Adaptive Functional Data Analysis

Statistical methods that adapt to individual observations or unknown population structures are attractive due to both numerical and theoretical advantages over their non-adaptive counterparts. In this thesis, we contribute to adaptive modeling of functional data, focusing on the fundamental aspects of representation and regression, where challenges arise from the infinite-dimensionality of their underlying spaces. For adaptive representation, the notion of mixture inner product spaces (MIPS) is developed, featuring an infinite-dimensional mixture of finite-dimensional subspaces. We show that MIPS provides a new perspective for representing functional data, in which each realization of the underlying process falls into a realization-specific component subspace whose dimension is larger for less smooth while smaller for smoother realizations. Moreover, MIPS also offers an alternative solution to the longstanding issue of nonexistent probability density for functional data. For adaptive functional regression, we are interested in the scenario that functional predictor process lies in a potentially nonlinear manifold that is intrinsically finite-dimensional but embedded in an infinite-dimensional function space. We propose a nonparametric estimator built upon local linear manifold smoothing that achieves a polynomial convergence rate and adapts to the intrinsic data geometry even when functional data are observed intermittently and contaminated by noise, in contrast to the logarithm rate in nonparametric functional regression literature. We demonstrate that both proposals enjoy favorable finite sample performance relative to commonly used methods via simulated and real data examples.Ph.D