An EM algorithm for estimation in Markov-modulated Poisson processes. (English) Zbl 0875.62405

Summary: It has recently been shown that the maximum-likelihood estimate of the parameters of a Markov-modulated Poisson process is consistent. We present an EM algorithm for computing such estimates and discuss how it may be implemented. We also compare it to the Nelder-Mead downhill simplex algorithm for some numerical examples, and the results show that the number of iterations the EM algorithm requires to converge is in general smaller than the number of likelihood evaluations required by the downhill simplex algorithm. An EM iteration is more complicated than a likelihood evaluation, though, and thus also implementation aspects must be taken into account to determine the efficiencies of the algorithms.


62M05 Markov processes: estimation; hidden Markov models
65C99 Probabilistic methods, stochastic differential equations
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[1] Asmussen, S.; Nerman, O.: Fitting phase-type distributions via the EM algorithm. Symposium i anvendt statistik, 335-346 (1991)
[2] Baum, L. E.: An inequality and associated maximization technique in statistical estimation for probabilistic functions of Markov processes. Inequalities, III: proc. Symp., 1-8 (1972)
[3] Baum, L. E.; Eagon, J. A.: An inequality with applications to statistical estimation for probabilistic functions of Markov processes and to a model for ecology. Bull. amer. Math. soc. 73, 360-363 (1967) · Zbl 0157.11101
[4] Baum, L. E.; Petrie, T.; Soules, G.; Weiss, N.: A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains. Ann. math. Statist. 41, 164-171 (1970) · Zbl 0188.49603
[5] Billingsley, P.: Statistical inference for Markov processes. (1961) · Zbl 0106.34201
[6] Davison, A. C.; Ramesh, N. I.: A stochastic model for times of exposures to air pollution from a point source. Statistics for the environment (1993)
[7] Dempster, A. P.; Laird, N. M.; Rubin, D. B.: Maximum likelihood from incomplete data via the EM algorithm. J. roy. Statist. soc. 39, 1-22 (1977) · Zbl 0364.62022
[8] Deng, L.; Mark, J. W.: Parameter estimation for Markov modulated Poisson processes via the EM algorithm with time discretization. Telecomm. syst. 1, 321-338 (1993)
[9] Fischer, W.; Meier-Hellstern, K.: The Markov-modulated Poisson process (MMPP) cookbook. Perf. eval. 18, 149-171 (1993) · Zbl 0781.60098
[10] Fredkin, D. R.; Rice, J. A.: Maximum likelihood estimation and identification directly from single-channel recordings. Proc. roy. Soc. lond. B 249, 125-132 (1992)
[11] Gusella, R.: Characterizing the variability of arrival processes with indexes of dispersion. IEEE J. Select. areas commun. 9, 203-211 (1991)
[12] Heffes, H.; Lucantoni, D.: A Markov modulated characterization of packetized voice and data traffic and related statistical multiplexer performance. IEEE J. Select. areas commun. 4, 856-867 (1986)
[13] Holst, U. and G. Lindgren, Recursive estimation of parameters in Markov modulated Poisson processes, to appear in: IEEE Trans. Comm., · Zbl 1054.62578
[14] Kawashima, K.; Saito, H.: Teletraffic issues in ATM networks. Comput. networks ISDN systems 20, 369-375 (1990)
[15] Levinson, S. E.; Rabiner, L. R.; Sondhi, M. M.: An introduction to the application of the theory of probabilistic functions of a Markov process in automatic speech recognition. Bell syst. Tech. J. 62, 1035-1074 (1983) · Zbl 0507.68058
[16] Lindgren, G.: Markov regime models for mixed distributions and switching regressions. Scand. J. Statist. 5, 81-91 (1978) · Zbl 0382.62073
[17] Louis, T. A.: Finding the observed information matrix when using the EM algorithm. J. roy. Statist. soc. B 44, 226-233 (1982) · Zbl 0488.62018
[18] Meier, K. S.: A statistical procedure for Fitting Markov-modulated Poisson processes. Ph.d. thesis (1984)
[19] Meier-Hellstern, K. S.: A Fitting algorithm for Markov-modulated Poisson processes having two arrival rates. European J. Oper. res. 29, 370-377 (1987) · Zbl 0615.62122
[20] Meilijson, I.: A fast improvement to the EM algorithm on its own terms. J. roy. Statist. soc. B 51, 127-138 (1989) · Zbl 0674.65118
[21] Neuts, M. F.: Structured stochastic matrices of M/G/1 type and their applications. (1989) · Zbl 0695.60088
[22] Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T.: Numerical recipes. (1989) · Zbl 0698.65001
[23] Ramesh, N. I.: Statistical analysis on Markov-modulated Poisson processes. (1993)
[24] Rossiter, M. H.: Characterizing a random point process by a switched Poisson process. Ph.d. thesis (1989) · Zbl 0717.60103
[25] Rydén, T.: Parameter estimation for Markov modulated Poisson processes. Stochastic models 10, 795-829 (1994) · Zbl 0815.62059
[26] Rydén, T.: On identifiability and order of continuous-time aggregated Markov chains, Markov modulated Poisson processes, and phase-type distributions. J. appl. Prob. 33 (1996) · Zbl 0865.60064
[27] Sundberg, R.: An iterative method for solution of the likelihood equations for incomplete data from exponential families. Comm. statist. Simulation comput. 5, 55-64 (1976) · Zbl 0352.62014
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