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Probability gains expected for renewal process models

Abstract

We usually use the Brownian distribution, lognormal distribution, Gamma distribution, Weibull distribution, and exponential distribution to calculate long-term probability for the distribution of time intervals between successive events. The values of two parameters of these distributions are determined by the maximum likelihood method. The difference in log likelihood between the proposed model and the stationary Poisson process model, which scores both the period of no events and instances of each event, is considered as the index for evaluating the effectiveness of the earthquake probability model. First, we show that the expected value of the log-likelihood difference becomes the expected value of the logarithm of the probability gain. Next, by converting the time unit into the expected value of the interval, the hazard is made to represent a probability gain. This conversion reduces the degrees of freedom of model parameters to 1. We then demonstrate that the expected value of the probability gain in observed parameter values ranges between 2 and 5. Therefore, we can conclude that the long-term probability calculated before an earthquake may become several times larger than that of the Poisson process model.

References

  • Daley, D. J. and D. Vere-Jones, An introduction to the theory of point processes, vol. 1, Elementary theory and methods, Second edition, 469 pp, Springer, New York, 2003.

    Google Scholar 

  • Earthquake Research Committee, the Headquarters for Earthquake Research Promotion, Government of Japan, Long-term evaluation of earthquakes in the sea off Miyagi Prefecture, URL http://www.jishin.go.jp/main/chousa/00nov4/miyagi.htm, 2000 (in Japanese).

  • Earthquake Research Committee, the Headquarters for Earthquake Research Promotion, Government of Japan, Regarding methods for evaluating long-term probability of earthquake occurrence, 46 pp, 2001 (in Japanese).

  • Earthquake Research Committee, the Headquarters for Earthquake Research Promotion, Government of Japan, Long-term evaluation of earthquakes in the sea off from Sanriku to Boso, URL http://www.jishin.go.jp/main/chousa/02julsanriku/index.htm, 2002 (in Japanese).

  • Evison, F. F. and D. A. Rhoades, The precursory earthquake swarm in New Zealand: Hypothesis tests, N. Z. J. Geol. Geophys., 36, 51–60, 1993.

    Article  Google Scholar 

  • Imoto, M., Information criterion of precursors, Zisin II, 47, 1994 (in Japanese).

  • Imoto, M., Quality factor of earthquake probability models in terms of mean information gain, Zisin II, 53, 2000 (in Japanese).

  • Imoto, M., Application of the stress release model to the Nankai earthquake sequence, southwest Japan, Tectonophysics, 338, 287–295, 2001.

    Article  Google Scholar 

  • Kagan, Y. Y. and L. Knopoff, Statistical short-term earthquake prediction, Science, 236, 4808, 1563–1567, 1987.

    Article  Google Scholar 

  • Matthews, M. V., W. L. Ellsworth, and P. A. Reasenberg, Brownian Model for recurrent earthquakes, Bull. Seism. Soc. Am., 92, 2233–2250, 2002.

    Article  Google Scholar 

  • Sakamoto, Y., M. Ishiguro, and G. Kitagawa, Akaike Information Criterion Statistics, 290 pp, D. Reidel, Dordrecht, 1983.

    Google Scholar 

  • Utsu, T., Estimation of parameters for recurrence models of earthquakes, Bull. Earthq. Res. Inst., 59, 53–66, 1984.

    Google Scholar 

  • Utsu, T., Zisin Katsudo Sosetsu, 876 pp, Univesty of Tokyo Press, 1999 (in Japanese).

    Google Scholar 

  • Vere-Jones, D., Probabilities and information gain for earthquake forecasting, Comput. Seismol., 30, 248–263, 1998.

    Google Scholar 

  • Working Group on California Earthquake Probabilities, Probabilities of large earthquakes occurring in California on the San Andreas Fault, U.S. Geological Survey Open-File Report 88-398, 1988.

  • Working Group on California Earthquake Probabilities, Probabilities of large earthquakes in the San Francisco Bay Region, California: U.S. Geological Survey Circular 1053, 51 pp, 1990.

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Correspondence to Masajiro Imoto.

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Imoto, M. Probability gains expected for renewal process models. Earth Planet Sp 56, 563–571 (2004). https://doi.org/10.1186/BF03352517

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  • DOI: https://doi.org/10.1186/BF03352517

Key words

  • Probability gain
  • information gain
  • renewal model
  • Kull-back Leibler
  • long-term probability
  • Gamma
  • Brownian
  • Weibull