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Analysis of multi-way functional data under weak separability, with application to brain connectivity studies

Lynch, Brian (2019) Analysis of multi-way functional data under weak separability, with application to brain connectivity studies. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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We develop statistical methods for two-way functional data, in which we observe a sample of functions of two continuous variables, for example space and time. Analysis of two-way functional data presents complexities not found in traditional one-way functional data, and this analysis can serve as a starting point in understanding multi-way functional data. Motivated by the concept of factorizing the signal into separate spatial and temporal components, we develop the concept of weak separability of the underlying random process. Compared to the traditional strong separability assumption, which models the covariance structure as the product of the space and time covariances, weak separability is more flexible yet still interpretable, modeling the covariance structure as a weighted sum of strongly separable components.

We propose asymptotic and bootstrap testing procedures for weak separability, and their performance is studied in simulations. We apply the testing procedures to brain imaging data, in which functional connectivity between two brain regions is measured as a function of frequency and time. We illustrate how, under weak separability, the functional process can be understood in terms of products of basis functions for frequency and time. We go on to develop methods to approximate the covariance structure using L-separability, defined as a class of decompositions of the covariance structure under weak separability, and show its relationship to nonnegative matrix factorization. Using psychiatric data as a case study, we illustrate the L-separable decomposition, as well as two-way localization methods for the basis functions.


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Item Type: University of Pittsburgh ETD
Status: Unpublished
CreatorsEmailPitt UsernameORCID
Lynch, Brianbcl28@pitt.edubcl280000-0001-8046-8612
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee ChairChen,
Committee MemberCheng,
Committee MemberRen,
Committee MemberLei,
Date: 30 January 2019
Date Type: Publication
Defense Date: 29 November 2018
Approval Date: 30 January 2019
Submission Date: 30 November 2018
Access Restriction: No restriction; Release the ETD for access worldwide immediately.
Number of Pages: 124
Institution: University of Pittsburgh
Schools and Programs: Dietrich School of Arts and Sciences > Statistics
Degree: PhD - Doctor of Philosophy
Thesis Type: Doctoral Dissertation
Refereed: Yes
Uncontrolled Keywords: asymptotics, functional principal component, hypothesis testing, marginal kernel, separable covariance, tensor product
Date Deposited: 31 Jan 2019 00:02
Last Modified: 31 Jan 2019 00:02

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