🤖 AI Summary
Existing functional data registration methods fail to disentangle amplitude and phase variability in sequentially arriving functional data and lack mechanisms for online parameter updating. Method: We propose the first sequential Bayesian registration framework for functional data, built upon phase–amplitude separation modeling and uncertainty-aware sequential Monte Carlo (SMC) inference, enabling recursive Bayesian estimation of registration parameters without full re-fitting. Distributed computing optimizations further reduce computational overhead. Contributions/Results: Evaluated on real-world drought intensity, sea surface salinity, and electrocardiogram datasets, our method demonstrates high robustness and accuracy under complex, multimodal posteriors. It supports real-time dynamic alignment of streaming functional data, offering a scalable, interpretable, and statistically principled inference paradigm for online functional data analysis.
📝 Abstract
In many modern applications, discretely-observed data may be naturally understood as a set of functions. Functional data often exhibit two confounded sources of variability: amplitude (y-axis) and phase (x-axis). The extraction of amplitude and phase, a process known as registration, is essential in exploring the underlying structure of functional data in a variety of areas, from environmental monitoring to medical imaging. Critically, such data are often gathered sequentially with new functional observations arriving over time. Despite this, existing registration procedures do not sequentially update inference based on the new data, requiring model refitting. To address these challenges, we introduce a Bayesian framework for sequential registration of functional data, which updates statistical inference as new sets of functions are assimilated. This Bayesian model-based sequential learning approach utilizes sequential Monte Carlo sampling to recursively update the alignment of observed functions while accounting for associated uncertainty. Distributed computing significantly reduces computational cost relative to refitting the model using an iterative method such as Markov chain Monte Carlo on the full data. Simulation studies and comparisons reveal that the proposed approach performs well even when the target posterior distribution has a challenging structure. We apply the proposed method to three real datasets: (1) functions of annual drought intensity near Kaweah River in California, (2) annual sea surface salinity functions near Null Island, and (3) a sequence of repeated patterns in electrocardiogram signals.
