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.

A Multi-Dimensional Functional Principal Components Analysis of EEG Data

“…Note that under the additional assumptions that modes of variability in the functional dimension stay the same across longitudinal times and electrode locations, or that modes of variability in the longitudinal dimension stay the same across functional times and electrode locations, more parsimonious versions of MD-FPCA can be derived using the marginal and product FPCA ideas of Park and Staicu (2015) and Chen et al (2016). These extensions would lead to a common set of eigenfunctions in functional time across longitudinal times and electrode locations and/or a common set of eigenfunctions in longitudinal time across functional time and electrode locations.…”

“…Greven et al (2010) proposed a decomposition based on a functional random intercept and slope to capture longitudinal variations, which we refer to as linear FPCA (LFPCA). Chen and Müller (2012) suggested a double decomposition (DFPCA) to capture potential nonlinear and nonparametric longitudinal trends within repeatedly observed functional data; parsimonious extensions of DFPCA have recently been proposed by Park and Staicu (2015) and Chen et al (2016). While ANOVA-FPCA models longitudinal repetitions as repeated measurements without a particular time ordering, similar to an ANOVA, LFPCA models longitudinal trends linearly, and DFPCA does not assume a parametric form.…”

Additive Function-on-Function Regression

“…In practice, we use estimates of FPC scores from functional principal component analysis (FPCA), as we show next. Using the eigenbasis of the marginal covariance of the response, rather than a spline basis, is appealing because of the resulting parsimonious representation of the response and has been often used in the literature; see for example, Aston et al (2010); Jiang and Wang (2010); Park and Staicu (2015). This choice of orthogonal basis also allows us to formulate the mean model for the conditional response profile, given scalar/vector covariates, based on mean models for the conditional FPC scores given the covariates: $\mathrm{normalE}false[Y_{i}false(tfalse)false|X_{i}false(·false)false]=true{\sum }_{k=1}^K{\varphi }_{k}false(tfalse)\mathrm{normalE}false[{\xi }_{ik}false|X_{i}false(·false)false]$, where $\mathrm{normalE}false[{\xi }_{ik}false|X_{i}false(·false)false]={{\displaystyle \int }}_{{\mathrm{scriptT}}_X}{G}_{k}false\{X_{i}false(sfalse),sfalse\}ds$, G k (·, ·) are unknown bivariate functions and ξ ik are the FPC scores of response.…”

New Insights into Activity Patterns in Children, Found Using Functional Data Analyses

“…Such applications are not uncommon in the functional data analysis literature, with methods developed for both FoSR and several related settings (19) (7) (17) (16) (22) (35). Several barriers to the broader adoption of such methods exist, and one goal of this article is to reduce those barriers by building awareness of functional data approaches and clearly interpreting the results of such analyses.…”