In the present project, we consider problem of estimating and forecasting earthquakes in Italy. We refer to real data data from the Catalogo Parametrico dei Terremoti Italiani 2015 (CPTI15) and a homogeneous Italian catalog in terms of moment magnitude (Gasperini et al., 2013).
Standard methods for dealing with this kind of data include nonparametric procedures such as kernel smoothing, which usually rely on too simplistic or unrealistic choices.
As an alternative, we propose a Bayesian nonparametric approach to study Italian seismicity from 1981 to 2009, based on finding Gaussian Markov random field solutions to a class of stochastic partial differential equations
(SPDEs) for adaptive smoothing splines. The model fits naturally within the Integrated Nested Laplace Approximation (INLA) framework, that allows to efficiently estimate the unknown parameters and speed up calculations.
In preliminary studies, we obtained promising results which, however, lack of optimal configurations. So, one first goal of this project is a more in-depth analysis of parameters tuning, for instance priors for the hyperparameters and support triangularization.
We also aim to model temporal features of data. Up to now, in fact, we have considered a "declustered" version of the catalog, in which foreshocks and aftershocks effects are removed, as well as temporal dependence. In real application would be useful to include both spatial and temporal nature of the phenomenon, and this could be done with spatio-temporal modeling.
References
-Gasperini, P., Lolli, B., & Vannucci, G. (2013) Empirical calibration of local magnitude data sets versus moment magnitude in Italy. Bulletin of the Seismological Society of America, 103(4), 2227-2246.
-Rovida, A., Locati, M., Camassi, R., Lolli, B., & Gasperini, P. (2016) CPTI15, the 2015 version of the Parametric Catalogue of Italian Earthquakes. Istituto Nazionale di Geofisica e Vulcanologia.
Time dependent earthquake model provides a more accurate representation of reality. For this reason, scientists encourage the development of testable models of time-varying earthquake occurrence. Learning how to forecast earthquakes is one of the most important problem in seismology. From a scientific perspective, our ability to forecast earthquakes is a measure of our understanding of how earthquakes are generated. From a practical perspective, foreknowledge of an increased hazard of earthquakes occurrence in a particular location would be useful for decision-making on the timing of mitigation measures (Savage et al., 2011).
It is common to use MCMC sampling to make inference for latent Gaussian models. However, it is well known that MCMC tends to exhibit poor performance when applied to such models because first, the latent Gaussian variables are strongly dependent on each other; second, the latent Gaussian variables and the hyperparameters are also strongly dependent, especially when the sample is large (Rue et al., 2009). On the other hand, INLA returns accurate approximation of marginal posterior densities due to the latent Gaussian prior and, also, computes estimates much faster than general MCMC techniques. INLA has been developed as a computationally efficient alternative to MCMC and the availability of an R package (R-INLA) allows researchers to easily apply this method.
Furthermore, INLA can be combined with the Stochastic Partial Differential Equation (SPDE) approach proposed by Lindgren et al. (2011) in order to implement spatial and spatio-temporal models for point-reference data.
References
-Lindgren, F., Rue, H., & Lindström, J. (2011) An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 73(4), 423-498.
-Rue, H., Martino, S., Chopin, N. (2009) Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations. Journal of the royal statistical society: Series b, 71(2), 319-392
-Savage, M., Rhoades, D. A., Smith, E. G., Gerstenberger, M. C., & Vere-Jones, D. (2010) Seismogenesis and Earthquake Forecasting. Springer.