Doctor of Philosophy, The Ohio State University, 2021, Statistics
Functional Data Analysis (FDA) and Statistical Shape Analysis (SA) are fields in which the data objects of interest vary over a continuum, such as univariate functions and planar curves. While observations are typically measured and stored discretely, there are inherent benefits in acknowledging the infinite-dimensional processes from which the data arise. The typical statistical goals in FDA and SA are summarization, visualization, inference, and prediction. However, the geometric structure of the data presents unique challenges.
In FDA, the observations exhibit two distinct forms of variability: amplitude, which describes the magnitude of features and phase, which describes the relative timing of amplitude features. In SA, objects are analyzed through their shape, which is a quantity that remains unchanged if the object is scaled, translated, rotated in space or reparametrized (referred to as shape-preserving transformations). Within both fields, analysis usually follows unrelated sequential steps. First, an estimation step is used to obtain an infinite-dimensional representation of the discretely measured observations. Then, a registration step is used to decouple amplitude and phase variability in the FDA setting, and remove variability in the observations associated with shape-preserving transformations in the SA setting. Finally, inference can be performed based on the registration results. There are two well-documented drawbacks to the sequential pipeline for analysis. (1) There is no formal uncertainty propagation between steps, which leads to overconfidence in inferential results. (2) There is a lack of flexibility under realistic observation regimes, such as sparsely sampled or fragmented observations. Previous methods that have attempted to overcome these drawbacks suffer from being too rigid or fail to account for misregistration of observations.
In this thesis, we develop flexible modelling frameworks for FDA and SA that simultaneously perform t (open full item for complete abstract)
Committee: Oksana Chkrebtii (Advisor); Sebastian Kurtek (Advisor); Peter Craigmile (Committee Member); Radu Herbei (Committee Member)
Subjects: Statistics