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Abstract Sparse functional/longitudinal data have attracted widespread interest due to the prevalence of such data in social and life sciences. A prominent scenario where such data are routinely encountered are accelerated longitudinal studies, where subjects are enrolled in the study at a random time and are only tracked for a short amount of time relative to the domain of interest. The statistical analysis of such functional snippets is challenging since information for far-off-diagonal regions of the covariance structure is missing. Our main methodological contribution is to address this challenge by bypassing covariance estimation and instead modelling the underlying process as the solution of a data-adaptive stochastic differential equation. Taking advantage of the interface between Gaussian functional data and stochastic differential equations makes it possible to efficiently reconstruct the target process by estimating its dynamic distribution. The proposed approach allows one to consistently recover forward sample paths from functional snippets at the subject level. We establish the existence and uniqueness of the solution to the proposed data-driven stochastic differential equation and derive rates of convergence for the corresponding estimators. The finite sample performance is demonstrated with simulation studies and functional snippets arising from a growth study and spinal bone mineral density data. 

Abstract The problem of modeling the relationship between univariate distributions and one or more explanatory variables lately has found increasing interest. Existing approaches proceed by substituting proxy estimated distributions for the typically unknown response distributions. These estimates are obtained from available data but are problematic when for some of the distributions only few data are available. Such situations are common in practice and cannot be addressed with currently available approaches, especially when one aims at density estimates. We show how this and other problems associated with density estimation such as tuning parameter selection and bias issues can be side-stepped when covariates are available. We also introduce a novel version of distribution-response regression that is based on empirical measures. By avoiding the preprocessing step of recovering complete individual response distributions, the proposed approach is applicable when the sample size available for each distribution varies and especially when it is small for some of the distributions but large for others. In this case, one can still obtain consistent distribution estimates even for distributions with only few data by gaining strength across the entire sample of distributions, while traditional approaches where distributions or densities are estimated individually fail, since sparsely sampled densities cannot be consistently estimated. The proposed model is demonstrated to outperform existing approaches through simulations and Environmental Influences on Child Health Outcomes data. 

Abstract The relationship between the early‐age activity of Mediterranean fruit flies (medflies) or other fruit flies and their lifespan has not been much studied, in contrast to the connections between lifespan and diet, sexual signaling, and reproduction. The objective of this study is to assess intra‐day and day‐to‐day activity profiles of female Mediterranean fruit flies and their role as biomarker of longevity as well as to explore the relationships between these activity profiles, diet, and age‐at‐death throughout the lifespan. We use advanced statistical methods from functional data analysis (FDA). Three distinct patterns of activity variations in early‐age activity profiles can be distinguished. A low‐caloric diet is associated with a delayed activity peak, while a high‐caloric diet is linked with an earlier activity peak. We find that age‐at‐death of individual medflies is connected to their activity profiles in early life. An increased risk of mortality is associated with increased activity in early age, as well as with a higher contrast between daytime and nighttime activity. Conversely, medflies are more likely to have a longer lifespan when they are fed a medium‐caloric diet and when their daily activity is more evenly distributed across the early‐age span and between daytime and nighttime. The before‐death activity profile of medflies displays two characteristic before‐death patterns, where one pattern is characterized by slowly declining daily activity and the other by a sudden decline in activity that is followed by death. 

Abstract Quantifying the association between components of multivariate random curves is of general interest and is a ubiquitous and basic problem that can be addressed with functional data analysis. An important application is the problem of assessing functional connectivity based on functional magnetic resonance imaging (fMRI), where one aims to determine the similarity of fMRI time courses that are recorded on anatomically separated brain regions. In the functional brain connectivity literature, the static temporal Pearson correlation has been the prevailing measure for functional connectivity. However, recent research has revealed temporally changing patterns of functional connectivity, leading to the study of dynamic functional connectivity. This motivates new similarity measures for pairs of random curves that reflect the dynamic features of functional similarity. Specifically, we introduce gradient synchronization measures in a general setting. These similarity measures are based on the concordance and discordance of the gradients between paired smooth random functions. Asymptotic normality of the proposed estimates is obtained under regularity conditions. We illustrate the proposed synchronization measures via simulations and an application to resting-state fMRI signals from the Alzheimer’s Disease Neuroimaging Initiative and they are found to improve discrimination between subjects with different disease status. 

Abstract Series of univariate distributions indexed by equally spaced time points are ubiquitous in applications and their analysis constitutes one of the challenges of the emerging field of distributional data analysis. To quantify such distributional time series, we propose a class of intrinsic autoregressive models that operate in the space of optimal transport maps. The autoregressive transport models that we introduce here are based on regressing optimal transport maps on each other, where predictors can be transport maps from an overall barycenter to a current distribution or transport maps between past consecutive distributions of the distributional time series. Autoregressive transport models and their associated distributional regression models specify the link between predictor and response transport maps by moving along geodesics in Wasserstein space. These models emerge as natural extensions of the classical autoregressive models in Euclidean space. Unique stationary solutions of autoregressive transport models are shown to exist under a geometric moment contraction condition of Wu & Shao [(2004) Limit theorems for iterated random functions. Journal of Applied Probability 41, 425–436)], using properties of iterated random functions. We also discuss an extension to a varying coefficient model for first-order autoregressive transport models. In addition to simulations, the proposed models are illustrated with distributional time series of house prices across U.S. counties and annual summer temperature distributions. 

Abstract Multivariate functional data present theoretical and practical complications that are not found in univariate functional data. One of these is a situation where the component functions of multivariate functional data are positive and are subject to mutual time warping. That is, the component processes exhibit a common shape but are subject to systematic phase variation across their domains in addition to subject‐specific time warping, where each subject has its own internal clock. This motivates a novel model for multivariate functional data that connect such mutual time warping to a latent‐deformation‐based framework by exploiting a novel time‐warping separability assumption. This separability assumption allows for meaningful interpretation and dimension reduction. The resulting latent deformation model is shown to be well suited to represent commonly encountered functional vector data. The proposed approach combines a random amplitude factor for each component with population‐based registration across the components of a multivariate functional data vector and includes a latent population function, which corresponds to a common underlying trajectory. We propose estimators for all components of the model, enabling implementation of the proposed data‐based representation for multivariate functional data and downstream analyses such as Fréchet regression. Rates of convergence are established when curves are fully observed or observed with measurement error. The usefulness of the model, interpretations, and practical aspects are illustrated in simulations and with application to multivariate human growth curves and multivariate environmental pollution data.