Nonparametric Statistics on Manifolds With Applications to Shape Spaces

Persistent Link:
http://hdl.handle.net/10150/194508
Title:
Nonparametric Statistics on Manifolds With Applications to Shape Spaces
Author:
Bhattacharya, Abhishek
Issue Date:
2008
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
This thesis presents certain recent methodologies and some new results for the statistical analysis of probability distributions on non-Euclidean manifolds. The notions of Frechet mean and variation as measures of center and spread are introduced and their properties are discussed. The sample estimates from a random sample are shown to be consistent under fairly broad conditions. Depending on the choice of distance on the manifold, intrinsic and extrinsic statistical analyses are carried out. In both cases, sufficient conditions are derived for the uniqueness of the population means and for the asymptotic normality of the sample estimates. Analytic expressions for the parameters in the asymptotic distributions are derived. The manifolds of particular interest in this thesis are the shape spaces of k-ads. The statistical analysis tools developed on general manifolds are applied to the spaces of direct similarity shapes, planar shapes, reflection similarity shapes, affine shapes and projective shapes. Two-sample nonparametric tests are constructed to compare the mean shapes and variation in shapes for two random samples. The samples in consideration can be either independent of each other or be the outcome of a matched pair experiment. The testing procedures are based on the asymptotic distribution of the test statistics, or on nonparametric bootstrap methods suitably constructed. Real life examples are included to illustrate the theory.
Type:
text; Electronic Dissertation
Keywords:
extrinsic analysis; Frechet analysis; intrinsic analysis; manifold; nonparametric inference; shapes of k-ads
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Mathematics; Graduate College
Degree Grantor:
University of Arizona
Advisor:
Bhattacharya, Rabi
Committee Chair:
Bhattacharya, Rabi

Full metadata record

DC FieldValue Language
dc.language.isoENen_US
dc.titleNonparametric Statistics on Manifolds With Applications to Shape Spacesen_US
dc.creatorBhattacharya, Abhisheken_US
dc.contributor.authorBhattacharya, Abhisheken_US
dc.date.issued2008en_US
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.abstractThis thesis presents certain recent methodologies and some new results for the statistical analysis of probability distributions on non-Euclidean manifolds. The notions of Frechet mean and variation as measures of center and spread are introduced and their properties are discussed. The sample estimates from a random sample are shown to be consistent under fairly broad conditions. Depending on the choice of distance on the manifold, intrinsic and extrinsic statistical analyses are carried out. In both cases, sufficient conditions are derived for the uniqueness of the population means and for the asymptotic normality of the sample estimates. Analytic expressions for the parameters in the asymptotic distributions are derived. The manifolds of particular interest in this thesis are the shape spaces of k-ads. The statistical analysis tools developed on general manifolds are applied to the spaces of direct similarity shapes, planar shapes, reflection similarity shapes, affine shapes and projective shapes. Two-sample nonparametric tests are constructed to compare the mean shapes and variation in shapes for two random samples. The samples in consideration can be either independent of each other or be the outcome of a matched pair experiment. The testing procedures are based on the asymptotic distribution of the test statistics, or on nonparametric bootstrap methods suitably constructed. Real life examples are included to illustrate the theory.en_US
dc.typetexten_US
dc.typeElectronic Dissertationen_US
dc.subjectextrinsic analysisen_US
dc.subjectFrechet analysisen_US
dc.subjectintrinsic analysisen_US
dc.subjectmanifolden_US
dc.subjectnonparametric inferenceen_US
dc.subjectshapes of k-adsen_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineMathematicsen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.advisorBhattacharya, Rabien_US
dc.contributor.chairBhattacharya, Rabien_US
dc.contributor.committeememberGlickenstein, Daviden_US
dc.contributor.committeememberKennedy, Thomas G.en_US
dc.contributor.committeememberPickrell, Douglas M.en_US
dc.contributor.committeememberShaked, Mosheen_US
dc.identifier.proquest10065en_US
dc.identifier.oclc659750606en_US
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