STATISTICAL METHODS FOR THE ANALYSIS OF FUNCTIONAL DATA UNDER MODELS WITH COMPLEX ASSOCIATION STRUCTURES

High dimensional data play an ever increasing role in the study of human health and behavior. Recent technological advances have presented researchers with increasing opportunities to collect and store vast amounts of data which are associated with health outcomes. In many cases, these high dimensional data are characterized by inherent correlations on some domain, for example, time, and can be thought of as realizations of some underlying smooth process, potentially measured with noise. In this scenario, statistical methods developed in the field of functional data analysis can be brought to bear. However, the analysis of high dimensional functional data presents a number of challenges. Some key challenges are: 1) data storage and manipulation; 2) identifying interpretable features of the data associated with an outcome of interest; 3) dynamic prediction where the predictor and response are high dimensional functional data which exhibit strong non-linear patterns or complex association structures; and 4) computational challenges associated with the size of the data. In this thesis we present three works which each address one or more of these problems in varied contexts. First, we propose a data storage and analysis pipeline for minute level accelerometry data recorded over the course of several days. We discuss methods for interpretable dimension reduction via functional principal component analysis and functional regression, and present a compelling application involving the prediction of mortality using patterns of physical activity. Next, we propose a method for dynamically predicting future values of an irregularly measured functional outcome using a concurrently measured functional predictor while allowing for complex, non-linear subject-specific random effects. The method is applied to child growth data to predict stunting. Finally, we propose a method for dynamic prediction of time-to-event outcomes using densely measured longitudinal predictors in models which allow for complex, non-linear associations between the entire history of the predictor and outcome. The method is specifically designed to handle very large datasets and is applied to Multiple Sclerosis neuroimaging data to predict lesion recovery. Throughout, we emphasize the development of statistical and analytic methods which are reproducible and implemented in high quality, open source software