| For functional data on a nonlinear low-dimensional space, we propose the notions of manifold mean, of manifold modes of functional variation and of functional manifold components, which constitute nonlinear representations of functional data that complement classical linear representations such as eigenfunctions and functional principal components. These new tools can be used to summarize functional manifold data and for dimension reduction. In simulations and applications, we study examples of functional data which lie on a manifold and validate the superior behavior of manifold mean and functional manifold components over traditional cross-sectional mean and functional principal components. Our estimating procedures borrow ideas from existing nonlinear dimension reduction methods, which we modify to address functional data settings. We also explore functional regression problems with these methods and discuss their consistency properties. |
