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| Titel |
Flexible space-time process for seismic data |
| VerfasserIn |
G. Adelfio, M. Chiodi |
| Konferenz |
EGU General Assembly 2009
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| Medientyp |
Artikel
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| Sprache |
Englisch
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| Digitales Dokument |
PDF |
| Erschienen |
In: GRA - Volume 11 (2009) |
| Datensatznummer |
250019302
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| Zusammenfassung |
| Introduction
Point processes are well studied objects in probability theory and a powerful tool in
statistics for modeling and analyzing the distribution of real phenomena, such as the seismic
one. Point processes can be specified mathematically in several ways, for instance, by
considering the joint distributions of the counts of points in arbitrary sets or defining a
complete intensity function. The conditional intensity function is a function of
the point history and it is itself a stochastic process depending on the past up to
timet.
In this paper some techniques to estimate the intensity function of space-time point
processes are developed, by following semi-parametric approaches and diagnostic methods to
assess their goodness of fit.
In particular, because of its particularly adaptive properties to anomalous behavior in data,
in this paper a nonparametric estimation approach is used to interpret dependence features of
seismic activity of a given area of observation; to justify the estimation approach a diagnostic
method for space-time point processes is also revised.
Flexible modeling and diagnostics for point processes
The definition of effective stochastic models to adequately describe the seismic
activity of a fixed area is of great interest in seismology, since a reliable description of
earthquakes occurrence might suggest useful ideas on the mechanism of a such complex
phenomena.
A number of statistical models have been proposed for representing the intensity function
of earthquakes. The simpler models assume that earthquakes occur in space and time
according to a stationary point process, such that conditional rate becomes a constant. In
seismology, however, the stationarity hypothesis might be acceptable only with respect to
time, because epicenters usually display a substantial degree of spatial heterogeneity and
clustering. Description of seismic events often requires the definition of more complex
models than stationary Poisson process and the relaxation of the assumption of statistical
independence of earthquakes. Therefore, second-order properties of point processes
may have a relevant role in the study and the comprehension of seismic process
and its realization. Indeed, when aggregation is present, it is useful to introduce
some generalizations of the simple Poisson process, such as self-exciting point
processes, to model events that are clustered together, and self-correcting processes
when regularities are observed, e.g. the strain-release model (Daley and Vere-Jones,
2003). A widely used model is ETAS model (Ogata, 1988), that is a self-exciting
point process, describing earthquakes activity, in a given region during a period
of time, through a branching structure. Also in this field, the parametric models
estimation suffers by many drawbacks, often related to the definition of a reliable
mathematical model from the geophysical theory and to the sensitivity of statistical
estimates to the composition of the sample, that is the space-time region under
study.
Many of the disadvantages of the parametric modeling can be avoided by making use
of nonparametric techniques, such as kernel density methods (Silverman, 1986).
Therefore a flexible model, that is useful in presence of several data for which a
not immediately obvious discrimination between principal and secondary events
is not reliable, estimated by nonparametric method is proposed. In particular, a
three dimensional Gaussian kernel estimator is used (Adelfio and Ogata, 2008),
developing a method of estimation with predictive features, taking into account the
dependence on the past history of a point process. For this purpose variation of
the likelihood function to measure the capability of the observations until a fixed
time to give information on the next observation is considered (Adelfio and Chiodi,
2008a).
To assess the goodness of fit of a given model diagnostic methods are necessary (Adelfio
and Chiodi, 2008b). Therefore, a new method for point processes is here considered (Adelfio
and Schoenberg, 2008); it is based on the interpretation of second-order statistics (such as
R/S, correlation integral, spectral density) weighted by a quantity proportional to the inverse
of the conditional intensity function. Such transformed statistics are useful to test the fit of
space-time point processes when features like self-similarity, long-range dependence and
fractal dimension have to be taken into account for a deeper comprehension of the observed
phenomena.
References
Adelfio, G. and Chiodi, M. (2008a). Nonparametric intensity estimation in space-time
point processes and application to seismological problems. Proceedings of The first joint
meeting of the Soci |
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