![]() Geological Survey Circular 1053, 51 pp, 1990. Working Group on California Earthquake Probabilities, Probabilities of large earthquakes in the San Francisco Bay Region, California: U.S. Geological Survey Open-File Report 88-398, 1988. Working Group on California Earthquake Probabilities, Probabilities of large earthquakes occurring in California on the San Andreas Fault, U.S. Vere-Jones, D., Probabilities and information gain for earthquake forecasting, Comput. Utsu, T., Zisin Katsudo Sosetsu, 876 pp, Univesty of Tokyo Press, 1999 (in Japanese). Utsu, T., Estimation of parameters for recurrence models of earthquakes, Bull. Kitagawa, Akaike Information Criterion Statistics, 290 pp, D. Chapter 4 offers detailed statistical analysis. Reasenberg, Brownian Model for recurrent earthquakes, Bull. Chapter 3 details the research methods used to test the hypotheses. ![]() The problem is to maximize the expected length of. Knopoff, Statistical short-term earthquake prediction, Science, 236, 4808, 1563–1567, 1987. As a variable is inspected, it can be either selected or rejected and this decision becomes at once final. Imoto, M., Application of the stress release model to the Nankai earthquake sequence, southwest Japan, Tectonophysics, 338, 287–295, 2001. Imoto, M., Quality factor of earthquake probability models in terms of mean information gain, Zisin II, 53, 2000 (in Japanese). Imoto, M., Information criterion of precursors, Zisin II, 47, 1994 (in Japanese). Rhoades, The precursory earthquake swarm in New Zealand: Hypothesis tests, N. 1, Elementary theory and methods, Second edition, 469 pp, Springer, New York, 2003.Įarthquake Research Committee, the Headquarters for Earthquake Research Promotion, Government of Japan, Long-term evaluation of earthquakes in the sea off Miyagi Prefecture, URL, 2000 (in Japanese).Įarthquake 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).Įarthquake 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, 2002 (in Japanese).Įvison, F. Vere-Jones, An introduction to the theory of point processes, vol. 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.ĭaley, D. We then demonstrate that the expected value of the probability gain in observed parameter values ranges between 2 and 5. F is arithmetic with span d (0, ), and h is a multiple of d. For h > 0, U(t, t + h U(t + h) U(t) h / as t in each of the following cases: F is non-arithmetic. This conversion reduces the degrees of freedom of model parameters to 1. Next we have the renewal theorem for the delayed renwal process, also known as Blackwells theorem, named for David Blackwell. ![]() Next, by converting the time unit into the expected value of the interval, the hazard is made to represent a probability gain. First, we show that the expected value of the log-likelihood difference becomes the expected value of the logarithm of the probability gain. 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. The values of two parameters of these distributions are determined by the maximum likelihood method. A renewal-reward process additionally has a random sequence of rewards incurred at each holding time, which are IID but need not be independent of the holding times.Ī renewal process has asymptotic properties analogous to the strong law of large numbers and central limit theorem.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. Instead of exponentially distributed holding times, a renewal process may have any independent and identically distributed (IID) holding times that have finite mean. Renewal theory is the branch of probability theory that generalizes the Poisson process for arbitrary holding times. ![]()
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