Distributional Representation of Longitudinal Data: Visualization, Regression and Prediction

We develop a representation of Gaussian distributed sparsely sampled longitudinal data whereby the data for each subject are mapped to a multivariate Gaussian distribution; this map is entirely data-driven. The proposed method utilizes functional principal component analysis and is nonparametric, assuming no prior knowledge of the covariance or mean structure of the longitudinal data. This approach naturally connects with a deeper investigation of the behavior of the functional principal component scores obtained for longitudinal data, as the number of observations per subject increases from sparse to dense. We show how this is reflected in the shrinkage of the distribution of the conditional scores given noisy longitudinal observations towards a point mass located at the true but unobservable FPCs. Mapping each subject's sparse observations to the corresponding conditional score distribution leads to useful visualizations and representations of sparse longitudinal data. Asymptotic rates of convergence as sample size increases are obtained for the 2-Wasserstein metric between the true and estimated conditional score distributions, both for a $K$-truncated functional principal component representation as well as for the case when $K=K(n)$ diverges with sample size $n\to\infty$. We apply these ideas to construct predictive distributions aimed at predicting outcomes given sparse longitudinal data.