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Fisher’s legacy of directional statistics, and beyond to statistics on manifolds

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  • Mardia, Kanti V.

Abstract

It is not an exaggeration to say that R.A. Fisher is the Albert Einstein of Statistics. He pioneered almost all the main branches of statistics, but it is not as well known that he opened the area of Directional Statistics with his 1953 paper introducing a distribution on the sphere which is now known as the Fisher distribution. He stressed that for spherical data one should take into account that the data is on a manifold. We will describe this Fisher distribution and reanalyze his geological data. We also comment on the two goals he set himself in that paper, and on how he reinvented the von Mises distribution on the circle. Since then, many extensions of this distribution have appeared bearing Fisher’s name such as the von Mises–Fisher distribution and the matrix Fisher distribution. In fact, the subject of Directional Statistics has grown tremendously in the last two decades with new applications emerging in life sciences, image analysis, machine learning and so on. We give a recent new method of constructing the Fisher type distributions on manifolds which has been motivated by some problems in machine learning. The number of directional distributions has increased since then, including the bivariate von Mises distribution and we describe its connection to work resulting in the 2024 Nobel-winning AlphaFold (in Chemistry). Further, the subject has evolved as Statistics on Manifolds which also includes the new field of Shape Analysis, and finally, we end with a historical note pointing out some correspondence between D’Arcy Thompson and R.A. Fisher related to Shape Analysis.

Suggested Citation

  • Mardia, Kanti V., 2025. "Fisher’s legacy of directional statistics, and beyond to statistics on manifolds," Journal of Multivariate Analysis, Elsevier, vol. 207(C).
    • Handle: RePEc:eee:jmvana:v:207:y:2025:i:c:s0047259x24001118
    DOI: 10.1016/j.jmva.2024.105404
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    References listed on IDEAS

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    1. Ewen Callaway, 2024. "Chemistry Nobel goes to developers of AlphaFold AI that predicts protein structures," Nature, Nature, vol. 634(8034), pages 525-526, October.
    2. Janice L. Scealy & David Heslop & Jia Liu & Andrew T. A. Wood, 2022. "Directions Old and New: Palaeomagnetism and Fisher (1953) Meet Modern Statistics," International Statistical Review, International Statistical Institute, vol. 90(2), pages 237-258, August.
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    4. Andrew W. Senior & Richard Evans & John Jumper & James Kirkpatrick & Laurent Sifre & Tim Green & Chongli Qin & Augustin Žídek & Alexander W. R. Nelson & Alex Bridgland & Hugo Penedones & Stig Petersen, 2020. "Improved protein structure prediction using potentials from deep learning," Nature, Nature, vol. 577(7792), pages 706-710, January.
    5. Kanti V. Mardia & Karthik Sriram, 2023. "Families of Discrete Circular Distributions with Some Novel Applications," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(1), pages 1-42, February.
    6. Mardia, Kanti V. & Wiechers, Henrik & Eltzner, Benjamin & Huckemann, Stephan F., 2022. "Principal component analysis and clustering on manifolds," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    7. Kanti V. Mardia, 2013. "Statistical approaches to three key challenges in protein structural bioinformatics," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 62(3), pages 487-514, May.
    8. Mardia, Kanti V., 2024. "Fisher’s pioneering work on discriminant analysis and its impact on Artificial Intelligence," Journal of Multivariate Analysis, Elsevier, vol. 203(C).
    9. Kanti V. Mardia & Charles C. Taylor & Ganesh K. Subramaniam, 2007. "Protein Bioinformatics and Mixtures of Bivariate von Mises Distributions for Angular Data," Biometrics, The International Biometric Society, vol. 63(2), pages 505-512, June.
    10. Arthur Pewsey & Eduardo García-Portugués, 2021. "Rejoinder 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 76-82, March.
    11. 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.
    12. Arthur Pewsey & Eduardo García-Portugués, 2021. "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 1-58, March.
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