Registration for exponential family functional data

We introduce a novel method for separating amplitude and phase variability in exponential family functional data. Our method alternates between two steps: the first uses generalized functional principal components analysis to calculate template functions, and the second estimates smooth warping functions that map observed curves to templates. Existing approaches to registration have primarily focused on continuous functional observations, and the few approaches for discrete functional data require a pre-smoothing step; these methods are frequently computationally intensive. In contrast, we focus on the likelihood of the observed data and avoid the need for preprocessing, and we implement both steps of our algorithm in a computationally efficient way. Our motivation comes from the Baltimore Longitudinal Study on Aging, in which accelerometer data provides valuable insights into the timing of sedentary behavior. We analyze binary functional data with observations each minute over 24 hours for 592 participants, where values represent activity and inactivity. Diurnal patterns of activity are obscured due to misalignment in the original data but are clear after curves are aligned. Simulations designed to mimic the application indicate that the proposed methods outperform competing approaches in terms of estimation accuracy and computational efficiency. Code for our method and simulations is publicly available.