Statistical Analysis on Representation Spaces of Affine Lines and Affine Planes

This dissertation contains two related parts. The first part focuses on affine lines, and the second part is about affine planes. In the first part, it develops tools for statistical analysis on spaces of affine lines. Motivated by the problem of dimension reduction for high-dimensional dynamic data, it develops a natural mathematical representation of affine lines, endowed with a convenient Riemannian structure for computing basic quantities such as geodesics, sample means, and tangent PCA. It adapts recent works for constructing Be ́zier curves on Riemannian manifolds to this representation space and develops a tool for fitting smooth trajectories in the space of affine lines. Finally, it provides techniques to compare such trajectories resulting from different measurements in a manner that is invariant to the parameterizations of these trajectories. These individual tools together form a comprehensive toolbox for representing and analyzing large dynamical systems as low-dimensional objects. We demonstrate this toolbox using a number of examples from real and simulated data. In the second part, we apply a similar statistical analysis procedure on affine planes, such as developing a representation space of affine planes, equipping a Riemannian structure on it, and illustrating statistical analysis through some real and simulated datasets.