Editing Directional statistics (section)

==Distributions on higher-dimensional manifolds==
[[File:Point sets from Kent distributions mapped onto a sphere - journal.pcbi.0020131.g004.svg|thumb|250px|Three points sets sampled from different Kent distributions on the sphere.]]

There also exist distributions on the [[two-dimensional sphere]] (such as the [[Kent distribution]]<ref>Kent, J (1982) [https://web.archive.org/web/20190721162933/https://apps.dtic.mil/dtic/tr/fulltext/u2/a097475.pdf The Fisher–Bingham distribution on the sphere]. J Royal Stat Soc, 44, 71–80.</ref>), the [[n-sphere|''N''-dimensional sphere]] (the [[von Mises–Fisher distribution]]<ref>Fisher, RA (1953) Dispersion on a sphere. Proc. Roy. Soc. London Ser. A., 217, 295–305</ref>) or the [[torus]] (the [[bivariate von Mises distribution]]<ref>{{cite journal | last1 = Mardia | first1 = KM. Taylor | last2 = CC | last3 = Subramaniam | first3 = GK. | year = 2007 | title = Protein Bioinformatics and Mixtures of Bivariate von Mises Distributions for Angular Data | journal = Biometrics | volume = 63 | issue = 2| pages = 505–512 | doi=10.1111/j.1541-0420.2006.00682.x| pmid = 17688502 | s2cid = 14293602 }}</ref>).

The [[matrix von Mises–Fisher distribution]]<ref>{{cite journal |last1=Pal |first1=Subhadip |last2=Sengupta |first2=Subhajit |last3=Mitra |first3=Riten |last4=Banerjee |first4=Arunava |title=Conjugate Priors and Posterior Inference for the Matrix Langevin Distribution on the Stiefel Manifold |journal=Bayesian Analysis |date=September 2020 |volume=15 |issue=3 |pages=871–908 |doi=10.1214/19-BA1176 |s2cid=209974627 |issn=1936-0975|doi-access=free }}</ref> is a distribution on the [[Stiefel manifold]], and can be used to construct probability distributions over [[rotation matrix|rotation matrices]].<ref>{{cite journal | last1 = Downs | year = 1972 | title = Orientational statistics | journal = Biometrika | volume = 59 | issue = 3| pages = 665–676 | doi=10.1093/biomet/59.3.665}}</ref>

The [[Bingham distribution]] is a distribution over axes in ''N'' dimensions, or equivalently, over points on the (''N''&nbsp;−&nbsp;1)-dimensional sphere with the antipodes identified.<ref>{{cite journal | last1 = Bingham | first1 = C. | author-link = Christopher Bingham | year = 1974 | title = An Antipodally Symmetric Distribution on the Sphere | journal = Ann. Stat. | volume = 2 | issue = 6| pages = 1201–1225 | doi=10.1214/aos/1176342874| doi-access = free }}</ref> For example, if ''N''&nbsp;=&nbsp;2, the axes are undirected lines through the origin in the plane. In this case, each axis cuts the unit circle in the plane (which is the one-dimensional sphere) at two points that are each other's antipodes. For ''N''&nbsp;=&nbsp;4, the Bingham distribution is a distribution over the space of unit [[quaternions]] ([[versor]]s). Since a versor corresponds to a rotation matrix, the Bingham distribution for ''N''&nbsp;=&nbsp;4 can be used to construct probability distributions over the space of rotations, just like the Matrix-von Mises–Fisher distribution.

These distributions are for example used in [[geology]],<ref>{{cite journal | last1 = Peel | first1 = D. | last2 = Whiten | first2 = WJ. | last3 = McLachlan | first3 = GJ. | year = 2001 | title = Fitting mixtures of Kent distributions to aid in joint set identification | url =http://www.maths.uq.edu.au/~gjm/pwm_jasa01.pdf | journal = J. Am. Stat. Assoc. | volume = 96 | issue = 453| pages = 56–63 | doi=10.1198/016214501750332974| s2cid = 11667311 }}</ref> [[crystallography]]<ref>{{cite journal | last1 = Krieger Lassen | first1 = N. C. | last2 = Juul Jensen | first2 = D. | last3 = Conradsen | first3 = K. | year = 1994 | title = On the statistical analysis of orientation data | journal = Acta Crystallogr | volume = A50 | issue = 6| pages = 741–748 | doi = 10.1107/S010876739400437X | bibcode = 1994AcCrA..50..741K }}</ref> and [[bioinformatics]].<ref name="compbiol.plosjournals.org"/>
<ref>Kent, J.T., Hamelryck, T. (2005). [http://www.amsta.leeds.ac.uk/statistics/workshop/lasr2005/Proceedings/kent.pdf Using the Fisher–Bingham distribution in stochastic models for protein structure] {{Webarchive|url=https://web.archive.org/web/20240120062215/http://www.amsta.leeds.ac.uk/statistics/workshop/lasr2005/Proceedings/kent.pdf |date=2024-01-20 }}. In S. Barber, P.D. Baxter, K.V.Mardia, & R.E. Walls (Eds.), Quantitative Biology, Shape Analysis, and Wavelets, pp. 57–60. Leeds, Leeds University Press</ref>
<ref>{{cite journal|title= A generative, probabilistic model of local protein structure| doi=10.1073/pnas.0801715105| pmid=18579771|volume=105| issue=26|journal=Proceedings of the National Academy of Sciences|pages=8932–8937|pmc=2440424| year=2008| last1=Boomsma| first1=Wouter| last2=Mardia| first2=Kanti V.| last3=Taylor| first3=Charles C.| last4=Ferkinghoff-Borg| first4=Jesper| last5=Krogh| first5=Anders| last6=Hamelryck| first6=Thomas| bibcode=2008PNAS..105.8932B| doi-access=free}}</ref>