Modelling multi‐scale, state‐switching functional data with hidden Markov models

Abstract

Data sets composed of sequences of curves sampled at high frequencies in time are increasingly common in practice, but they can exhibit complicated dependence structures that cannot be modelled using common methods in functional data analysis. We detail a hierarchical approach that treats the curves as observations from a hidden Markov model. The distribution of each curve is then defined by another fine‐scale model that may involve autoregression and require data transformations using moving‐window summary statistics or Fourier analysis. This approach is broadly applicable to sequences of curves exhibiting intricate dependence structures. As a case study, we use this framework to model the fine‐scale kinematic movements of a northern resident killer whale (Orcinus orca) off the western coast of Canada. Through simulations, we show that our model produces more interpretable state estimation and more accurate parameter estimates compared to existing methods.