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Covariance on Manifolds

Title: Covariance on Manifolds.
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Name(s): Balov, Nikolay H. (Nikolay Hristov), 1970-, author
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
Type of Resource: text
Genre: Text
Issuance: monographic
Date Issued: 2009
Publisher: Florida State University
Place of Publication: Tallahassee, Florida
Physical Form: computer
online resource
Extent: 1 online resource
Language(s): English
Abstract/Description: 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.
Identifier: FSU_migr_etd-1045 (IID)
Submitted Note: A Dissertation submitted to the Department of Statistics in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
Degree Awarded: Spring Semester, 2009.
Date of Defense: March 25, 2009.
Keywords: Statistics, Manifolds, Covariance
Bibliography Note: Includes bibliographical references.
Advisory Committee: Anuj Srivastava, Professor Directing Dissertation; Eric Klassen, Outside Committee Member; Victor Patrangenaru, Committee Member; Daniel McGee, Committee Member.
Subject(s): Statistics
Probabilities
Persistent Link to This Record: http://purl.flvc.org/fsu/fd/FSU_migr_etd-1045
Owner Institution: FSU

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Balov, N. H. (N. H. ). (2009). Covariance on Manifolds. Retrieved from http://purl.flvc.org/fsu/fd/FSU_migr_etd-1045