverman, B.W. (1986). Density Estimation for Statistics and Data Analysis. Boca Raton , FL : Chapman & Hall/CRC. · Zbl 0617.62042 |
| [80] | Simonoff, J.S. (1996). Smoothing Methods in Statistics. New York : Springer. · Zbl 0859.62035 |
| [81] | Sittler, R.W. (1956). Systems analysis of discrete Markov processes. IRE Trans. Circuit Theory , 3(4), 257-266. |
| [82] | Stute, W. & Wang, J.‐L. (1993). The strong law under random censorship. Ann. Stat. , 21, 1591-1607. · Zbl 0785.60020 |
| [83] | Therneau, T.M. & Grambsch, P.M. (2000). Modeling Survival Data: Extending the Cox Model. New York : Springer. · Zbl 0958.62094 |
| [84] | Tortorella, M. (2005). Numerical solutions of renewal‐type equations. INFORMS J. Comput. , 17(1), 66-74. · Zbl 1239.65083 |
| [85] | Vere‐Jones, D. (1970). Stochastic models for earthquake occurrence. J. R. Stat. Soc. B , 32, 1-62. · Zbl 0201.27702 |
| [86] | Warr, R.L. (2012). Numerical approximation of probability mass functions via the inverse discrete Fourier transform. Technical Report LA‐UR‐12‐24885. Los Alamos, NM: Los Alamos National Laboratory. |
| [87] | Warr, R.L. & Collins, D.H. (2011). A comprehensive method for solving general finite‐state semi‐Markov processes. Technical Report LA‐UR‐11‐01596. Los Alamos , NM : Los Alamos National Laboratory. |
| [88] | Williams, B.J. & Huzurbazar, A.V. (2006). Posterior sampling with constructed likelihood functions: an application to flowgraph models. Appl. Stoch. Models Bus. Ind. , 22(2), 127-137. · Zbl 1114.62027 |
| [89] | Wolfram, S. (2003). The Mathematica Book, 5th ed. Champaign , IL : Wolfram Media/Cambridge University Press. |
| [90] | Yau, C.L. & Huzurbazar, A.V. (2002). Analysis of censored and incomplete survival data using flowgraph models. Stat. Med., 21, 3727-3743. |
| [91] | Zio, E. (2009). Computational Methods for Reliability and Risk Analysis. Singapore : World Scientific. · Zbl 1291.62019 |
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.
