Markov-modulated Hawkes process with stepwise decay

  • Ting Wang
  • Mark Bebbington
  • David Harte


This paper proposes a new model—the Markov-modulated Hawkes process with stepwise decay (MMHPSD)—to investigate the variation in seismicity rate during a series of earthquake sequences including multiple main shocks. The MMHPSD is a self-exciting process which switches among different states, in each of which the process has distinguishable background seismicity and decay rates. Parameter estimation is developed via the expectation maximization algorithm. The model is applied to data from the Landers–Hector Mine earthquake sequence, demonstrating that it is useful for modelling changes in the temporal patterns of seismicity. The states in the model can capture the behavior of main shocks, large aftershocks, secondary aftershocks, and a period of quiescence with different background rates and decay rates.


Markov-modulated Hawkes process with stepwise decay EM algorithm ETAS model Simulation Landers 


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  1. Akaike H. (1974) A new look at the statistical model identification. IEEE Transactions on Automatic Control, 19(6): 716–723MathSciNetzbMATHCrossRefGoogle Scholar
  2. Akaike, H. (1978). A Bayesian analysis of the minimum AIC procedure. Annals of the Institute of Statistical Mathematics, 30(1), 9–14; also included in E. Parzen et al. (Eds.) (1998), Selected papers of Hirotugu Akaike (pp. 275–280). Berlin: Springer.Google Scholar
  3. Bebbington M.S. (2007) Identifying volcanic regimes using hidden Markov models. Geophysical Journal International, 171: 921–942CrossRefGoogle Scholar
  4. Bebbington M.S. (2008) Estimating rate- and state-fraction parameters using a two-node stochastic model for aftershocks. Tectonophysics, 457: 71–85CrossRefGoogle Scholar
  5. Bebbington M.S., Harte D.S. (2001) On the statistics of the linked stress release model. Journal of Applied Probability, 38A: 176–187MathSciNetzbMATHCrossRefGoogle Scholar
  6. Bebbington M.S., Harte D.S., Jaumé S.C. (2010) Repeated intermittent earthquake cycles in the San Francisco Bay Region. Pure and Applied Geophysics, 167: 801–818CrossRefGoogle Scholar
  7. Borovkov K., Bebbington M.S. (2003) A stochastic two-node stress transfer model reproducing Omori’s law. Pure and Applied Geophysics, 160: 1429–1445CrossRefGoogle Scholar
  8. Bowsher C.G. (2007) Modelling security market events in continuous time: intensity based, multivariate point process models. Journal of Econometrics, 141: 876–912MathSciNetCrossRefGoogle Scholar
  9. Brémaud P., Massoulié L. (1996) Stability of nonlinear Hawkes processes. Annals of Probability, 24: 1563–1588MathSciNetzbMATHCrossRefGoogle Scholar
  10. Bufe C.G., Varnes D.J. (1993) Predictive modeling of the seismic cycle of the greater San Francisco Bay region. Journal of Geophysical Research, 98: 9871–9883CrossRefGoogle Scholar
  11. Daley D.J., Vere-Jones D. (2003) Introduction to the theory of point processes (2nd ed). Springer, New YorkzbMATHGoogle Scholar
  12. Fedotov S.A. (1968) The seismic cycle, quantitative seismic zoning, and long-term seismic forecasting. In: Medvedev S.V. (eds) Seismic zoning in the USSR. Izdatel’stvo Nauka, Moscow, pp 133–166Google Scholar
  13. Fischer W., Meier-Hellstern K.S. (1993) The Markov-modulated Poisson process (MMPP) cookbook. Performance Evaluation, 18(2): 149–171MathSciNetzbMATHCrossRefGoogle Scholar
  14. Fletcher R., Powell M.J.D. (1963) A rapidly convergent method for minimization. The Computer Journal, 6: 163–168MathSciNetzbMATHGoogle Scholar
  15. Harte, D. S. (2005). Package “HiddenMarkov”: discrete time hidden Markov models. R statistical program routines. Wellington: Statistics Research Associates.
  16. Hawkes A.G. (1971) Spectra of some self-exciting and mutually exciting point processes. Biometrika, 58: 83–90MathSciNetzbMATHCrossRefGoogle Scholar
  17. Hawkes A.G., Adamopoulos L. (1973) Cluster models for earthquakes-regional comparisons. Bulletin of the International Statistical Institute, 45: 454–461Google Scholar
  18. Heffes H., Lucantoni D. (1986) A Markov modulated characterization of packetized voice and data traffic related statistical performance. IEEE Journal on Selected Areas in Communications, 4: 856–868CrossRefGoogle Scholar
  19. Helmstetter, A., Sornette, D. (2002). Subcritical and supercritical regimes in epidemic models of earthquake aftershocks. Journal of Geophysical Research, 107. doi: 10.1029/2001JB001580
  20. Hill, D. P., Reasenberg, P. A., Michael, A., Arabaz, W., Beroza, G. C., Brune, J. N., Brumbaugh, D., Davis, S., DePolo, D., Ellsworth, W. L., Gomberg, J., Harmsen, S., House, L., Jackson, S. M., Johnston, M., Jones, L., Keller, R., Malone, S., Nava, S., Pechmann, J. C., Sanford, A., Simpson, R. W., Smith, R. S., Stark, M., Stickney, M., Walter, S., Zollweg, J. (1993). Seismicity in the western United States remotely triggered by the M7.4 Landers, California, earthquake of June 28, 1992. Science, 260, 1617–1623.Google Scholar
  21. Hill D.P., Johnston M.J.S., Langbein J.O., Bilham R. (1995) Response of Long Valley caldera to the M w = 7.3 Landers, California, earthquake. Journal of Geophysical Research, 100: 12985–13005CrossRefGoogle Scholar
  22. Hughes J.P., Guttorp P. (1994) A class of stochastic models for relating synoptic atmospheric patterns to regional hydrologic phenomena. Water Resources Research, 30: 1535–1546CrossRefGoogle Scholar
  23. Jaumé, S. C., Bebbington, M. S. (2004). Accelerating seismic release from a self-correcting stochastic model. Journal of Geophysical Research, 109, B12301. doi: 10.1029/2003JB002867
  24. MacDonald I., Zucchini W. (1997) Hidden-Markov and other models for discrete-valued time series. Chapman and Hall, New YorkzbMATHGoogle Scholar
  25. Marsan D. (2003) Triggering of seismicity at short timescales following Californian earthquakes. Journal of Geophysical Research, 108: 2266. doi: 10.1029/2002JB001946 CrossRefGoogle Scholar
  26. Marsan D., Nalbant S.S. (2005) Methods for measuring seismicity rate changes: a review and a study of how the M w 7.3 Landers earthquake affected the aftershock sequence of the M w 6.1 Joshua Tree earthquake. Pure and Applied Geophysics, 162: 1151–1185CrossRefGoogle Scholar
  27. Mogi K. (1968) Source locations of elastic shocks in the fracturing process in rocks (1). Bulletin of Earthquake Research Institute, 46: 1103–1125Google Scholar
  28. Ogata Y. (1988) Statistical models for earthquake occurrences and residual analysis for point processes. Journal of the American Statistical Association, 83(401): 9–27CrossRefGoogle Scholar
  29. Ogata Y. (1998) Space-time point-process models for earthquake occurrences. Annals of the Institute of Statistical Mathematics, 50: 379–402zbMATHCrossRefGoogle Scholar
  30. Ogata, Y., Jones, L. M., Toda, S. (2003). When and where the aftershock activity was depressed: contrasting decay patterns of the proximate large earthquakes in southern California. Journal of Geophysical Research, 108(B6), 2318. doi: 10.1029/2002JB002009 (ESE1-12).
  31. Pievatolo A., Rotondi R. (2008) Statistical identification of seismic phases. Geophysical Journal International, 173: 942–957CrossRefGoogle Scholar
  32. Rabiner L.R. (1989) A tutorial on hidden Markov models and selected applications in speech recognition. Proceedings of the IEEE, 77: 257–286CrossRefGoogle Scholar
  33. Roberts W.J.J., Ephraim Y., Dieguez E. (2006) On Rydén’s EM algorithm for estimating MMPPs. IEEE Signal Processing Letters, 13(6): 373–376CrossRefGoogle Scholar
  34. Rydén T. (1994) Parameter estimation for Markov modulated Poisson processes. Communications in Statistics–Stochastic Models, 10(4): 795–829MathSciNetzbMATHCrossRefGoogle Scholar
  35. Rydén T. (1996) An EM algorithm for estimation in Markov-modulated Poisson processes. Computational Statistics & Data Analysis, 21: 431–447MathSciNetzbMATHCrossRefGoogle Scholar
  36. Schwarz G. (1978) Estimating the dimension of a model. Annals of Statistics, 6: 461–464MathSciNetzbMATHCrossRefGoogle Scholar
  37. Shibata R. (1980) Asymptotically efficient selection of the order of the model for estimating parameters of a linear process. The Annals of Statistics, 8: 147–164MathSciNetzbMATHCrossRefGoogle Scholar
  38. Shibata R. (1981) An optimal selection of regression variables. Biometrika, 68(1): 45–54MathSciNetzbMATHCrossRefGoogle Scholar
  39. Utsu T., Ogata Y., Matsu’ura R.S. (1995) The centenary of the Omori formula for a decay law of aftershock activity. Journal of Physics of the Earth, 43: 1–33CrossRefGoogle Scholar
  40. Van Loan C.F. (1978) Computing integrals involving the matrix exponential. IEEE Transactions on Automatic Control, AC- 23(3): 395–404zbMATHCrossRefGoogle Scholar
  41. Vere-Jones D., Robinson R., Yang W. (2001) Remarks on the accelerated moment release model: problems of model formulation, simulation and estimation. Geophysical Journal International, 144: 517–531CrossRefGoogle Scholar
  42. Wang, T. (2010). Statistical models for earthquakes incorporating ancillary data. PhD thesis, New Zealand: Massey University.Google Scholar
  43. Zhuang J. (2000) Statistical modeling of seismicity patterns before and after the 1990 Oct 5 Cape Palliser earthquake, New Zealand. New Zealand Journal of Geology and Geophysics, 43: 447–460CrossRefGoogle Scholar
  44. Zucchini W., Guttorp P. (1991) A hidden Markov model for space-time precipitation. Water Resources Research, 27(8): 1917–1923CrossRefGoogle Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 2010

Authors and Affiliations

  1. 1.Volcanic Risk SolutionsMassey UniversityPalmerston NorthNew Zealand
  2. 2.Statistics Research AssociatesWellingtonNew Zealand

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