Estimation of Functional Principal Components from Sparse Functional Data

Music City Center 

This work introduces a novel method for extracting functional principal components from sparse, univariate functional data, commonly seen in longitudinal studies with irregular sampling and measurement errors. The approach utilizes basis expansion for estimation and includes an approximate Generalized Cross-Validation (GCV) for optimally selecting the number of basis functions and principal components. Crucially, the methodology preserves essential mathematical properties: eigenfunction orthogonality, eigenvalue positivity, and a positive estimate of the error variance. Using conditional estimation, it then recovers complete individual trajectories and principal scores across the domain. Simulation studies demonstrate the method's superior performance in estimating eigenfunctions, eigenvalues, and error variance compared to existing techniques. Its practical utility is highlighted through an application to CD4 cell count data from the Multicenter AIDS Cohort Study.

Functional Data Analysis

Longitudinal Data

Functional Principal Components

Modified Gram-Schmidt Orthonormalizing

Basis Functions

Maximum Likelihood Estimate 

IMS