Longitudinal Functional Data Analysis

Abstract

We consider dependent functional data that are correlated because of a longitudinal-based design: each subject is observed at repeated times and at each time a functional observation (curve) is recorded. We propose a novel parsimonious modeling framework for repeatedly observed functional observations that allows to extract low dimensional features. The proposed methodology accounts for the longitudinal design, is designed to study the dynamic behavior of the underlying process, allows prediction of full future trajectory, and is computationally fast. Theoretical properties of this framework are studied and numerical investigations confirm excellent behavior in finite samples. The proposed method is motivated by and applied to a diffusion tensor imaging study of multiple sclerosis.

Keywords: Dependent functional data; Diffusion Tensor Imaging; Functional principal component analysis; Longitudinal design; Multiple Sclerosis.

Figures

Figure 1

Left panel: 95% pointwise and joint confidence bands of the slope function βT(s) of μ(s, T) using bootstrap; Right: final mean estimate, μ^(s,T)=μ^0(s)

Figure 2

Top: First three eigenfunctions of the estimated marginal covariance; Bottom: estimated mean function μ^0(s) (gray line) ± 2λ^kϕ^k(s) (+ and − signs, respectively)

Figure 3

Estimated time-varying coefficients ξ^ik(T) for k = 1, 2 and 3 using REM

Figure 4

Predicted values of FA for the last visits of three randomly selected subjects; actual observations (gray); predictions using our model (black solid) and using the naive approach (black dashed)