Department of Mathematics - Mathematics Colloquium - Theory of FPCA for Discretized Functional Data

Functional data analysis is an important research field in statistics which treats data as random functions drawn from some infinite-dimensional functional space, and functional principal component analysis (FPCA) plays a central role for data reduction and representation. After nearly three decades of research, there remains a key problem unsolved, namely, the perturbation analysis of covariance operator for diverging number of eigencomponents obtained from noisy and discretely observed data. This is fundamental for studying models and methods based on FPCA, while there has not been much progress since the result obtained by Hall et al. (2006) for a fixed number of eigenfunction estimates. In this work, we establish a unified theory for this problem, deriving the moment bounds of eigenfunctions and asymptotic distributions of eigenvalues for a wide range of sampling schemes. We also exploit double truncation to derive the uniform convergence of such estimated eigenfunctions. The technical arguments in this work are useful for handling the perturbation series of discretely observed functional data and can be applied in models and methods involving inverse using FPCA as regularization, such as functional linear regression.