Detection of temporal changes in earthquake rates

Many statistical analyses of earthquake rates and time-dependent forecasting of future rates involve the detection of changes in the basic rate of events, independent of the fluctuations caused by aftershock sequences. We examine some of the statistical techniques for inferring these changes, using both real and synthetic earthquake data to check the statistical significance of these inferences. One common method is to use the Akaike Information Criterion (AIC) to choose between a single model and a double model with a changepoint; this criterion evaluates the strength of the fit and incorporates a penalty for the extra parameters. We test this method on many realisations of the ETAS model, with and without changepoints present, to see how often it chooses the correct model. A more rigorous method is to calculate the Bayesian evidence, or marginal likelihood, for each model and then compare these. The evidence is essentially the likelihood of the model integrated over the whole of the model space, giving a measure of how likely the data is for that model. It does not rely on estimation of best-fit parameters, making it a better comparator than the AIC; Occam's razor also arises naturally in this process due to the fact that more complex models tend to be able to explain a larger range of observations, and therefore the relative likelihood of any particular observations will be smaller than for a simpler model. Evidence can be calculated using Markov Chain Monte Carlo techniques. We compare these two approaches on synthetic data. We also look at the 1997-98 Colfiorito sequence in Umbria-Marche, Italy, using maximum likelihood to fit the ETAS model and then simulating the ETAS model to create synthetic versions of the catalogue for comparison. We simulate using ensembles of parameter values sampled from the posterior for each parameter, with the largest events artificially inserted, to compare the resultant event rates, inter-event time distributions and other metrics with those of the real data. We also look at the currently topical question of the clustering of global megaquakes: can these be explained as a Poisson process or an ETAS-type clustering, or does the recent spate of worldwide large events deviate significantly from these hypotheses?