Statistical Modelling of Multivariate Functional and Distributional Data

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Modern recording devices are collecting data of greater complexity and, when these data are measured over space or time, also at ever-increasing resolution. While providing more detailed information about the associated physical phenomena occurring around us, they also pose statistical challenges related to interpretable modeling and feasible computation for such data. Data measured over space, time, or some other continuum, are fittingly termed functional data, and constitute an important subfield of modern statistics. This project will develop important methodology for the analysis of two types of functional data. The first set of projects aim at the estimation of dependency patterns between components of so-called multivariate functional data, where a common set of features is measured over time for each subject in a study, such as neuroimaging scans where signals are recorded over time at a variety of spatial locations. Another important class of functional data are samples of distributions or histograms, which regularly arise in the analysis of demographic data as mortality distributions, for example, but also in other important fields such as neuroscience and finance. This project outlines methods for dimension reduction and regression that respect the well-known positivity and area-under-the-curve constraints for distributions, yielding interpretable data summaries and model fits that provide the practitioner with a clearer understanding of the information contained in their data.Both fMRI and EEG yield time-dependent signals at multiple brain locations, resulting in multivariate functional data. Quantifying connectivity patterns to define brain networks, for example in order to identify normal and pathological characteristics, is an important neuroscientific problem that can be addressed using multivariate functional data techniques. This project seeks to advance the use of functional graphical models to estimate underlying brain dependency networks, including improved computational efficiency compared to existing methods. These methods are equally applicable in other domains that produce data of similar structure, such as longitudinal medical studies, where a common set of measurements is recorded repeatedly over time. Also considered in this proposal are methods for distributional data, which can be thought of as collections of curves or surfaces, each corresponding to a probability distribution. For example, neuroimaging data naturally provide such distributional samples, as levels of myelination or signal correlations within brain regions are high-dimensional data that can be effectively summarized at the subject level by a histogram or distribution. Given a sample of such distributional data, this project investigates statistical methods of interpretable dimension reduction and dependency of distributional response functions on relevant covariates through distributional regression. A key tool is the Wasserstein metric for distributions, which has been widely successful in applied settings, but has not been utilized to its full extent in statistics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.