MSc Thesis Defence: Pernille Emma Hartung Hansen
Methods in Geometric Statistics
This thesis is on the topic of Geometric Statistics. The thesis explores methods for generalizing linear statistics to complete connected smooth finite dimensional Riemannian manifolds. One powerful method for such generalizations is through linearization of the manifold in question. We study this technique in depth and generalize several concepts from linear statistics. For each resulting generalized definition, we discuss the limitations that curvature of the space places upon them and analyze how well they describe the distribution in question. Lastly, we explore a different method for generalizing linear statistics: stochastic processes on the frame bundle of a manifold.
A main technique for generalizing definitions to Riemannian manifolds through linearization is to identify characteristic properties of the Euclidean statistical definitions that carry over to the theory of Riemannian geometry. This is used to define the Fréchet means and the Normal law, generalizations of the mean and normal distribution. Fréchet means are defined as global minima of the variance as this is a characterizing property of the Euclidean mean. The existence and uniqueness of such minima is then explored and commonly used sufficient conditions are derived. Similarly, the Normal Law is defined through the maximal entropy characterization of the Gaussian Normal distribution. Furthermore, two methods for quantifying variance of distributions are presented and analyzed: the covariance matrix and Principal Geodesic analysis.
The thesis also touches upon the method of generalizing distributions through stochastic processes on the frame bundle of a manifold. The vertical and horizontal frame bundles are defined for general fiber bundles and the construction is applied to the frame bundle. The horizontal lift is defined, yielding a global frame field of the horizontal frame bundle. It is this frame field that allows us to define stochastic process on the frame bundle through solutions to stochastic differential equations.
Vejleder: Stefan Sommer
Censor: Poul G. Hjorth, DTU