We apply tools from functional data analysis to model cumulative trajectories of COVID-19 cases across countries, establishing a framework for quantifying and comparing cases and deaths across countries longitudinally. It emerges that a country’s trajectory during an initial first month “priming period” largely determines how the situation unfolds subsequently. We also propose a method for forecasting case counts, which takes advantage of the common, latent information in the entire sample of curves, instead of just the history of a single country. Our framework facilitates to quantify the effects of demographic covariates and social mobility on doubling rates and case fatality rates through a time-varying regression model. Decreased workplace mobility is associated with lower doubling rates with a roughly 2 week delay, and case fatality rates exhibit a positive feedback pattern.

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We study the dynamic behavior of cross-sectional ranks over time for functional data and show that the ranks of the observed curves at each time point and their evolution over time can yield valuable insights into the time-dynamics of functional data. This approach is of particular interest in sports statistics in addition to other areas where functional data arise. For the analysis of the dynamics of ranks, we obtain estimates of the cross-sectional ranks of functional data and introduce several statistics of interest for ranked functional data. To quantify the evolution of ranks over time, we develop a model for rank derivatives, in which we decompose rank dynamics into two components, where one component corresponds to population changes and the other to individual changes. We establish the joint asymptotic normality for suitable estimates of these two components. These approaches are illustrated with simulations and three longitudinal data sets: Growth curves obtained from the Zürich Longitudinal Growth Study, monthly house price data in the U.S from 1980 to 2015, and Major League Baseball offensive data for the 2017 season.

Many studies have been conducted on flows of probability measures, often in terms of gradient flows. We utilize a generalized notion of derivatives with respect to time to model the instantaneous evolution of empirically observed one-dimensional distributions that vary over time and develop consistent estimates for these derivatives. Employing local Fréchet regression and working in local tangent spaces with regard to the Wasserstein metric, we derive the rate of convergence of the proposed estimators. The resulting time dynamics are illustrated with time-varying distribution data that include yearly income distributions and the evolution of mortality over calendar years.