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A representation of Gaussian distributed sparsely sampled longitudinal datain terms of predictive distributions for their functional principal componentscores (FPCs) maps available data for each subject to a multivariate Gaussianpredictive distribution. Of special interest is the case where the number ofobservations per subject increases in the transition from sparse (longitudinal)to dense (functional) sampling of underlying stochastic processes. We study theconvergence of the predicted scores given noisy longitudinal observationstowards the true but unobservable FPCs, and under Gaussianity demonstrate theshrinkage of the entire predictive distribution towards a point mass located atthe true FPCs and also extensions to the shrinkage of functional $K$-truncatedpredictive distributions when the truncation point $K=K(n)$ diverges withsample size $n$. To address the problem of non-consistency of pointpredictions, we construct predictive distributions aimed at predicting outcomesfor the case of sparsely sampled longitudinal predictors in functional linearmodels and derive asymptotic rates of convergence for the $2$-Wassersteinmetric between true and estimated predictive distributions. Predictivedistributions are illustrated for longitudinal data from the BaltimoreLongitudinal Study of Aging.

Predictive Distributions and the Transition from Sparse to Dense Functional Data