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

Abstract

Dealing with data from a space-time point process, the estimation of the conditional intensity function is a crucial issue even if a complete definition of a parametric model is not available. In particular, in case of exploratory contexts or if we want to assess the adequacy of a specific parametric model, some kind of nonparametric estimation procedure could be useful.Often, for these purposes kernel estimators are used and the estimation of the intensity function depends on the estimation of bandwidth parameters. In some fields, like for instance the seismological one, predictive properties of the estimated intensity function are pursued. Since a direct ML approach cannot be used, we propose an estimation procedure based on the subsequent increments of likelihood obtained adding an observation one at a time. Simulated results and some applications to statistical seismology are provided.

A Review of Self-Exciting Spatio-Temporal Point Processes and Their Applications

“…To avoid this problem, Chiodi and Adelfio (2011) introduced the Forward Likelihood-based Predictive approach (FLP). Rather than directly maximizing the likelihood, consider increments in the log-likelihood, using the first k observations to predict the (k + 1)th:…”

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

“…In this package, we use the method proposed in Chiodi and Adelfio (2011), that measures the ability of the observations and estimates until t k to give information on the next observation.…”

“…The etasFLP package is mainly based on those ideas. In Chiodi (2013, 2015a), we classified events according to their probability of being a background or an offspring event, as proposed by Zhuang et al (2002), and then estimated the space-time intensity of the generating point process of the different components by mixing non-parametric and parametric approaches, applying a forward predictive likelihood estimation approach to semi-parametric models (Chiodi and Adelfio 2011;Adelfio and Chiodi 2015a). The probabilities of being a background event are used as weights in non-parametric intensity estimation of the background seismicity.…”

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

“…In this paper, we use the method proposed in Chiodi and Adelfio (2011) that measures the ability of the observations and estimation until t k to give information on the next observation.…”