New methods for fixed-margin binary matrix sampling, Fréchet covariance, and MANOVA tests for random objects in multiple metric spaces
| dc.contributor.author | Fout, Alex M., author | |
| dc.contributor.author | Fosdick, Bailey, advisor | |
| dc.contributor.author | Kaplan, Andee, committee member | |
| dc.contributor.author | Cooley, Daniel, committee member | |
| dc.contributor.author | Adams, Henry, committee member | |
| dc.date.accessioned | 2022-08-29T10:17:06Z | |
| dc.date.available | 2022-08-29T10:17:06Z | |
| dc.date.issued | 2022 | |
| dc.description.abstract | Many approaches to the analysis of network data essentially view the data as Euclidean and apply standard multivariate techniques. In this dissertation, we refrain from this approach, exploring two alternate approaches to the analysis of networks and other structured data. The first approach seeks to determine how unique an observed simple, directed network is by comparing it to like networks which share its degree distribution. Generating networks for comparison requires sampling from the space of all binary matrices with the prescribed row and column margins, since enumeration of all such matrices is often infeasible for even moderately sized networks with 20-50 nodes. We propose two new sampling methods for this problem. First, we extend two Markov chain Monte Carlo methods to sample from the space non-uniformly, allowing flexibility in the case that some networks are more likely than others. We show that non-uniform sampling could impede the MCMC process, but in certain special cases is still valid. Critically, we illustrate the differential conclusions that could be drawn from uniform vs. nonuniform sampling. Second, we develop a generalized divide and conquer approach which recursively divides matrices into smaller subproblems which are much easier to count and sample. Each division step reveals interesting mathematics involving the enumeration of integer partitions and points in convex lattice polytopes. The second broad approach we explore is comparing random objects in metric spaces lacking a coordinate system. Traditional definitions of the mean and variance no longer apply, and standard statistical tests have needed reconceptualization in terms of only distances in the metric space. We consider the multivariate setting where random objects exist in multiple metric spaces, which can be thought of as distinct views of the random object. We define the notion of Fréchet covariance to measure dependence between two metric spaces, and establish consistency for the sample estimator. We then propose several tests for differences in means and covariance matrices among two or more groups in multiple metric spaces, and compare their performance on scenarios involving random probability distributions and networks with node covariates. | |
| dc.format.medium | born digital | |
| dc.format.medium | doctoral dissertations | |
| dc.identifier | Fout_colostate_0053A_17272.pdf | |
| dc.identifier.uri | https://hdl.handle.net/10217/235685 | |
| dc.language | English | |
| dc.language.iso | eng | |
| dc.publisher | Colorado State University. Libraries | |
| dc.relation.ispartof | 2020- | |
| dc.rights | Copyright and other restrictions may apply. User is responsible for compliance with all applicable laws. For information about copyright law, please see https://libguides.colostate.edu/copyright. | |
| dc.title | New methods for fixed-margin binary matrix sampling, Fréchet covariance, and MANOVA tests for random objects in multiple metric spaces | |
| dc.type | Text | |
| dcterms.rights.dpla | This Item is protected by copyright and/or related rights (https://rightsstatements.org/vocab/InC/1.0/). 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). | |
| thesis.degree.discipline | Statistics | |
| thesis.degree.grantor | Colorado State University | |
| thesis.degree.level | Doctoral | |
| thesis.degree.name | Doctor of Philosophy (Ph.D.) |
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