Multidimensional Residual Analysis of Point Process Models for Earthquake Occurrences

Residual analysis methods for examining the fit of multidimensional point process models are applied to point process models for the space–time–magnitude distribution of earthquake occurrences, using, in particular, the multidimensional version of Ogata's epidemic-type aftershock sequence (ETAS) model and a 30-year catalog of 580 earthquakes occurring in Bear Valley, California. One method involves rescaled residuals, obtained by transforming points along one coordinate to form a homogeneous Poisson process inside a random, irregular boundary. Another method involves thinning the point process according to the conditional intensity to form a homogeneous Poisson process on the original, untransformed space. The thinned residuals suggest that the fit of the model may be significantly improved by using an anisotropic spatial distance function in the estimation of the spatially varying background rate. Using rescaled residuals, it is shown that the temporal–magnitude distribution of aftershock activity is not separable, and that, in particular, in contrast to the ETAS model, the triggering density of earthquakes appears to depend on the magnitude of the secondary events in question. The residual analysis highlights that the fit of the space–time ETAS model may be improved by allowing the parameters governing the triggering density to vary for earthquakes of different magnitudes. Such modifications may be important because the ETAS model is widely used in seismology for hazard analysis.

[1]  Earthquake risk and insurance , 1993 .

[2]  Yosihiko Ogata,et al.  On Lewis' simulation method for point processes , 1981, IEEE Trans. Inf. Theory.

[3]  Yan Y. Kagan,et al.  Likelihood analysis of earthquake catalogues , 1991 .

[4]  A. Hawkes.  Point Spectra of Some Mutually Exciting Point Processes , 1971 .

[5]  D. Veee-Jones.  STATISTICAL ASPECTS OF THE ESTIMATION OF EARTHQUAKE RISK1,2 , 1975 .

[6]  Yosihiko Ogata,et al.  Statistical Models for Earthquake Occurrences and Residual Analysis for Point Processes , 1988 .

[7]  L Song,et al.  STOCHASTIC MODELS FOR EARTHQUAKE OCCURRENCE , 1980 .

[8]  G. Shedler,et al.  Simulation of Nonhomogeneous Poisson Processes by Thinning , 1979 .

[9]  M. Gopalan Nair,et al.  Random Space Change for Multiparameter Point Processes , 1990 .

[10]  A. Karr,et al.  Point Processes and Their Statistical Inference. 2nd edn. , 1993 .

[11]  David Nualart,et al.  A Characterization of the Spatial Poisson Process and Changing Time , 1986 .

[12]  P. Brémaud.  Point processes and queues, martingale dynamics , 1983 .

[13]  Andrew B. Lawson,et al.  On tests for spatial trend in a non-homogeneous Poisson process , 1988 .

[14]  Timothy C. Brown,et al.  A simple proof of the multivariate random time change theorem for point processes , 1988, Journal of Applied Probability.

[15]  Frederic Paik Schoenberg,et al.  On Rescaled Poisson Processes and the Brownian Bridge , 2002 .

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

[17]  R. Cowan.  An introduction to the theory of point processes , 1978 .

[18]  Peter J. Diggle,et al.  On parameter estimation and goodness-of-fit testing for spatial point patterns , 1979 .

[19]  Peter Breeze,et al.  Point Processes and Their Statistical Inference , 1991 .

[20]  J. Jacod.  Multivariate point processes: predictable projection, Radon-Nikodym derivatives, representation of martingales , 1975 .

[21]  Y. Ogata.  Space-Time Point-Process Models for Earthquake Occurrences , 1998 .

[22]  Yan Y. Kagan,et al.  Problems in the Modelling and Statistical Analysis of Earthquakes , 1996 .

[23]  Niels Keiding,et al.  Statistical Models Based on Counting Processes , 1993 .

[24]  E. Khmaladze,et al.  An innovation approach to goodness of fit tests in $R m$ , 1988 .

[25]  F. Schoenberg,et al.  Short-term Exciting, Long-term Correcting Models for Earthquake Catalogs , 2000 .

[26]  Donald L. Snyder,et al.  The statistical analysis of space-time point processes , 1976, IEEE Trans. Inf. Theory.

[27]  B. Gutenberg,et al.  Frequency of Earthquakes in California , 1944, Nature.

[28]  D. Vere-Jones,et al.  Stochastic Declustering of Space-Time Earthquake Occurrences , 2002 .

[29]  D. Nualart,et al.  A Martingale Approach to Point Processes in the Plane , 1988 .

[30]  F. Papangelou.  Integrability of expected increments of point processes and a related random change of scale , 1972 .

[31]  Daryl J. Daley,et al.  An Introduction to the Theory of Point Processes , 2013 .

[32]  Odd O. Aalen,et al.  Random time changes for multivariate counting processes , 1978 .

[33]  L. Heinrich,et al.  Goodness-of-fit tests for the second moment funciton of a stationary multidimensional poisson process , 1991 .

[34]  B. Ripley.  Tests of 'Randomness' for Spatial Point Patterns , 1979 .

[35]  P. Meyer,et al.  Demonstration simplifiee d'un theoreme de Knight , 1971 .

[36]  O. Kallenberg.  Random time change and an integral representation for marked stopping times , 1990 .

[37]  Frederic Paik Schoenberg,et al.  Rescaling Marked Point Processes , 2004 .

[38]  Y. Ogata,et al.  Likelihood Analysis of Spatial Point Patterns , 1984 .

[39]  M. Bebbington,et al.  A linked stress release model for historical Japanese earthquakes: coupling among major seismic regions , 1999 .

[40]  Frederic Paik Schoenberg,et al.  Transforming Spatial Point Processes into Poisson Processes , 1999 .