On some special-purpose hidden Markov models

Abstract

Hidden Markov models (HMMs) provide flexible devices for modelling time series of observations that depend on underlying serially correlated states. They constitute a specific class of dependent mixtures that have proved useful in many application fields. This thesis exploits the flexible mathematical structure of HMMs to develop three types of special-purpose HMMs, i.e. HMMs that differ from the standard setting and that are designed to address special demands. The first main part of the thesis considers HMMs whose matrix of state transition probabilities is structured such that it allows for arbitrary state dwell-time distributions while preserving the Markov property of the latent process. Such HMMs represent a convenient tool for approximating more flexible, but also more complicated hidden semi Markov models (HSMMs). They require fewer assumptions and enable the fitting of stationary HSMMs. Several applications illustrate the feasibility of the proposed method. In the second main part of the thesis it is shown that general-type state-space models (SSMs) can be approximated arbitrarily accurately by suitably defined HMMs. The proposed approximation method, based on HMMs, has the important advantage that it is easy to implement. Unlike the case of SSMs, where the likelihood is given by a multiple integral which cannot be evaluated directly, the likelihood of the proposed model is easy to compute; numerical maximization thus is feasible. That makes it possible to experiment with variations of models with relatively little programming effort. This is illustrated by a substantial investigation of several new variations of the well-known stochastic volatility model that were applied to series of daily returns. With reference to the recent financial crisis it is shown that a moderate increase in the flexibility, particularly of the log-volatility process, appears to enhance the model's ability to cope with extreme fluctuations of returns. Several other applications illustrate the ease with which the method can be applied to several types of SSMs. The final part of the thesis considers the modelling of sleep EEG signals via HMMs. The proposed method is applied to populations of sleep EEG time series related to well-matched subjects with and without sleep disordered breathing. The analysis confirms results from studies on sleep stage time series obtained by labour-intensive visual classification.