Type of Document	Dissertation
Author	Balov, Nikolay Hristov
Author's Email Address	balov@stat.fsu.edu
URN	etd-04152009-130953
Title	Covariance on Manifolds
Degree	Doctor of Philosophy
Department	Statistics, Department of
Advisory Committee	Advisor Name Title Anuj Srivastava Committee Chair Daniel McGee Committee Member Victor Patrangenaru Committee Member Eric Klassen Outside Committee Member
Keywords	Statistics Manifolds Covariance
Date of Defense	2009-03-25
Availability	unrestricted
Abstract With ever increasing complexity of observational and theoretical data models, the sufficiency of the classical statistical techniques, designed to be applied only on vector quantities, is being challenged. Nonlinear statistical analysis has become an area of intensive research in recent years. Despite the impressive progress in this direction, a unified and consistent framework has not been reached. In this regard, the following work is an attempt to improve our understanding of random phenomena on non-Euclidean spaces. More specifically, the motivating goal of the present dissertation is to generalize the notion of distribution covariance, which in standard settings is defined only in Euclidean spaces, on arbitrary manifolds with metric. We introduce a tensor field structure, named covariance field, that is consistent with the heterogeneous nature of manifolds. It not only describes the variability imposed by a probability distribution but also provides alternative distribution representations. The covariance field combines the distribution density with geometric characteristics of its domain and thus fills the gap between these two.We present some of the properties of the covariance fields and argue that they can be successfully applied to various statistical problems. In particular, we provide a systematic approach for defining parametric families of probability distributions on manifolds, parameter estimation for regression analysis, nonparametric statistical tests for comparing probability distributions and interpolation between such distributions. We then present several application areas where this new theory may have potential impact. One of them is the branch of directional statistics, with domain of influence ranging from geosciences to medical image analysis. The fundamental level at which the covariance based structures are introduced, also opens a new area for future research.
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