Hidden Markov models with state-dependent mixtures: minimal representation, model testing and applications to clustering

Finite-state hidden Markov models (HMMs), also called Markov-dependent finite mixtures, form a popular, frequently used model class for serially dependent observations with unobserved heterogeneity. We consider HMMs in which the state-dependent distributions are themselves finite mixtures. In such models, the parametrization is not unique, since components from the state-dependent mixtures may also be represented as states in the underlying Markov chain. We determine a unique (up to label switching) representation for the HMM in which the Markov chain has a minimal number of states. Further, we propose a likelihood-ratio test for the hypothesis that the number of states in the Markov chain can be reduced without changing the distribution of the time-series model. Our method has important applications in cluster analysis and model selection. After highlighting the relevance of serial dependence for clustering, we propose two-step clustering algorithms. Starting with a BIC choice for a standard HMM (with simple state-dependent distributions), in the first step we determine the minimal representation of the HMM by testing, and in the second step we merge components in the resulting state-dependent finite mixtures by using either a local entropy or a modality-based criterion. The states in the resulting Markov chain, potentially split according to the remaining state-dependent components, are then interpreted as clusters. For model selection, we illustrate our method on a series of logarithmic returns of gold prices using normal HMMs. The AIC choice is a six-state HMM, while the BIC choice has four states. When starting with the AIC choice, successive testing results in a four-state Markov chain, with two state-dependent distributions consisting of two-component normal mixtures.