Shape-based functional data analysis

Abstract Functional data analysis (FDA) is a fast-growing area of research and development in statistics. While most FDA literature imposes the classical $$\mathbb {L}^2$$ L 2 Hilbert structure on function spaces, there an emergent need for different, shape-based approach analyzing functional data. This paper reviews develops fundamental geometrical concepts that help connect traditionally diverse fields shape analyses. It showcases focusing shapes often more appropriate when structural features (number peaks valleys their heights) carry salient information recaps recent mathematical representations associated procedures comparing, summarizing, testing functions. Specifically, it discusses three tasks: fitting, fPCA, regression models. The latter refers to models separate functions from phases use them individually analysis. ensuing results provide better interpretations tend preserve geometric structures. also extension where are not real-valued but manifold-valued. article presents several examples this shape-centric using simulated real