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Spherical regression

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  • T. D. Downs

Abstract

Methods are introduced for regressing points on the surface of one sphere on points on another. Complex variables and stereographic projection are used to deal with theoretical problems of directional statistics much as they have been used historically to deal with problems in non-Euclidean geometry. The complex plane harbours the group of Möbius transformations, and stereographic projection is used as a bridge to map these Möbius transforms to regression link functions on the surface of a unit sphere. A special form for these links is introduced which employs the complex plane and stereographic projection to effect angular scale changes on the sphere. The family of special forms is closed under orthogonal transformations of the dependent variable and Möbius transformations of the independent variable, and incorporates independence and proper and improper rotations as special cases. Parameter estimation and inference are exemplified using the von Mises--Fisher spherical distribution and vectorcardiogram data. All statistical results and calculations have been formulated in the real domain. Copyright Biometrika Trust 2003, Oxford University Press.

Suggested Citation

  • T. D. Downs, 2003. "Spherical regression," Biometrika, Biometrika Trust, vol. 90(3), pages 655-668, September.
    • Handle: RePEc:oup:biomet:v:90:y:2003:i:3:p:655-668
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    Cited by:

    1. Davy Paindaveine & Thomas Verdebout, 2019. "Inference for Spherical Location under High Concentration," Working Papers ECARES 2019-02, ULB -- Universite Libre de Bruxelles.
    2. Jayant Jha & Atanu Biswas, 2020. "Orientation relationship in finite dimensional space," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 47(3), pages 1011-1034, September.
    3. Zhu, Changbo & Müller, Hans-Georg, 2024. "Spherical autoregressive models, with application to distributional and compositional time series," Journal of Econometrics, Elsevier, vol. 239(2).
    4. Davy Paindaveine & Thomas Verdebout, 2017. "Detecting the Direction of a Signal on High-dimensional Spheres: Non-null and Le Cam Optimality Results," Working Papers ECARES ECARES 2017-40, ULB -- Universite Libre de Bruxelles.
    5. Kanti V. Mardia, 2021. "Comments on: Recent advances in directional statistics," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(1), pages 59-63, March.
    6. Laha, A. K. & Putatunda, Sayan, 2017. "Real Time Location Prediction with Taxi-GPS Data Streams," IIMA Working Papers WP 2017-03-02, Indian Institute of Management Ahmedabad, Research and Publication Department.

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