Quantifying Infinite-Dimensional Data: Functional Data Analysis in Action

Functional data analysis (FDA) is concerned with inherently infinite-dimensional data objects and therefore can be viewed as part of the methodology for big data. The size of functional data may vary from terabytes as encountered in functional magnetic resonance imaging (fMRI) and other applications in brain imaging to just a few kilobytes in longitudinal data with small or modest sample sizes. In this contribution, we highlight some applications of FDA methodology through various data illustrations. We briefly review some basic computational tools that can be used to accelerate implementations of FDA methodology. The analyses presented in this paper illustrate the principal analysis by conditional expectation (PACE) package for FDA, where our applications include both relatively simple and more complex functional data from the biomedical sciences. The data we discuss range from functional data that result from daily movement profile tracking and that are modeled as repeatedly observed functions per subject, to medfly longitudinal behavior profiles, where the goal is to predict remaining lifetime of individual flies. We also discuss the quantification of connectivity of fMRI signals that is of interest in brain imaging and the prediction of continuous traits from high-dimensional SNPs in genomics. The methods of FDA that we demonstrate for these analyses include functional principal component analysis, functional regression and correlation, the modeling of dependent functional data, and the stringing of high-dimensional data into functional data and can be implemented with the PACE package.