Wasserstein regression with empirical measures and density estimation for sparse data

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.

Functional Data Analysis: An Introduction and Recent Developments

Functional data analysis (FDA) is a statistical framework that allows for the analysis of curves, images, or functions on higher dimensional domains. The goals of FDA, such as descriptive analyses, classification, and regression, are generally the same as for statistical analyses of scalar-valued or multivariate data, but FDA brings additional challenges due to the high- and infinite dimensionality of observations and parameters, respectively. This paper provides an introduction to FDA, including a description of the most common statistical analysis techniques, their respective software implementations, and some recent developments in the field. The paper covers fundamental concepts such as descriptives and outliers, smoothing, amplitude and phase variation, and functional principal component analysis. It also discusses functional regression, statistical inference with functional data, functional classification and clustering, and machine learning approaches for functional data analysis. The methods discussed in this paper are widely applicable in fields such as medicine, biophysics, neuroscience, and chemistry and are increasingly relevant due to the widespread use of technologies that allow for the collection of functional data. Sparse functional data methods are also relevant for longitudinal data analysis. All presented methods are demonstrated using available software in R by analyzing a dataset on human motion and motor control. To facilitate the understanding of the methods, their implementation, and hands-on application, the code for these practical examples is made available through a code and data supplement and on GitHub.

Autoregressive optimal transport models

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.

Modeling sparse longitudinal data in early neurodevelopment

Early childhood is a period marked by rapid brain growth accompanied by cognitive and motor development. However, it remains unclear how early developmental skills relate to neuroanatomical growth across time with no growth quantile trajectories of typical brain development currently available to place and compare individual neuroanatomical development. Even though longitudinal neuroimaging data have become more common, they are often sparse, making dynamic analyses at subject level a challenging task. Using the Principal Analysis through Conditional Expectation (PACE) approach geared towards sparse longitudinal data, we investigate the evolution of gray matter, white matter and cerebrospinal fluid volumes in a cohort of 446 children between the ages of 1 and 120 months. For each child, we calculate their dynamic age-varying association between the growing brain and scores that assess cognitive functioning, applying the functional varying coefficient model. Using local Fréchet regression, we construct age-varying growth percentiles to reveal the evolution of brain development across the population. To further demonstrate its utility, we apply PACE to predict individual trajectories of brain development.

Point process models for COVID-19 cases and deaths

The study of events distributed over time which can be quantified as point processes has attracted much interest over the years due to its wide range of applications. It has recently gained new relevance due to the COVID-19 case and death processes associated with SARS-CoV-2 that characterize the COVID-19 pandemic and are observed across different countries. It is of interest to study the behavior of these point processes and how they may be related to covariates such as mobility restrictions, gross domestic product per capita, and fraction of population of older age. As infections and deaths in a region are intrinsically events that arrive at random times, a point process approach is natural for this setting. We adopt techniques for conditional functional point processes that target point processes as responses with vector covariates as predictors, to study the interaction and optimal transport between case and death processes and doubling times conditional on covariates.

Registration for exponential family functional data

We introduce a novel method for separating amplitude and phase variability in exponential family functional data. Our method alternates between two steps: the first uses generalized functional principal components analysis to calculate template functions, and the second estimates smooth warping functions that map observed curves to templates. Existing approaches to registration have primarily focused on continuous functional observations, and the few approaches for discrete functional data require a pre-smoothing step; these methods are frequently computationally intensive. In contrast, we focus on the likelihood of the observed data and avoid the need for preprocessing, and we implement both steps of our algorithm in a computationally efficient way. Our motivation comes from the Baltimore Longitudinal Study on Aging, in which accelerometer data provides valuable insights into the timing of sedentary behavior. We analyze binary functional data with observations each minute over 24 hours for 592 participants, where values represent activity and inactivity. Diurnal patterns of activity are obscured due to misalignment in the original data but are clear after curves are aligned. Simulations designed to mimic the application indicate that the proposed methods outperform competing approaches in terms of estimation accuracy and computational efficiency. Code for our method and simulations is publicly available.