IDEAS home Printed from https://ideas.repec.org/a/spr/compst/v29y2014i3p661-683.html
   My bibliography  Save this article

Holonomic gradient descent for the Fisher–Bingham distribution on the $$d$$ d -dimensional sphere

Author

Listed:
  • Tamio Koyama
  • Hiromasa Nakayama
  • Kenta Nishiyama
  • Nobuki Takayama

Abstract

We propose an accelerated version of the holonomic gradient descent and apply it to calculating the maximum likelihood estimate (MLE) of the Fisher–Bingham distribution on a $$d$$ d -dimensional sphere. We derive a Pfaffian system (an integrable connection) and a series expansion associated with the normalizing constant with an error estimation. These enable us to solve some MLE problems up to dimension $$d=7$$ d = 7 with a specified accuracy. Copyright Springer-Verlag Berlin Heidelberg 2014

Suggested Citation

  • 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.
    • Handle: RePEc:spr:compst:v:29:y:2014:i:3:p:661-683
    DOI: 10.1007/s00180-013-0456-z
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s00180-013-0456-z
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s00180-013-0456-z?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    2. 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.
    3. Hashiguchi, Hiroki & Numata, Yasuhide & Takayama, Nobuki & Takemura, Akimichi, 2013. "The holonomic gradient method for the distribution function of the largest root of a Wishart matrix," Journal of Multivariate Analysis, Elsevier, vol. 117(C), pages 296-312.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Bee, Marco & Benedetti, Roberto & Espa, Giuseppe, 2017. "Approximate maximum likelihood estimation of the Bingham distribution," Computational Statistics & Data Analysis, Elsevier, vol. 108(C), pages 84-96.
    2. Ian L. Dryden, 2014. "Comment on Geodesic Monte Carlo on Embedded Manifolds by Byrne and Girolami," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 41(1), pages 8-9, March.
    3. Christophe Ley & Thomas Verdebout, 2014. "Skew-rotsymmetric Distributions on Unit Spheres and Related Efficient Inferential Proceedures," Working Papers ECARES ECARES 2014-46, ULB -- Universite Libre de Bruxelles.
    4. Jean-Luc Dortet-Bernadet & Nicolas Wicker, 2018. "A Note on Inverse Stereographic Projection of Elliptical Distributions," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 80(1), pages 138-151, February.
    5. Daya K. Nagar & Raúl Alejandro Morán-Vásquez & Arjun K. Gupta, 2015. "Extended Matrix Variate Hypergeometric Functions and Matrix Variate Distributions," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2015, pages 1-15, January.
    6. Ley, Christophe & Verdebout, Thomas, 2017. "Skew-rotationally-symmetric distributions and related efficient inferential procedures," Journal of Multivariate Analysis, Elsevier, vol. 159(C), pages 67-81.
    7. Kume, A. & Walker, S.G., 2014. "On the Bingham distribution with large dimension," Journal of Multivariate Analysis, Elsevier, vol. 124(C), pages 345-352.
    8. Tamio Koyama & Akimichi Takemura, 2016. "Holonomic gradient method for distribution function of a weighted sum of noncentral chi-square random variables," Computational Statistics, Springer, vol. 31(4), pages 1645-1659, December.
    9. Kume, A. & Wood, Andrew T.A., 2007. "On the derivatives of the normalising constant of the Bingham distribution," Statistics & Probability Letters, Elsevier, vol. 77(8), pages 832-837, April.
    10. 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).
    11. Shimizu, Koki & Hashiguchi, Hiroki, 2021. "Heterogeneous hypergeometric functions with two matrix arguments and the exact distribution of the largest eigenvalue of a singular beta-Wishart matrix," Journal of Multivariate Analysis, Elsevier, vol. 183(C).
    12. Takasu, Yuya & Yano, Keisuke & Komaki, Fumiyasu, 2018. "Scoring rules for statistical models on spheres," Statistics & Probability Letters, Elsevier, vol. 138(C), pages 111-115.
    13. Hashiguchi, Hiroki & Takayama, Nobuki & Takemura, Akimichi, 2018. "Distribution of the ratio of two Wishart matrices and cumulative probability evaluation by the holonomic gradient method," Journal of Multivariate Analysis, Elsevier, vol. 165(C), pages 270-278.
    14. Marco Bee & Roberto Benedetti & Giuseppe Espa, 2015. "Approximate likelihood inference for the Bingham distribution," DEM Working Papers 2015/02, Department of Economics and Management.
    15. Toru Kitagawa & Jeff Rowley, 2022. "von Mises-Fisher distributions and their statistical divergence," Papers 2202.05192, arXiv.org, revised Nov 2022.
    16. Takayama, Nobuki & Jiu, Lin & Kuriki, Satoshi & Zhang, Yi, 2020. "Computation of the expected Euler characteristic for the largest eigenvalue of a real non-central Wishart matrix," Journal of Multivariate Analysis, Elsevier, vol. 179(C).
    17. 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.
    18. Tianlu Yuan, 2021. "The 8-parameter Fisher–Bingham distribution on the sphere," Computational Statistics, Springer, vol. 36(1), pages 409-420, March.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:compst:v:29:y:2014:i:3:p:661-683. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.