Using Negative Binomial Hidden Markov models to extrapolate past states of seismicity into the future

Over the years numerous attempts have been made to obtain the distribution of earthquake numbers. The most popular distribution that has been widely used to describe earthquake numbers is the Poisson distribution due to its simplicity and relative ease of application. Another distribution that has been used to approximate the earthquake number distribution is the Negative Binomial. However, for small-time intervals, both the Poisson and Negative binomial distributions fail to fit observed earthquake frequencies. We propose an extension of mixture models that is Hidden Markov Models (HMMs) with Poisson and Negative Binomial state-specific probability distributions and thus derive Poisson (P-HMMs) and Negative Binomial Hidden Markov Models (NB-HMMs), respectively. We use the parametrization of the Negative Binomial distribution in which the probability density function is expressed in terms of the mean and the shape parameter. In this parametrization, a variance is a quadratic form of the mean and the Negative Binomial distribution tends to the Poisson distribution when the shape parameter tends to infinity. Three-time units have been selected to count the number of earthquakes, namely 1-day, 5-day, and 10-days counting intervals resulting in daily, 5-day, and 10-day time series.

The region of Killini, Western Greece has been selected to apply the proposed methodology. All earthquakes with Local Magnitude ML 3:2 have been selected in the time interval from 1990 to 2007, inclusive. This time interval is divided into two sub-intervals that correspond to the learning and the testing periods. In the learning period from 1990 to 2004, inclusive the parameters of the models are estimated while in the testing period from 2005 to 2007, inclusive the ability of the models is tested to extrapolate past states of seismicity into the future. We applied both models with a different number of states to the daily, 5-day and 10-day time series of earthquakes that occurred in the Killini region during the learning period. Based on the Bayesian Information Criterion (BIC) for all three counting intervals the NB-HMMs model with three components was selected. The best-fitting model was used to estimate through simulations the number of earthquakes expected to occur in the study area during 1-day, 5-day, and 10-day intervals for the testing period. From the results obtained it appears that regardless of the selected time unit the models are able to capture the future variations
of seismic activity.