Local Clustering for Functional Data

In functional data analysis, unsupervised clustering has been extensively conducted and has important implications. In most of the existing functional clustering analyses, it is assumed that there is a single clustering structure across the whole domain of measurement (say, time interval). In some data analyses, for example, the analysis of normalized COVID-19 daily confirmed cases for the U.S. states, it is observed that functions can have different clustering patterns in different time subintervals. To tackle the lack of flexibility of the existing functional clustering techniques, we develop a local clustering approach, which can fully data-dependently identify subintervals, where, in different subintervals, functions have different clustering structures. This approach is built on the basis expansion technique and has a novel penalization form. It simultaneously achieves subinterval identification, clustering, and estimation. Its estimation and clustering consistency properties are rigorously established. In simulation, it significantly outperforms multiple competitors. In the analysis of the COVID-19 case trajectory data, it identifies sensible subintervals and clustering structures. Supplementary materials for this article are available online.