Statistical Models for Earthquake Occurrences and Residual Analysis for Point Processes

Mixed Non-Parametric and Parametric Estimation Techniques in R Package etasFLP for Earthquakes' Description

“…The ETAS model, introduced by Ogata (1988), is a space time branching process for earthquake description, widely used in seismological context. Seismic events are usually collected in seismic catalogs, that represent the time-space-magnitude dimensions of n observed events, where the ith event is identified by its occurrence time T i , its hypocentral coordinates (x i , y i , z i ) and its magnitude m i .…”

“…In the space domain, the integral involved in the likelihood is solved through a transformation to polar coordinates centered on each observed point (Ogata 1988): The round angle is then divided in ntheta parts; the default value (ntheta = 100) is enough to obtain a very good approximation. Each of the n × ntheta slices is then integrated on the time domain.…”

“…Often the conditional intensity function presents a superposition of a parametric component over a non-parametric component, e.g., in the epidemic type aftershock sequence (ETAS) model (Ogata 1988), widely used in statistical seismology. Indeed if we want to predict large earthquakes in presence of clusters of aftershocks, these may complicate the statistical analysis of the background seismic activity (Adelfio, Chiodi, De Luca, Luzio, and Vitale 2006) so it could be useful to study the features of independent events separately from the study of the strongly correlated ones in order to describe the seismicity of an area in space, time and magnitude domains.…”

Mathematical and computer modeling of seismic processes based on a soliton approach

“…При цьому відома низка емпіричних законів. Так, наприклад, відомою є формула Оморі [2], яка описує залежність кількості поштовхів від заданої нижньої границі магнітуди та часу: …”

Evidence of Precursory Patterns in Aggregated Time Series

“…K, c and p are empirical constants. Later, Ogata (1988) introduced the ETAS model where all aftershocks can produce their own aftershocks, where the seismicity is described by Equation 2 -Formula for ETAS λ(t)=μ+K*exp(a(Mi-Mmin))/(c+t-ti) p μ represents the background seismicity, tj are the occurrence times of the events with magnitudes mj that took place before time t, m0 is the cut-off magnitude of the data (usually equal to the completeness magnitude mc) above which all events can produce secondary aftershocks, and α is a measure of the efficiency of a shock in generating aftershock activity relative to its magnitude. The parameters are estimated using the maximum likelihood method and the goodness of fit is usually tested with residual analysis (Ogata, 1999).…”