Random Fixed Boundary Flows

Zhigang Yao, Yuqing Xia and Zengyan Fan

Journal of the American Statistical Association, 2024, vol. 119, issue 547, 2356-2368

Abstract: We consider fixed boundary flows with canonical interpretability as principal components extended on nonlinear Riemannian manifolds. We aim to find a flow with fixed start and end points for noisy multivariate datasets lying near an embedded nonlinear Riemannian manifold. In geometric terms, the fixed boundary flow is defined as an optimal curve that moves in the data cloud with two fixed end points. At any point on the flow, we maximize the inner product of the vector field, which is calculated locally, and the tangent vector of the flow. The rigorous definition is derived from an optimization problem using the intrinsic metric on the manifolds. For random datasets, we name the fixed boundary flow the random fixed boundary flow and analyze its limiting behavior under noisy observed samples. We construct a high-level algorithm to compute the random fixed boundary flow, and provide the convergence of the algorithm. We show that the fixed boundary flow yields a concatenate of three segments, one of which coincides with the usual principal flow when the manifold is reduced to the Euclidean space. We further prove that the random fixed boundary flow converges largely to the population fixed boundary flow with high probability. Finally, we illustrate how the random fixed boundary flow can be used and interpreted, and demonstrate its application in real datasets. Supplementary materials for this article are available online.

Date: 2024
References: Add references at CitEc
Citations:

Downloads: (external link)
http://hdl.handle.net/10.1080/01621459.2023.2257892 (text/html)
Access to full text is restricted to subscribers.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:taf:jnlasa:v:119:y:2024:i:547:p:2356-2368

Ordering information: This journal article can be ordered from
http://www.tandfonline.com/pricing/journal/UASA20

DOI: 10.1080/01621459.2023.2257892

Access Statistics for this article

Journal of the American Statistical Association is currently edited by Xuming He, Jun Liu, Joseph Ibrahim and Alyson Wilson

More articles in Journal of the American Statistical Association from Taylor & Francis Journals
Bibliographic data for series maintained by Chris Longhurst ().