A Geometric Approach to Visualization of Variability in Univariate and Multivariate Functional Data

This dissertation describes a new method for the construction and visualization of geometrically-motivated displays for univariate functional data and multivariate curve data. For univariate functional data, we use a recent functional data analysis framework, based on a representation of functions called square-root slope functions. We decompose observed variation in functional data into three main components: amplitude, phase, and vertical translation. For multivariate curve data, we use a similar recent elastic curve data analysis framework, based on square-root velocity functions, to decompose variability in curve data into five main components: location, scale, shape, orientation, and reparametrization. We then construct separate displays for each component, using the geometry and metric of each representation space, based on a novel definition of the median, the two quartiles, and extreme observations. The outlyingness of function and curve data is a very complex concept. Thus, we propose to separately identify outliers based on each of the main components after decomposition. We provide a variety of visualization tools for the proposed displays, including surface plots for the amplitude and phase components of univariate functional data, and circular plots for the orientation and seed components of bivariate curve data, among others. We evaluate the proposed methods using extensive simulations and then focus our attention on multiple real data applications, including exploratory data analysis of sea surface temperature functions, electrocardiogram biosignals, growth ii curves, gait functions, and respiration functions. We also study variability in simulated 3D spirals, handwritten signatures, 3D fibers from Diffusion Tensor Magnetic Resonance Imaging (DT-MRI), and 3D Lorenz attractor curves.