A Bayesian spatial–temporal model with latent multivariate log‐gamma random effects with application to earthquake magnitudes

Abstract

We introduce a Bayesian spatial-temporal model for analyzing earthquake magnitudes. Specifically, we define a spatial-temporal Pareto regression model with latent multivariate log-Gamma random vectors to analyze earthquake magnitudes. This represents a marked departure from the traditional spatial generalized linear regression model, which uses latent Gaussian random effects. The multivariate log-Gamma distribution results in a full-conditional distribution that can be easily sampled from, which leads to a fast mixing Gibbs sampler. Thus, our proposed model is a computationally efficient approach for modelling Pareto spatial data. The empirical results suggest similar estimation properties between the latent Gaussian model and latent multivariate log-Gamma model, but our proposed model has stronger predictive properties. Additionally, we analyze a small US earthquake data set as an illustration of the effectiveness of our approach.

Bayesian Variable Selection for Pareto Regression Models with Latent Multivariate Log Gamma Process with Applications to Earthquake Magnitudes

“…In earthquake data, most recorded earthquakes have a magnitude around 3-5, but sometime there will have some significant earthquakes with large magnitude. Hu [5] used Pareto regression to model earthquake magnitudes, since the Pareto distribution is a heavy tailed distribution with a threshold. Earthquake magnitude data also has a threshold, since people consider earthquake only over a certain magnitude.…”

“…They just built simple linear regression models or generalized linear models to explore covariates effects on earthquake magnitudes [4]. Hu and Bradley [5] proposed using the Pareto regression with spatial random effects for earthquake magnitudes, but they did not consider the model selection problems. In order to have more explicit understanding of dependent covariates of earthquake magnitudes, variable selection approaches should be considered in a Pareto regression model with spatial random effects.…”

“…However, there are some difficulties for Bayesian variable selection to carry out because of the challenge in assigning prior distributions for the parameters. In order to tackle this issue, we consider the multivariate log-Gamma distribution (MLG) based on Bradley et al [15], which is conjugate with the Pareto distribution [5]. Hence, the Bayesian approach to variable selection is straightforward for our model.…”

Estimating Spatial Changes Over Time of Arctic Sea Ice using Hidden 2×2 Tables

“…In this article, we focus on a low-rank spatial linear mixed effects model (Cressie and Johannesson, 2006; to achieve dimension-reduction for the latent random effects in the spatio-temporal GLM, where the spatio-temporal correlations are modeled by a dynamic spatio-temporal model (e.g., Wikle et al, 2001;Kang et al, 2010;Katzfuss and Cressie, 2011). Recent developments on efficient Bayesian inference based on spatio-temporal GLMs can be found in Holan and Wikle (2016), Bradley et al (2018), Hu and Bradley (2018) and references therein. Through pre-specified basis functions, the spatial linear mixed effects model induces a non-stationary spatial field at different time points, which is very flexible and, in regional, oceanic, and global 290 B. ZHANG AND N. CRESSIE applications, may be preferred over parametric (stationary) covariance models.…”

Interseismic deformation transients and precursory phenomena: Insights from stick-slip experiments with a granular fault zone

“…Current models for earthquake recurrence incorporate mathematical models of earthquake statistics (Gutenberg-Richter, Omori-Utsu-Aftershocks, Brownian-First-Passage-Time), numerical models of earthquakes and rupture processes (Rate-and-State-Friction), interseismic stress built-up and the interaction of multiple faults over a larger area via stress transfer (e.g. Brinkman et al, 2016;Ellsworth et al, 1999;Field et al, 2014;Hainzl et al, 2013;Hu & Bradley, 2018;Kawamura et al, 2012;Lapusta & Rice, 2003;Parsons, 2005;Zöller et al, 2011). These models inherently rely on the accurate description and characterization of fault properties and behaviour, as well as extensive catalogues of slip events.…”