Modeling Continuous Time Series With Many Zeros

Abstract

Earthquake activity is generally modeled using point processes as earthquake events usually occur at random times and locations. Recent studies have found it mathematically challenging and computationally complex to incorporate a point process model into a hidden Markov model to describe long-term seismicity. Given that earthquake data can be discretized in time to consider daily or hourly energy release, time series models could be a useful method for earthquake data analysis. Time series models can account for the autocorrelation of earthquakes. However, one issue that arises with the earthquake occurrence data is that there is a substantial proportion of time when no earthquake is recorded. This thesis proposes a class of two-part autoregressive (2PAR) models for continuous time series data with excess zeros. We employ a Bernoulli variable to model the excess zeros in the data, and use autoregressive processes to describe the serial correlation. Using this class of 2PAR models, we can model correlations that exist in either zeros or nonzeros in the data. We have proposed a class of residual analysis to check the goodness-of-fit of the proposed models. We also introduced a forecasting procedure using simulation to check the performance of the models.

We carried out a simulation study which shows that the estimators are unbiased and consistent, and the residual analysis and forecasting procedure for the proposed models are promising. We applied the proposed models to the energy indices obtained from the total stress release per hour from the 2010 Darfield earthquake sequence. The results reveal that the 2PAR models with serial correlation in both the presence probability and the earthquake energy indices captured the main features of the data. A retrospective forecasting experiment suggested that the proposed models provide higher information gain against a reference model.