Addressing phase discrepancies in functional data: A Bayesian approach for accurate alignment and smoothing

In many real-world applications, functional data exhibit considerable variability in both amplitude and phase. This is especially true in biomechanical data, such as the knee flexion angle dataset motivating our work where timing differences across curves can obscure meaningful comparisons. The curves of this study also exhibit substantial variability from one another. These pronounced differences make the dataset particularly challenging to align properly without distorting or losing some of the individual curve characteristics. Our alignment model addresses these challenges by eliminating phase discrepancies while preserving the individual characteristics of each curve and avoiding distortion, thanks to its flexible smoothing component. Additionally, the model accommodates group structures through a dedicated parameter. By leveraging a Bayesian approach, the new prior on the warping parameters ensures that the resulting warping functions automatically satisfy all necessary validity conditions. We applied our model to the knee flexion dataset, demonstrating excellent performance in both smoothing and alignment, particularly in the presence of high inter curve variability and complex group structures.