Unsupervised clustering of multilevel functional data

We propose a hierarchical k–centers functional clustering (HiKCFC) method for identifying structure in multilevel functional data. HiKCFC extends subspace-projection-based clustering to hierarchical settings, where clusters are characterized jointly at multiple levels through their mean functions and covariance eigenfunctions. The method is accompanied by theoretical conditions characterizing when hierarchical clusters become non-identifiable under a multilevel functional model, thereby clarifying the structural requirements for cluster separation. Simulation studies demonstrate that HiKCFC achieves accurate cluster recovery and stable performance relative to conventional hierarchical clustering procedures. In an application to freeway traffic flow data, HiKCFC reveals interpretable spatial and temporal patterns by location and day type, illustrating its ability to capture complex multilevel variability. The proposed framework provides an interpretable approach to unsupervised clustering of multilevel functional data.