Hidden Markov models for longitudinal circular data

The use of longitudinal data has become increasingly popular in statistics over the last decade. The central question in modelling longitudinal data is how to formalize the unobserved heterogeneity of the data, that is, the heterogeneity that cannot be explained by means of observable covariates. This is often accomplished by introducing subject-specific random effects that can have either a continuous or a discrete distribution. However, time-constant unobserved heterogeneity only is accounted for. These considerations are obviously pertinent when we deal with circular longitudinal data. In this paper we propose an approach for modelling longitudinal circular data base on discrete time-varying random effects. Indeed, the problem of adequately representing the unobserved heterogeneity is addressed by including time-varying subject-specific random effects which follow a finite-state first-order Markov chain. The proposed approach may be cast in the literature about hidden Markov models (HMMs) for longitudinal data. Under the model we propose, the conditional (with respect to random effects and covariates) distribution of the circular response is projected normal or von Mises. We further introduce a joint approach to time-varying clustering and bad points detection under a longitudinal setting, extending the Hidden Markov model for circular data. We replace the state-dependent distribution with a two-component mixture where one mixture (reference) component represents the data we would expect from the given state (i.e. good points) while the other mixture component clusters the atypical data and has a small prior probability, the same component-specific mean and an inflated scale parameter. This change makes the model more robust. We estimate model parameters by using an ad hoc version of the expectation-conditional-maximization (ECM) algorithm, extending the Baum-Welch iterative procedure to deal with contaminated distribution.