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Distribution on warp maps for alignment of open and closed curves

Bharath, Karthik; Kurtek, Sebastian

Authors

Sebastian Kurtek



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.

Journal Article Type Article
Publication Date Jul 22, 2019
Journal Journal of the American Statistical Association
Print ISSN 0162-1459
Electronic ISSN 1537-274X
Publisher Taylor & Francis Open
Peer Reviewed Peer Reviewed
Volume 115
Issue 531
Pages 1378-1392
APA6 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
DOI https://doi.org/10.1080/01621459.2019.1632066
Keywords Statistics, Probability and Uncertainty; Statistics and Probability
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....0/01621459.2019.1632066

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