Model selection and model checking for hidden Markov models applied to non-volcanic tremor data

Abstract

Hidden Markov models (HMMs) are commonly used to model time series data and are now widely applied in many fields. In the field of seismology, HMMs have recently been applied to data collected from non-volcanic tremor events. As new HMMs such as these are developed, the issues of model selection and model checking must be considered. One open question regarding model selection for HMMs is how to estimate the number of hidden states in the Markov chain. Currently, AIC and BIC are commonly used for this purpose despite some evidence of poor performance. There is also a need for new tools to check the validity of HMMs with complex structures. Here, motivated by the HMMs developed to model non-volcanic tremor, we consider both the issue of selecting the number of hidden states in an HMM and how to identify and diagnose lack of fit for a selected model. Through simulation studies, we compare the performance of various model selection information criteria when used to select the number of hidden states in HMMs. We find that AIC and BIC are not always reliable tools for selecting the number of hidden states in HMMs and that other information criteria can perform better, depending on factors such as sample size and sojourn times in each state. In addition to addressing the model selection issue, we propose new residual analysis and stochastic reconstruction methods for checking the fit of HMMs. The new methods are adapted from model checking techniques for point process models and enhance current model checking methods for HMMs. We find that 1) our residual analysis is particularly useful for models where pseudo-residuals are difficult to interpret, and 2) stochastic reconstruction is useful for diagnostic purposes. We apply the new model checking methods to our selected HMM fitted to non-volcanic tremor data and discuss future improvements of the current model for the classification of tremor events.