Point process diagnostics based on weighted second-order statistics and their asymptotic properties

Abstract

A new approach for point process diagnostics is presented. The method is\ud based on extending second-order statistics for point processes by weighting each point\ud by the inverse of the conditional intensity function at the point’s location. The result\ud is generalized versions of the spectral density, R/S statistic, correlation integral and\ud K-function, which can be used to test the fit of a complex point process model with\ud an arbitrary conditional intensity function, rather than a stationary Poisson model.\ud Asymptotic properties of these generalized second-order statistics are derived, using\ud an approach based on martingale theory

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

“…In the ETAS model, non-parametric components are associated mainly to background seismicity, while parametric ones are related to induced or triggered seismicity. Adelfio, Chiodi, and Luzio (2010) presented a seismic sequences detection technique that identifies the conditional intensity function of a model describing the seismic activity as an ETAS model; Adelfio (2010a) used non-parametric methods to estimate the intensity function of a space-time point process and interpreted clustering results by a second-order diagnostic approach; see also Adelfio and Schoenberg (2009), as well as Adelfio and Chiodi (2009). Console, Jackson, and Kagan (2010) proposed a stochastic method associating to each event a probability to be either a background event or an offspring generated by other events; Marsan and Lengliné (2008) used the concept of cascade triggering without using models.…”

Alternated estimation in semi-parametric space-time branching-type point processes with application to seismic catalogs

“…Adelfio et al (2010) presented a seismic sequences detection technique based on MLE of parameters, that identifies the conditional intensity function of a model describing the seismic activity as a clustering-process, like ETAS model. In Adelfio (2010) nonparametric methods are used to estimate the intensity function of a space-time point process and clustering results are interpreted by a secondorder diagnostic approach (Adelfio and Schoenberg 2009;Adelfio and Chiodi 2009). Console et al (2010) proposed a stochastic method associating to each event a probability to be either a background event or an offspring generated by other events; Marsan and Lenglin (2008) used the concept of cascade triggering without using models; Diaz-Avalos et al (2013) used also a nonparametric approach to check the separability of a point process.…”

Forward likelihood‐based predictive approach for space–time point processes

“…Therefore, more complex models than stationary Poisson process are usually considered to relax the assumption of statistical independence of earthquakes (for diagnostics for general space-time point processes see Adelfio and Chiodi, 2009;Adelfio and Schoenberg, 2009). For instance, self-exciting point processes are often used to model events that are clustered together; ETAS model (Ogata, 1998) is a well known case of self-exciting point process used to describe earthquakes activity, in a given region during a period of time, through a branching structure.…”