Functional k-means clustering with elastic distances accounting for phase and amplitude variation

Functional data analysis (FDA) has experienced substantial advancements in statistical methodologies, with numerous functional statistical techniques being developed and refined to enhance analytical performance. A key feature of functional data is that it comprises two distinct sources of variation: phase and amplitude, each contributing uniquely to the data structure. As a result, a central challenge in FDA is managing phase variation, which is often treated as noise and removed through registration or alignment procedures to improve statistical efficiency. Nonetheless, indiscriminately eliminating all phase variation can be problematic, as it may represent either noise or an intrinsic functional characteristic, or occasionally both. Recognizing the origin of phase variation is therefore essential. In this study, we propose a functional k-means clustering framework based on an elastic distance metric that simultaneously incorporates both phase and amplitude variations when grouping functional data. By integrating the Fisher–Rao metric, our framework achieves robustness against variability in both dimensions. This approach consistently outperforms conventional functional k-means algorithms, as demonstrated through a range of simulated and real-world datasets.