Hidden semi-Markov models with inhomogeneous state dwell-time distributions

Hidden Markov models are powerful tools for analysing time series data that depend on underlying but unobserved states. Owing to their flexible hierarchical framework, which separates the noisy observation process from the unobserved state process, they have gained prominence across numerous empirical disciplines. In movement ecology in particular, inference on an animal’s behavioural state can be made based on observed movement. However, in basic HMMs governed by a homogeneous first-order Markov chain, the time spent in a hidden state — the dwell time — necessarily follows a geometric distribution. This memoryless distribution implies that the most likely duration of a state is one sampling unit, which can be highly unrealistic in many practical contexts, such as modelling resting or sleeping behaviour. Hidden semi-Markov models (HSMMs) relax this limitation by explicitly modelling the dwell-time distribution within each state, yet existing formulations do not accommodate covariate effects on these dwell times. In ecological settings, however, the average duration of behaviours can vary substantially with external factors or the time of day. We extend the established approach of representing HSMMs as HMMs with expanded state spaces to incorporate covariate-dependent dwell-time distributions. This formulation retains full compatibility with the extensive methodological and computational toolkit developed for HMMs, most importantly efficient parameter estimation and state decoding. We examine in detail the special case of periodically varying covariate effects, derive key model properties such as the time-varying stationary distribution and overall dwell-time distribution, and illustrate the methodology using movement data from an Arctic muskox.