A0156
Title: Functional PCA with increasing dimensions: Phase transition from sparse to dense designs
Authors: Fang Yao - Peking University, University of Toronto (Canada) [presenting]
Abstract: Functional data analysis is an important research field that treats data as random functions drawn from some infinite-dimensional functional space, and functional principal component analysis (FPCA) based on eigen-decomposition plays a central role in data reduction and representation. After nearly three decades of research, a key problem remains unsolved, namely, the perturbation analysis of covariance operator for diverging number of eigen-components obtained from noisy and discretely observed data. This is fundamental for studying models and methods based on FPCA, while there has not been substantial progress since the result for a fixed number of eigenfunction estimates. A unified theory is established for this problem, obtaining upper bounds for eigenfunctions with diverging indices in both the L2 and supremum norms and deriving the asymptotic distributions of eigenvalues for a wide range of sampling schemes. The results provide insight into the phenomenon when the L2 bound of eigenfunction estimates with diverging indices is minimax optimal as if the curves are fully observed and reveal the transition of convergence rates from nonparametric to parametric regimes in connection to sparse or dense sampling. The technical arguments are useful for handling the perturbation series with noisy and discretely observed data and can be applied in models or those involving inverse problems based on FPCA as regularization, such as functional linear regression.
