Date of Award

Spring 1-1-2017

Document Type

Thesis

Degree Name

Master of Science (MS)

Department

Applied Mathematics

First Advisor

François G. Meyer

Second Advisor

Bengt Fornberg

Third Advisor

Jem N. Corcoran

Abstract

In many applications high-dimensional observations are assumed to arrange on or near a low-dimensional manifold embedded in an ambient Euclidean space. In this thesis, ideas from differential geometry are extended to equation-free data analysis to better understand high-dimensional datasets. In particular, two questions are addressed: (1) how can the intrinsic dimensionality of a manifold-valued dataset be determined? and (2) how can this intrinsic dimensionality be leveraged to obtain a better notion of centrality? For (1), two common methods for estimating global dimensionality are stated and a novel approach is proposed to obtain an estimator for local dimensionality. Then, for (2), a novel approach is presented to estimate the geometric median on manifolds of which no prior knowledge of the underlying

geometry is known. These methods are first applied to synthetic datasets and then to real world neurological measurements to create a biomarker for the development of epilepsy in an animal model.

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