Modeling Repeated Functional Observations

Abstract

We introduce a new methodological framework for repeatedly observed and thus dependent functional data, aiming at situations where curves are recorded repeatedly for each subject in a sample. Our methodology covers the case where the recordings of the curves are scheduled on a regular and dense grid and also situations more typical for longitudinal studies, where the timing of recordings is often sparse and random. The proposed models lead to an interpretable and straightforward decomposition of the inherent variation in repeatedly observed functional data and are implemented through a straightforward two-step functional principal component analysis. We provide consistency results and asymptotic convergence rates for the estimated model components. We compare the proposed model with an alternative approach via a two-dimensional Karhunen-Loève expansion and illustrate it through the analysis of longitudinal mortality data from period lifetables that are repeatedly observed for a sample of countries over many years, and also through simulation studies. This article has online supplementary materials.

Keywords:

• Asymptotics
• Functional data analysis
• Functional principal components
• Hierarchical model
• Lifetable
• Longitudinal data
• Mortality
• Rate of convergence
• Repeated measures
• Uniform convergence

SUPPLEMENTARY MATERIALS

A: Auxiliary Results and Proofs

B: Comparisons with the Karhunen-Loève Expansion

C: List of Countries Included in Mortality Data Analysis

D: Additional Simulations

E: Eigenanalysis of the Random Functions ξ₁(s) and ξ₂(s) for the Mortality Data

This research was supported by NSF grants DMS08-0619 and DMS11-04426. The authors gratefully acknowledge the very constructive comments of two referees, an associate editor, and the editor.