Flexible Modelling of Diel and Other Periodic Variation in Hidden Markov Models

Abstract

Animal behaviour is often characterised by periodic patterns such as seasonality or diel variation. Such periodic variation can be comprehensively studied from the increasingly detailed ecological time series that are nowadays collected, e.g. using GPS tracking. Within the class of hidden Markov models (HMMs), which is a popular tool for modelling time series driven by underlying behavioural modes, periodic variation is commonly modelled by including trigonometric functions in the linear predictors for the state transition probabilities. This parametric modelling can be too inflexible to capture complex periodic patterns, e.g. featuring multiple activity peaks per day. Here, we explore an alternative approach using penalised splines to model periodic variation in the state-switching dynamics of HMMs. The challenge of estimating the corresponding complex models is substantially reduced by the expectation–maximisation algorithm, which allows us to make use of the existing machinery (and software) for nonparametric regression. The practicality and potential usefulness of our approach is demonstrated in two real-data applications, modelling the movements of African elephants and of common fruit flies.