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.

翻译：针对稀疏采样的纵向高斯分布数据，通过其函数主成分得分（FPCs）的预测分布进行表征，可将每个受试者的可用数据映射为多元高斯预测分布。特别值得关注的是，当每个受试者的观测数量增加时，底层随机过程的采样模式将从稀疏（纵向）过渡到密集（函数型）。我们研究了在含噪声纵向观测条件下预测得分向真实但不可观测的FPCs的收敛性，并在高斯性假设下证明了整个预测分布向位于真实FPCs处的点质量的收缩现象，同时将结论推广至当截断点$K=K(n)$随样本量$n$发散时函数型$K$截断预测分布的收缩特性。针对点预测的非一致性问题，我们构建了用于预测函数线性模型中稀疏采样纵向预测变量结果的预测分布，并推导了真实预测分布与估计预测分布之间$2$-Wasserstein距离的渐近收敛速率。本文通过巴尔的摩老龄化纵向研究数据对预测分布进行了实例演示。