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. / A Dissertation submitted to the Department of Statistics in partial fulfillment of the requirements for the degree of
Doctor of Philosophy. / Spring Semester, 2009. / March 25, 2009. / Statistics, Manifolds, Covariance / Includes bibliographical references. / Anuj Srivastava, Professor Directing Dissertation; Eric Klassen, Outside Committee Member; Victor Patrangenaru, Committee Member; Daniel McGee, Committee Member.
| Identifer | oai:union.ndltd.org:fsu.edu/oai:fsu.digital.flvc.org:fsu_253917 |
| Contributors | Balov, Nikolay H. (Nikolay Hristov), 1970- (authoraut), Srivastava, Anuj (professor directing dissertation), Klassen, Eric (outside committee member), Patrangenaru, Victor (committee member), McGee, Daniel (committee member), Department of Statistics (degree granting department), Florida State University (degree granting institution) |
| Publisher | Florida State University, Florida State University |
| Source Sets | Florida State University |
| Language | English, English |
| Detected Language | English |
| Type | Text, text |
| Format | 1 online resource, computer, application/pdf |
| Rights | This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s). The copyright in theses and dissertations completed at Florida State University is held by the students who author them. |
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