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The Statistics of Circular Optimal Transport

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Directional Statistics for Innovative Applications

Part of the book series: Forum for Interdisciplinary Mathematics ((FFIM))

Abstract

Empirical optimal transport (OT) plans and distances provide effective tools to compare and statistically match probability measures defined on a given ground space. Fundamental to this are distributional limit laws, and we derive a central limit theorem for the empirical OT distance of circular data. Our limit results require only mild assumptions in general and include prominent examples such as the von Mises or wrapped Cauchy family. Most notably, no assumptions are required when data are sampled from the probability measure to be compared with, which is in strict contrast to the real line. A bootstrap principle follows immediately as our proof relies on Hadamard differentiability of the OT functional. This paves the way for a variety of statistical inference tasks and is exemplified for asymptotic OT-based goodness of fit testing for circular distributions. We discuss numerical implementation, consistency and investigate its statistical power. For testing uniformity, it turns out that this approach performs particularly well for unimodal alternatives and is almost as powerful as Rayleigh’s test, the most powerful invariant test for von Mises alternatives. For regimes with many modes, the circular OT test is less powerful which is explained by the shape of the corresponding transport plan.

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Notes

  1. 1.

    A function \(g:\mathbb {R}\rightarrow \mathbb {R}\) is called coercive if \(g(x)\rightarrow \infty \) as \(|x| \rightarrow \infty \).

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Acknowledgements

The authors gratefully acknowledge support for the DFG Research Training Group 2088 Discovering Structure in Complex Data: Statistics Meets Optimization and Inverse Problems and the DFG Cluster of Excellence 2067 Multiscale Bioimaging: From Molecular Machines to Networks of Excitable Cells. The authors would also like to thank the editor and the anonymous referees for their comments that improved the quality of this paper. In particular, credit is given to one referee for the suggestion to consider rotationally invariant distributions for the spherical OT distance.

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Authors and Affiliations

  1. Institute for Mathematical Stochastics, University of Göttingen, Goldschmidtstraße 7, Göttingen, 37077, Germany

    Shayan Hundrieser, Marcel Klatt & Axel Munk

  2. Max Planck Institute for Biophysical Chemistry, Am Faßberg 11, Göttingen, 37077, Germany

    Axel Munk

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  1. Shayan Hundrieser
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  2. Marcel Klatt
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Correspondence to Shayan Hundrieser .

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Editors and Affiliations

  1. Applied Statistics Unit, Indian Statistical Institute, Kolkata, West Bengal, India

    Ashis SenGupta

  2. Department of Statistics, University of California, Riverside, CA, USA

    Barry C. Arnold

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Hundrieser, S., Klatt, M., Munk, A. (2022). The Statistics of Circular Optimal Transport. In: SenGupta, A., Arnold, B.C. (eds) Directional Statistics for Innovative Applications. Forum for Interdisciplinary Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-19-1044-9_4

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