We propose a Riemannian factor model (RFM), a novel framework for analyzing potentially high-dimensional time series data observed on Riemannian manifolds. Such time series are encountered in various applications, including economics, finance, medical imaging, and genomics and microbiome research. The proposed model is geometry-aware and accounts for the inherent nonlinearity in the data. In a high-dimensional asymptotic regime, where the manifold dimension is allowed to diverge with the sample size n, we establish convergence rates for the estimated loading space. In particular, under short-memory and strong factor conditions, we obtain a dimension-free n^-1/2 rate, which matches the convergence rate of the high-dimensional linear factor model. Finite-sample performance of the proposed RFM is demonstrated with simulated time series on the Bures–Wasserstein manifolds and products of spheres, as well as an application to monthly realized covariances of selected U.S. stock returns—modeled as time series in the Bures–Wasserstein manifold, where the RFM provides demonstrably interpretable factors and yields competitive predictive performance.

This paper develops a novel unified framework for testing mutual independence among random objects residing in possibly different metric spaces. The framework generalizes existing methodologies and introduces new measures of mutual independence, and proposes associated tests that achieve minimax rate optimality and exhibit strong empirical power. The foundation of the proposed tests is the new concept of joint distance profiles, which uniquely characterize the joint law of random objects under a mild condition on either the joint law or the metric spaces. Our test statistics quantify the difference of the joint distance profiles of each data point with respect to the joint law and the product of marginal laws of the vector of random objects. To enhance power, we consider integrating this difference with respect to different measures and incorporate flexible data-adaptive weight profiles in the test statistics. We derive the limiting distribution of the test statistics under the null hypothesis of mutual independence and show that the proposed tests with certain weight profiles are asymptotically distribution-free if the marginal distance profiles are continuous. Furthermore, we establish the consistency of the tests under sequences of alternative hypotheses converging to the null. For practical implementations, we employ a permutation scheme to approximate the p-values and provide theoretical guarantees that the permutation-based tests maintain type I error control under the null and achieve consistency under alternatives. We demonstrate the power of the proposed tests across various types of data objects through simulations and real data applications, where the new tests exhibit better performance compared with popular existing approaches.

Regression with non-Euclidean responses -- e.g., probability distributions, networks, symmetric positive-definite matrices, and compositions -- has become increasingly important in modern applications. In this paper, we propose deep Fréchet neural networks (DFNNs), an end-to-end deep learning framework for predicting non-Euclidean responses -- which are considered as random objects in a metric space -- from Euclidean predictors. Our method leverages the representation-learning power of deep neural networks (DNNs) to the task of approximating conditional Fréchet means of the response given the predictors, the metric-space analogue of conditional expectations, by minimizing a Fréchet risk. The framework is highly flexible, accommodating diverse metrics and high-dimensional predictors. We establish a universal approximation theorem for DFNNs, advancing the state-of-the-art of neural network approximation theory to general metric-space-valued responses without making model assumptions or relying on local smoothing. Empirical studies on synthetic distributional and network-valued responses, as well as a real-world application to predicting employment occupational compositions, demonstrate that DFNNs consistently outperform existing methods.

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.

This article provides an overview on the statistical modeling of complex data as increasingly encountered in modern data analysis. It is argued that such data can often be described as elements of a metric space that satisfies certain structural conditions and features a probability measure. We refer to the random elements of such spaces as random objects and to the emerging field that deals with their statistical analysis as metric statistics. Metric statistics provides methodology, theory and visualization tools for the statistical description, quantification of variation, centrality and quantiles, regression and inference for populations of random objects, inferring these quantities from available data and samples. In addition to a brief review of current concepts, we focus on distance profiles as a major tool for object data in conjunction with the pairwise Wasserstein transports of the underlying one-dimensional distance distributions. These pairwise transports lead to the definition of intuitive and interpretable notions of transport ranks and transport quantiles as well as two-sample inference. An associated profile metric complements the original metric of the object space and may reveal important features of the object data in data analysis. We demonstrate these tools for the analysis of complex data through various examples and visualizations.

Brain growth in early childhood is reflected in the evolution of proportional cerebrospinal fluid volumes (pCSF), grey matter (pGM), and white matter (pWM). We study brain development as reflected in the relative fractions of these three tissues for a cohort of 388 children that were longitudinally followed between the ages of 18 and 96 months. We introduce statistical methodology (Riemannian Principal Analysis through Conditional Expectation, RPACE) that addresses major challenges that are of general interest for the analysis of longitudinal neuroimaging data, including the sparsity of the longitudinal observations over time and the compositional structure of the relative brain volumes. Applying the RPACE methodology, we find that longitudinal growth as reflected by tissue composition differs significantly for children of mothers with higher and lower maternal education levels.

The maturation of regional brain volumes from birth to preadolescence is a critical developmental process that underlies emerging brain structural connectivity and function. Regulated by genes and environment, the coordinated growth of different brain regions plays an important role in cognitive development. Current knowledge about structural network evolution is limited, partly due to the sparse and irregular nature of most longitudinal neuroimaging data. In particular, it is unknown how factors such as mother’s education or sex of the child impact the structural network evolution. To address this issue, we propose a method to construct evolving structural networks and study how the evolving connections among brain regions as reflected at the network level are related to maternal education and biological sex of the child and also how they are associated with cognitive development. Our methodology is based on applying local Fréchet regression to longitudinal neuroimaging data acquired from the RESONANCE cohort, a cohort of healthy children (245 females and 309 males) ranging in age from 9 weeks to 10 years. Our findings reveal that sustained highly coordinated volume growth across brain regions is associated with lower maternal education and lower cognitive development. This suggests that higher neurocognitive performance levels in children are associated with increased variability of regional growth patterns as children age.

The first 1000 days are essential for a child’s development, consisting of critical time windows for brain and cognitive development but also vulnerability. In this large multi-cohort study, we investigated typical development of 478 children with ≤6 timepoints across two different scanners with different acquisition protocols. PACE brain-for-age growth percentiles and regression slope functions with 95% pointwise confidence intervals identified critical windows of neurocognitive development. Our results could allow for more appropriate neurodevelopmental burden estimates across multi-scanner cohorts, and help identify primary risk factors and age-specific intervention impact.

The analysis of samples of random objects that do not lie in a vector space is gaining increasing attention in statistics. An important class of such object data is univariate probability measures defined on the real line. Adopting the Wasserstein metric, we develop a class of regression models for such data, where random distributions serve as predictors and the responses are either also distributions or scalars. To define this regression model, we utilize the geometry of tangent bundles of the space of random measures endowed with the Wasserstein metric for mapping distributions to tangent spaces. The proposed distribution-to-distribution regression model provides an extension of multivariate linear regression for Euclidean data and function-to-function regression for Hilbert space valued data in functional data analysis. In simulations, it performs better than an alternative transformation approach where one maps distributions to a Hilbert space through the log quantile density transformation and then applies traditional functional regression. We derive asymptotic rates of convergence for the estimator of the regression operator and for predicted distributions and also study an extension to autoregressive models for distribution-valued time series. The proposed methods are illustrated with data on human mortality and distributional time series of house prices.

We propose new tools for the geometric exploration of data objects taking values in a general separable metric space (Ω, d). Given a probability measure on Ω, we introduce depth profiles, where the depth profile of an element ω ∈ Ω refers to the distribution of the distances between ω and the other elements of Ω. Depth profiles can be harnessed to define transport ranks, which capture the centrality of each element in Ω with respect to the entire data cloud based on the optimal transport maps between the depth profiles. We study the properties of transport ranks and show that they provide an effective device for detecting and visualizing patterns in samples of random objects. Specifically, we study estimates of depth profiles and transport ranks based on samples of random objects and establish the convergence of the empirical estimates to the population targets using empirical process theory. We demonstrate the usefulness of depth profiles and associated transport ranks and visualizations for distributional data through a sample of age-at-death distributions for various countries, for compositional data through energy usage for U.S. states and for network data through New York taxi trips.

Local Frechet regression is a nonparametric regression method for metric space valued responses and Euclidean predictors, which can be utilized to obtain estimates of smooth trajectories taking values in general metric spaces from noisy metric space valued random objects. We derive uniform rates of convergence, which so far have eluded theoretical analysis of this method, for both fixed and random target trajectories, where we utilize tools from empirical processes. These results are shown to be widely applicable in metric space valued data analysis. In addition to simulations, we provide two pertinent examples where these results are important: The consistent estimation of the location of properly defined extrema in metric space valued trajectories, which we illustrate with the problem of locating the age of minimum brain connectivity as obtained from fMRI data; and time warping for metric space valued trajectories, illustrated with yearly age-at-death distributions for different countries.

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.

Delay differential equations form the underpinning of many complex dynamical systems. The forward problem of solving random differential equations with delay has received increasing attention in recent years. Motivated by the challenge to predict the COVID-19 caseload trajectories for individual states in the U.S., we target here the inverse problem. Given a sample of observed random trajectories obeying an unknown random differential equation model with delay, we use a functional data analysis framework to learn the model parameters that govern the underlying dynamics from the data. We show the existence and uniqueness of the analytical solutions of the population delay random differential equation model when one has discrete time delays in the functional concurrent regression model and also for a second scenario where one has a delay continuum or distributed delay. The latter involves a functional linear regression model with history index. The derivative of the process of interest is modeled using the process itself as predictor and also other functional predictors with predictor-specific delayed impacts. This dynamics learning approach is shown to be well suited to model the growth rate of COVID-19 for the states that are part of the U.S., by pooling information from the individual states, using the case process and concurrently observed economic and mobility data as predictors.

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.

The study of the dynamic behavior of cross-sectional ranks over time for functional data and the ranks of the observed curves at each time point and their temporal evolution can yield valuable insights into the time dynamics of functional data. This approach is of interest in various application areas. For the analysis of the dynamics of ranks, estimation of the cross-sectional ranks of functional data is a first step. Several statistics of interest for ranked functional data are proposed. To quantify the evolution of ranks over time, a model for rank derivatives is introduced, where rank dynamics are decomposed into two components. One component corresponds to population changes and the other to individual changes that both affect the rank trajectories of individuals. The joint asymptotic normality for suitable estimates of these two components is established. The proposed approaches are illustrated with simulations and three longitudinal datasets: Growth curves obtained from the Zürich Longitudinal Growth Study, monthly house price data in the US from 1996 to 2015 and Major League Baseball offensive data for the 2017 season.