Predictive Distributions and the Transition from Sparse to Dense Functional Data

TL;DR

This paper develops a framework for representing and analyzing Gaussian predictive distributions of functional principal component scores in longitudinal data, focusing on the transition from sparse to dense sampling and their convergence properties.

Contribution

It introduces a method to model predictive distributions for FPC scores in sparse-to-dense data transition, including convergence analysis and extensions to functional linear models.

Findings

Predictive distributions converge to true FPCs as data become denser.

Shrinkage of predictive distributions towards true scores demonstrated.

Asymptotic rates of convergence in Wasserstein metric derived.

Abstract

A representation of Gaussian distributed sparsely sampled longitudinal data in terms of predictive distributions for their functional principal component scores (FPCs) maps available data for each subject to a multivariate Gaussian predictive distribution. Of special interest is the case where the number of observations per subject increases in the transition from sparse (longitudinal) to dense (functional) sampling of underlying stochastic processes. We study the convergence of the predicted scores given noisy longitudinal observations towards the true but unobservable FPCs, and under Gaussianity demonstrate the shrinkage of the entire predictive distribution towards a point mass located at the true FPCs and also extensions to the shrinkage of functional $K$-truncated predictive distributions when the truncation point $K=K(n)$ diverges with sample size $n$. To address the problem of non-consistency of point predictions, we construct predictive distributions aimed at predicting outcomes for the case of sparsely sampled longitudinal predictors in functional linear models and derive asymptotic rates of convergence for the $2$-Wasserstein metric between true and estimated predictive distributions. Predictive distributions are illustrated for longitudinal data from the Baltimore Longitudinal Study of Aging.