Distribution on warp maps for alignment of open and closed curves

Bharath, Karthik; Kurtek, Sebastian

Abstract

Alignment of curve data is an integral part of their statistical analysis, and can be achieved using model-or optimization-based approaches. The parameter space is usually the set of monotone, continuous warp maps of a domain. Infinite-dimensional nature of the parameter space encourages sampling based approaches, which require a distribution on the set of warp maps. Moreover, the distribution should also enable sampling in the presence of important landmark information on the curves which constrain the warp maps. For alignment of closed and open curves in R d , d = 1, 2, 3, possibly with landmark information, we provide a constructive, point-process based definition of a distribution on the set of warp maps of [0, 1] and the unit circle S that is (1) simple to sample from, and (2) possesses the desiderata for decomposition of the alignment problem with landmark constraints into multiple unconstrained ones. For warp maps on [0, 1], the distribution is related to the Dirichlet process. We demonstrate its utility by using it as a prior distribution on warp maps in a Bayesian model for alignment of two univariate curves, and as a proposal distribution in a stochastic algorithm that optimizes a suitable alignment functional for higher-dimensional curves. Several examples from simulated and real datasets are provided.

Citation

Bharath, K., & Kurtek, S. (2019). Distribution on warp maps for alignment of open and closed curves. Journal of the American Statistical Association, 115(531), 1378-1392. https://doi.org/10.1080/01621459.2019.1632066

• Journal Article Type: Article
• Acceptance Date: Feb 1, 2019
• Online Publication Date: Jul 22, 2019
• Publication Date: Jul 22, 2019
• Deposit Date: Mar 8, 2019
• Publicly Available Date: Jul 23, 2020
• Journal: Journal of the American Statistical Association
• Print ISSN: 0162-1459
• Electronic ISSN: 1537-274X
• Publisher: Taylor and Francis
• Peer Reviewed: Peer Reviewed
• Volume: 115
• Issue: 531
• Pages: 1378-1392
• DOI: https://doi.org/10.1080/01621459.2019.1632066
• Keywords: Statistics, Probability and Uncertainty; Statistics and Probability
• Public URL: https://nottingham-repository.worktribe.com/output/1619682
• Publisher URL: https://www.tandfonline.com/doi/full/10.1080/01621459.2019.1632066
• Additional Information: This is an Accepted Manuscript of an article published by Taylor & Francis in Journal of the American Statistical Association on 14.06.2019, available online: http://www.tandfonline.com/10.1080/01621459.2019.1632066
• Contract Date: Mar 8, 2019