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：Gaussian 分布得很稀少的抽样纵向数据代表了以其功能性主要部分分数(FCCs)的预测分布方式分布的分散式长距离数据,显示每个受多变量高斯预测分布的主体的可用数据,特别令人感兴趣的是,在从原始随机过程从稀少(纵向)到密集(功能性)抽样的转变过程中,每个主体的观测数量从稀少(纵向)向密集(功能性)抽样过程的转变增加。我们研究了由于对真实但不可观测的FPC进行吵闹的纵向观测而预测分数的趋同情况,以及测量下显示整个预测分布向真正FPC点质量点的缩放,以及当调差点(K) $=K(n) 与抽样规模(美元) 差异时,功能性连续预测分布的缩放率扩大。为了解决点预测的不一致性问题,我们在功能性线性模型中,我们构建了预测性长距离预测值预测结果的预测分布,并且从2美元预测性长期预测性流度研究中得出了预测性长期趋近度数据。