We conclude with a summary of the main results along with brief discussions of possi-ble future research related to the individual topics. The three main parts of the thesis, Chapters 2 – 4, deal with HMMs that address special needs. The first part, Chapter 2, is concerned with HMMs that have dwell-time distributions other than geometric.
In Chapter 3 it is demonstrated that HMMs are useful tools for fitting nonlinear and non-Gaussian state-space models. Lastly, in Chapter 4, an HMM for populations of sleep EEG time series is developed. Details of the findings are as follows.
A restrictive feature of standard HMMs is that the state dwell-time distributions are necessarily geometric. In Chapter 2 it is shown how this restriction can be relaxed to allow for arbitrary dwell-time distributions while preserving the Markov property of the latent process. This is done by implementing an existing idea, the use of state aggre-gates, in a new way. The resulting class of HMMs can represent any given dwell-time distribution, either exact or approximately, where, in general, the approximation can be made arbitrarily accurate. The models described in Chapter 2 can either be regarded as approximations to HSMMs or as extensions of ordinary HMMs that offer additional flexibility for the state dwell-time distributions.
The range of methodology that is currently available for HMMs is much more extensive than that for HSMMs. HMMs are easier to apply and to adapt to meet the needs of applications having special features. In particular one can easily incorporate covariates, in the state-dependent process as well as in the latent process. Furthermore, unlike in the case of HSMMs, it is simple to fit stationary models of the proposed type.
The literature on hidden semi-Markov modelling contains relatively few applications.
Generally, in the author’s view, HSMMs have not yet attracted the attention they de-serve. Perhaps the ease of the HMM approximation method can make a contribution in that it makes them more conveniently accessible to practitioners.
Summary and outlook
HMMs with finite state space are nested in the broader family of SSMs. However, in general the likelihood of SSMs can not be evaluated directly; statistical inference for nonlinear and non-Gaussian SSMs with infinite state space is usually much more challenging than for (standard) HMMs. The material in Chapter 3 illustrates that structured HMMs provide convenient and flexible devices for accurately approximat-ing a diverse variety of SSMs. More precisely, it is shown that general-type SSMs can be approximated by suitably structured HMMs. The approximation can be made arbitrarily accurate at the cost of increasing numerical complexity. One of the main benefits compared to competing approaches, in particular to Monte Carlo methods, is that the programming effort involved in fitting the structured HMMs is very modest (cf. Appendix A4, which contains the functions used to compute and to maximize the likelihood for the SVt model; fitting other SSMs with the proposed method generally involves no more than straightforward changes to that code.). As in case of the HMM approximations to HSMMs, the proposed method enables one to apply all standard HMM techniques.
One of the most important applications of the approximation method via structured HMMs is stochastic volatility modelling. The evidence presented in Section 3.2 sup-ports the claim that nonstandard SV models can outperform the standard SV models SV0andSVtin terms of the AIC, goodness of fit (as assessed by the behaviour of resid-uals) and also the type of backtesting that is applied by central banks and regulatory authorities in order to assess the accuracy of models in terms of the Basel Accords.
Series of daily returns are sufficiently long for it to be worthwhile to “invest” in the few additional parameters required for such extensions, in the hope of improving the fit and the forecasting performance.
Future research could involve the exploration of other nonstandard state-space models, for SV modelling as well as in other scenarios. Due to the high flexibility of HMMs there are countless possible applications. One could also attempt to apply structured HMMs to estimate SSMs with higher-dimensional state spaces. The two-and three-dimensional cases are likely to be of most interest — numerous applications with states representing locations are imaginable. As the method becomes more involved in higher dimensions, alternative ways of choosing appropriate grids would need to be explored.
The purpose of population HMMs, discussed in Chapter 4, is to enable comparisons between a number of HMMs fitted to longitudinal data. The proposed two-stage fitting process is easy to use and, unlike joint maximum likelihood, it scales to large studies and integrates well with cluster computing. Numerical studies demonstrate good agreement between the proposed two-stage fitting method and full maximum likelihood, while also
demonstrating substantial decreases in computing time. The proposed model is applied to a novel study of sleep and its correlates. Despite being based entirely on the EEG signal, our results confirm established hypotheses derived from hypnograms that are obtained by visual classification of the polysomnogram data.
Generally speaking, HMMs prove useful to extract features and study sleep phenomena for epidemiological studies. On the other hand the proposed model represents a rather basic first approach to modelling of EEG series via HMMs. Important future research would include covariate adjusted and nonhomogeneous variations of the model. Further-more, the population HMM should be compared to alternative modelling approaches, in particular to models that incorporate random effects to explain the heterogeneity across the time series (see e.g. Altman 2007). The main challenge here will be to overcome the computational problems. The models proposed by Maruotti (2007) might offer a way forward.
Summary and outlook