ABSTRACT
This chapter discusses an alternative and probably more important relaxation of the assumption. Hidden semi-Markov models are designed to relax this restrictive, and in some applications unrealistic, condition. In such a model the latent process is a 'semi-Markov' process rather than a Markov process. An important benefit of using the hidden Markov models (HMM) formulation is that this enables to apply all the well-established methods available for HMMs. In particular, since the likelihood function of the HMM representing or approximating the HSMM of interest is available in the standard HMM form, the parameters can be estimated by direct numerical maximization of likelihood. The chapter considers a stationary three-state HSMM for the earthquake series, assuming Poisson state-dependent distributions and shifted Poisson dwell-time distributions. A disadvantage of that generalization is that the number of parameters increases rapidly as the order of the Markov chain increases.
