Analysing the interevent time distribution to identify seismicity phases: a Bayesian nonparametric approach to the multiple‐changepoint problem

In the study of earthquakes, several aspects of the underlying physical process, such as the time non‐stationarity of the process, are not yet well understood, because we lack clear indications about its evolution in time. Taking as our point of departure the theory that the seismic process evolves in phases with different activity patterns, we have attempted to identify these phases through the variations in the interevent time probability distribution within the framework of the multiple‐changepoint problem. In a nonparametric Bayesian setting, the distribution under examination has been considered a random realization from a mixture of Dirichlet processes, the parameter of which is proportional to a generalized gamma distribution. In this way we could avoid making precise assumptions about the functional form of the distribution. The number and location in time of the phases are unknown and are estimated at the same time as the interevent time distributions. We have analysed the sequence of main shocks that occurred in Irpinia, a particularly active area in southern Italy: the method consistently identifies changepoints at times when strong stress releases were recorded. The estimation problem can be solved by stochastic simulation methods based on Markov chains, the implementation of which is improved, in this case, by the good analytical properties of the Dirichlet process.

[1]  Charles F. Richter,et al.  Earthquake magnitude, intensity, energy, and acceleration , 1942 .

[2]  R. STONELEY,et al.  An Introduction to the Theory of Seismology , 1949, Nature.

[3]  D. D. Filippo,et al.  MAGNITUDO ED ENERGIA DEI TERREMOTI IN ITALIA , 1950 .

[4]  B. Gutenberg,et al.  Colloquium on plastic flow and deformation within the Earth , 1951 .

[5]  T. Ferguson.  A Bayesian Analysis of Some Nonparametric Problems , 1973 .

[6]  C. Antoniak.  Mixtures of Dirichlet Processes with Applications to Bayesian Nonparametric Problems , 1974 .

[7]  L. Knopoff,et al.  Is the sequence of earthquakes in Southern California, with aftershocks removed, Poissonian? , 1974, Bulletin of the Seismological Society of America.

[8]  A. F. Smith.  A Bayesian approach to inference about a change-point in a sequence of random variables , 1975 .

[9]  Adrian F. M. Smith.  Change-Point problems: approaches and applications , 1980 .

[10]  Stefano Tinti,et al.  Completeness analysis of a seismic catalog , 1985 .

[11]  Marco Scarsini,et al.  Change-point problems: a Bayesian nonparametric approach , 1985 .

[12]  A. Raftery,et al.  Bayesian analysis of a Poisson process with a change-point , 1986 .

[13]  R. E. Habermann.  Man-made changes of seismicity rates , 1987 .

[14]  Anne S. Kiremidjian,et al.  A review of earthquake occurrence models for seismic hazard analysis , 1988 .

[15]  P. Reasenberg,et al.  Statistical methods for investigating quiescence and other temporal seismicity patterns , 1988 .

[16]  David R. Brillinger,et al.  A more precise chronology of earthquakes produced by the San Andreas fault in southern California , 1989 .

[17]  J. Sethuraman.  A CONSTRUCTIVE DEFINITION OF DIRICHLET PRIORS , 1991 .

[18]  N. Kashiwagi.  Bayesian detection of structural changes , 1991 .

[19]  W. Gilks,et al.  Adaptive Rejection Sampling for Gibbs Sampling , 1992 .

[20]  Adrian F. M. Smith,et al.  Hierarchical Bayesian Analysis of Changepoint Problems , 1992 .

[21]  Marco Scarsini,et al.  Some Aspects of Change-Point Problems , 1993 .

[22]  Richard L. Tweedie,et al.  Markov Chains and Stochastic Stability , 1993, Communications and Control Engineering Series.

[23]  J. Hartigan,et al.  A Bayesian Analysis for Change Point Problems , 1993 .

[24]  Cinna Lomnitz,et al.  Fundamentals of Earthquake Prediction , 1994 .

[25]  D. Stephens.  Bayesian Retrospective Multiple‐Changepoint Identification , 1994 .

[26]  P. Green.  Reversible jump Markov chain Monte Carlo computation and Bayesian model determination , 1995 .

[27]  David Vere-Jones,et al.  Forecasting earthquakes and earthquake risk , 1995 .

[28]  Carlo Gaetan,et al.  Change-point analysis in stochastic processes , 1996 .

[29]  Antonietta Mira,et al.  Bayesian hierarchical nonparametric inference for change-point problems , 1996 .

[30]  A. Raftery,et al.  A note on the Dirichlet process prior in Bayesian nonparametric inference with partial exchangeability , 1997 .

[31]  Statistical Analysis of Temporal Variations of Seismicity Level in Some Italian Regions , 1999 .

[32]  Sanford Weisberg,et al.  Computing science and statistics : proceedings of the 30th Symposium on the Interface, Minneapolis, Minnesota, May 13-16, 1998 : dimension reduction, computational complexity and information , 1998 .