Model checking for hidden Markov models

Hidden Markov models (HMMs) were first introduced in the late 1960s, and later applied widely in areas including speech recognition, bioinformatics, finance and seismology. An HMM is used to model time series data when the observations are dependent on an underlying unobserved Markov chain, with estimation of the parameters achieved by directly maximising the likelihood or by using the Expectation-Maximization algorithm. In the end, one finds the most likely sequence of hidden states over time given the data and model parameters. An important part of any statistical study, however, is checking the fit of the model to assess how well it captures key features of the data. Unfortunately, as LML External Fellow Jiancang Zhuang and colleagues argue in a new paper, research on model checking for HMMs is underdeveloped and there is a need for new methods.
In the paper, the authors examine new model checking techniques in the context of a recently introduced two-dimensional HMM with extra zeros. This model was initially developed to classify non-volcanic tremor in the Kii and Shikoku regions of Japan. Observations of detected tremor are clustered in time and space interspersed with long periods of quiescence. Previously, classification of such data was a manual process, but HMMs developed by Zhuang and colleagues (Wang et al. 2017, 2018) automated this process. Here Zhuang and colleagues develop new tools for checking the fit of HMM, and specifically investigate a model checking theory for point process models developed by Baddeley et al.(2005) and Zhuang (2006).
Their analysis tests whether the selected 17 state 2-D HMM with extra zeros was a good fit for the data. They found that estimates of some key parameters were significantly lower than the empirical values and confirmed findings in Wang et al.(2018) of a violation of the assumption of a stationary Bernoulli element of the mixture emission, a key target for future improved modelling. The analysis also found that some states exhibited a clear bi-modal distribution given that a tremor occurred, again suggesting a need for an alternative approach to model selection. As the study concludes, non-volcanic tremor data is complex and capturing all features of the data is challenging but methods under development offer clear scope for improvement in model testing.
The paper is available here.

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