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Simulation and Visualization of 3D-Spherical Distributions

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Directional Statistics for Innovative Applications

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

Abstract

In this paper, we consider simulation and visualization of spherical distributions as well as plotting of densities and histograms on the sphere, in support of the dictum, “A picture is worth a thousand words.” This is in many ways a companion paper to the recently published theoretical article by the authors on spherical harmonics (Jammalamadaka and Terdik in J Multivariate Anal 171:436–451, 2019 [1]). We provide computational algorithms to simulate several of the spherical models discussed there and provide alternate and improved methods in some cases. This allows a user to choose between alternate approaches for generating such random variates, and in depicting them via plots and histograms. The algorithms are made available in a MATLAB package titled “3D-Directional Statistics, Simulation, and Visualization” abbreviated 3D-Directional-SSV which is available at MATLAB, File Exchange, and also posted on GitHub. This work is especially appropriate in this volume celebrating Florence Nightingale, the progenitor of what we now call the rose diagram in two dimensions.

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Notes

  1. 1.

    Note: In [2], the Procedure for FB\(_{4}^{-}\) on p.890 has a misprint. The correct expressions should be \(\mu _{1}=({2\gamma -\kappa })/{\sqrt{-2\gamma }},\quad \mu _{2}=-({2\gamma +\kappa })/{\sqrt{-2\gamma }}\).

  2. 2.

    See Sect. 2.2.1 for details.

  3. 3.

    Our algorithm corrects an error in [11].

References

  1. Jammalamadaka, S. R., Terdik, G.: Harmonic analysis and distribution-free inference for spherical distributions. J. Multivariate Anal. 171, 436–451 (2019)

    Google Scholar 

  2. Wood, A.T.A.: The simulation of spherical distributions in the Fisher-Bingham family. Commun. Stat.-Simul. Comput. 16(3), 885–898 (1987)

    Article  MathSciNet  Google Scholar 

  3. Kent, J.T.: The Fisher-Bingham distribution on the sphere. J. Roy. Stat. Soc. Ser. B (Methodol.) 71–80 (1982)

    Google Scholar 

  4. Kent, J.T., Ganeiber, A.M., Mardia, K.V.: A new unified approach for the simulation of a wide class of directional distributions. J. Comput. Graph. Stat. 27, 291–301 (2018)

    Article  MathSciNet  Google Scholar 

  5. Gatto, R.: The generalized von Mises–Fisher distribution. In: Advances in Directional and Linear Statistics: A Festschrift for Sreenivasa Rao Jammalamadaka, pp. 51–68. Springer Science & Business Media, Berlin (2011)

    Google Scholar 

  6. Mardia, K.V., Jupp, P.E.: Directional Statistics, vol. 494. Wiley, Hoboken (2009)

    Google Scholar 

  7. Fisher, R.A.: Dispersion on a sphere. Proc. Roy. Soc. Lond. A: Math. Phys. Eng. Sci. 217(1130), 295–305 (1953)

    Article  MathSciNet  Google Scholar 

  8. Dimroth, E.: Untersiichungen aim wechanismiis von blastais und syntexis. Phylliterr and Hodelsen des siidm3lichen Fichtelgebirges I Die statistische Airswertung einfacher Giirteldiagranime Txheim Man Pctr Mitt. 8, 248–274 (1962)

    Google Scholar 

  9. Watson, G.S.: Equatorial distributions on a sphere. Biometrika 52(1/2), 193–201 (1965)

    Article  MathSciNet  Google Scholar 

  10. Bingham, C.: An antipodally symmetric distribution on the sphere. Ann. Stat. 1201–1225 (1974)

    Google Scholar 

  11. Kent, J.T., Hamelryck, T.: Using the Fisher-Bingham distribution in stochastic models for protein structure. Quant. Biol. Shape Anal. Wavelets 24, 57–60 (2005)

    Google Scholar 

  12. Watson, G.S.: Statistics on Spheres. University of Arkansas Lecture Notes in the Mathematical Sciences, 6. Wiley, New York (1983)

    Google Scholar 

  13. Gorski, K.M., Hivon, E., Banday, A.J., et al.: HEALPix a framework for high-resolution discretization and fast analysis of data distributed on the sphere. Astrophys. J. 622(2), 759 (2005)

    Article  Google Scholar 

  14. von Neumann, J.: Various techniques used in connection with random digits. US Nat. Bur. Stand. Appl. Math. Ser. 12, 36–38 (1951)

    Google Scholar 

  15. Rubinstein, R.Y.: Simulation and the Monte Carlo Method. Wiley, Hoboken (1981)

    Google Scholar 

  16. Wood, A.T.A.: Some notes on the Fisher-Bingham family on the sphere. Commun. Stat.-Theory Methods 17(11), 3881–3897 (1988)

    Article  MathSciNet  Google Scholar 

  17. Ulrich, G.: Computer generation of distributions on the m-sphere. Appl. Stat. 158–163 (1984)

    Google Scholar 

  18. Atkinson, A.C., Pearce, M.C.: The computer generation of beta, gamma and normal random variables. J. Roy. Stat. Soc. Ser. A (General) 431–461 (1976)

    Google Scholar 

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Acknowledgements

We thank Professors Ashis SenGupta and Barry Arnold for inviting us to present our work for this special volume and a reviewer for thoughtful comments. Dr. Brian Wainwright helped us with a preliminary version of this work.

The research of the second author is partially supported by the Project EFOP3.6.2-16-2017-00015 of the European Union and co-financed by the European Social Fund.

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Authors and Affiliations

  1. Department of Statistics and Applied Probability, University of California, Santa Barbara, USA

    S. Rao Jammalamadaka

  2. Faculty of Informatics, University of Debrecen, Debrecen, Hungary

    Gyorgy Terdik

Authors
  1. S. Rao Jammalamadaka
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  2. Gyorgy Terdik
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Corresponding author

Correspondence to S. Rao Jammalamadaka .

Editor information

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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Jammalamadaka, S.R., Terdik, G. (2022). Simulation and Visualization of 3D-Spherical Distributions. 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_7

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