Search for: All records

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.

 

We propose and investigate an additive regression model for symmetric positive-definite matrix-valued responses and multiple scalar predictors. The model exploits the Abelian group structure inherited from either of the log-Cholesky and log-Euclidean frameworks for symmetric positive-definite matrices and naturally extends to general Abelian Lie groups. The proposed additive model is shown to connect to an additive model on a tangent space. This connection not only entails an efficient algorithm to estimate the component functions, but also allows one to generalize the proposed additive model to general Riemannian manifolds. Optimal asymptotic convergence rates and normality of the estimated component functions are established, and numerical studies show that the proposed model enjoys good numerical performance, and is not subject to the curse of dimensionality when there are multiple predictors. The practical merits of the proposed model are demonstrated through an analysis of brain diffusion tensor imaging data.

 

Modern data collection often entails longitudinal repeated measurements that assume values on a Riemannian manifold. Analyzing such longitudinal Riemannian data is challenging, because of both the sparsity of the observations and the nonlinear manifold constraint. Addressing this challenge, we propose an intrinsic functional principal component analysis for longitudinal Riemannian data. Information is pooled across subjects by estimating the mean curve with local Fréchet regression and smoothing the covariance structure of the linearized data on tangent spaces around the mean. Dimension reduction and imputation of the manifold‐valued trajectories are achieved by utilizing the leading principal components and applying best linear unbiased prediction. We show that the proposed mean and covariance function estimates achieve state‐of‐the‐art convergence rates. For illustration, we study the development of brain connectivity in a longitudinal cohort of Alzheimer's disease and normal participants by modeling the connectivity on the manifold of symmetric positive definite matrices with the affine‐invariant metric. In a second illustration for irregularly recorded longitudinal emotion compositional data for unemployed workers, we show that the proposed method leads to nicely interpretable eigenfunctions and principal component scores. Data used in preparation of this article were obtained from the Alzheimer's Disease Neuroimaging Initiative database.

 

Multivariate functional data are becoming ubiquitous with advances in modern technology and are substantially more complex than univariate functional data. We propose and study a novel model for multivariate functional data where the component processes are subject to mutual time warping. That is, the component processes exhibit a similar shape but are subject to systematic phase variation across their time domains. To address this previously unconsidered mode of warping, we propose new registration methodology that is based on a shift‐warping model. Our method differs from all existing registration methods for functional data in a fundamental way. Namely, instead of focusing on the traditional approach to warping, where one aims to recover individual‐specific registration, we focus on shift registration across the components of a multivariate functional data vector on a population‐wide level. Our proposed estimates for these shifts are identifiable, enjoy parametric rates of convergence, and often have intuitive physical interpretations, all in contrast to traditional curve‐specific registration approaches. We demonstrate the implementation and interpretation of the proposed method by applying our methodology to the Zürich Longitudinal Growth data and study its finite sample properties in simulations.

 

Abstract 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. 

We study the U.S. Supreme Court dynamics by analyzing the temporal evolution of the underlying policy positions of the Supreme Court Justices as reflected by their actual voting data, using functional data analysis methods. The proposed fully flexible nonparametric method makes it possible to dissect the time-dynamics of policy positions at the level of individual Justices, as well as providing a comprehensive view of the ideology evolution over the history of Supreme Court since its establishment. In addition to quantifying individual Justice’s policy positions, we uncover average changes over time and also the major patterns of change over time. Additionally, our approach allows for representing highly complex dynamic trajectories by a few principal components which complements other models of analyzing and predicting court behavior.