Our view is that while some of the basic principles of data analysis are going to remain unchanged, others are to be gradually replaced with Geometry and Topology methods. Linear methods are still making sense for functional data analysis, or in the context of tangent bundles of object spaces. Complex nonstandard data is represented on object spaces. An object space admitting a manifold stratification may be embedded in an Euclidean space. One defines the extrinsic energy distance associated with two probability measures on an arbitrary object space embedded in a numerical space, and one introduces an extrinsic energy statistic to test for homogeneity of distributions of two random objects (r.o.'s) on such an object space. This test is validated via a simulation example on the Kendall space of planar k-ads with a Veronese-Whitney (VW) embedding. One considers an application to medical imaging, to test for the homogeneity of the distributions of Kendall shapes of the midsections of the Corpus Callosum in a clinically normal population vs a population of ADHD diagnosed individuals. Surprisingly, due to the high dimensionality, these distributions are not significantly different, although they are known to have highly significant VW-means. New spread and location parameters are to be added to reflect the nontrivial topology of certain object spaces. TDA is going to be adapted to object spaces, and hypothesis testing for distributions is going to be based on extrinsic energy methods. For a random point on an object space embedded in an Euclidean space, the mean vector cannot be represented as a point on that space, except for the case when the embedded space is convex. To address this misgiving, since the mean vector is the minimizer of the expected square distance, following Frechet (1948), on an embedded compact object space, one may consider both minimizers and maximizers of the expected square distance to a given point on the embedded object space as mean, respectively anti-mean of the random point. Of all distances on an object space, one considers here the chord distance associated with the embedding of the object space, since for such distances one can give a necessary and sufficient condition for the existence of a unique Frechet mean (respectively Frechet anti-mean). For such distributions these location parameters are called extrinsic mean (respectively extrinsic anti-mean), and the corresponding sample statistics are consistent estimators of their population counterparts. Moreover around the extrinsic mean ( anti-mean ) located at a smooth point, one derives the limit distribution of such estimators. / A Dissertation submitted to the Department of Statistics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Summer Semester 2017. / June 14, 2017. / Includes bibliographical references. / Vic Patrangenaru, Professor Directing Dissertation; Washington Mio, University Representative; Adrian Barbu, Committee Member; Jonathan Bradley, Committee Member.
| Identifer | oai:union.ndltd.org:fsu.edu/oai:fsu.digital.flvc.org:fsu_552073 |
| Contributors | Guo, Ruite (authoraut), Patrangenaru, Victor (professor directing dissertation), Mio, Washington (university representative), Barbu, Adrian G. (Adrian Gheorghe), 1971- (committee member), Bradley, Jonathan R. (committee member), Florida State University (degree granting institution), College of Arts and Sciences (degree granting college), Department of Statistics (degree granting departmentdgg) |
| Publisher | Florida State University |
| Source Sets | Florida State University |
| Language | English, English |
| Detected Language | English |
| Type | Text, text, doctoral thesis |
| Format | 1 online resource (65 pages), computer, application/pdf |
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