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Properties and applications of Fisher distribution on the rotation group

Author

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  • Sei, Tomonari
  • Shibata, Hiroki
  • Takemura, Akimichi
  • Ohara, Katsuyoshi
  • Takayama, Nobuki

Abstract

We study properties of Fisher distribution (von Mises–Fisher distribution, matrix Langevin distribution) on the rotation group SO(3). In particular we apply the holonomic gradient descent, introduced by Nakayama et al. (2011) [16], and a method of series expansion for evaluating the normalizing constant of the distribution and for computing the maximum likelihood estimate. The rotation group can be identified with the Stiefel manifold of two orthonormal vectors. Therefore from the viewpoint of statistical modeling, it is of interest to compare Fisher distributions on these manifolds. We illustrate the difference with an example of near-earth objects data.

Suggested Citation

  • Sei, Tomonari & Shibata, Hiroki & Takemura, Akimichi & Ohara, Katsuyoshi & Takayama, Nobuki, 2013. "Properties and applications of Fisher distribution on the rotation group," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 440-455.
    • Handle: RePEc:eee:jmvana:v:116:y:2013:i:c:p:440-455
    DOI: 10.1016/j.jmva.2013.01.010
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    References listed on IDEAS

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    1. A. Kume & Andrew T. A. Wood, 2005. "Saddlepoint approximations for the Bingham and Fisher–Bingham normalising constants," Biometrika, Biometrika Trust, vol. 92(2), pages 465-476, June.
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    Cited by:

    1. Toru Kitagawa & Jeff Rowley, 2022. "von Mises-Fisher distributions and their statistical divergence," Papers 2202.05192, arXiv.org, revised Nov 2022.
    2. Jeon, Jeong Min & Van Keilegom, Ingrid, 2023. "Density estimation for mixed Euclidean and non-Euclidean data in the presence of measurement error," Journal of Multivariate Analysis, Elsevier, vol. 193(C).
    3. Tamio Koyama & Hiromasa Nakayama & Kenta Nishiyama & Nobuki Takayama, 2014. "Holonomic gradient descent for the Fisher–Bingham distribution on the $$d$$ d -dimensional sphere," Computational Statistics, Springer, vol. 29(3), pages 661-683, June.

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