Editing Directional statistics

{{Short description|Subdiscipline of statistics}}
'''Directional statistics''' (also '''circular statistics''' or '''spherical statistics''') is the subdiscipline of [[statistics]] that deals with [[Direction (geometry)|directions]] ([[unit vector]]s in [[Euclidean space]], '''R'''<sup>''n''</sup>), [[Cartesian Coordinate System|axes]] ([[Line (geometry)|lines]] through the origin in '''R'''<sup>''n''</sup>) or [[Rotation (mathematics)|rotations]] in '''R'''<sup>''n''</sup>. More generally, directional statistics deals with observations on compact [[Riemannian manifold]]s including the [[Stiefel manifold]].

[[File:Fb5 cover.jpg|thumb|250px|right|The overall shape of a [[protein]] can be parameterized as a sequence of points on the unit  [[sphere]]. Shown are two views of the spherical [[histogram]] of such points for a large collection of protein structures. The statistical treatment of such data is in the realm of directional statistics.<ref name="compbiol.plosjournals.org">{{cite journal|title=Hamelryck, T., Kent, J., Krogh, A. (2006) Sampling realistic protein conformations using local structural bias. PLoS Comput. Biol., 2(9): e131|journal=PLOS Computational Biology|volume=2|issue=9|pages=e131|doi=10.1371/journal.pcbi.0020131|pmid=17002495|pmc=1570370|year = 2006|last1 = Hamelryck|first1 = Thomas|last2=Kent|first2=John T.|last3=Krogh|first3=Anders|bibcode=2006PLSCB...2..131H |doi-access=free }}</ref>]]

The fact that 0 [[degree (angle)|degrees]] and 360 degrees are identical [[angle]]s, so that for example 180 degrees is not a sensible [[average|mean]] of 2 degrees and 358 degrees, provides one illustration that special statistical methods are required for the analysis of some types of data (in this case, angular data). Other examples of data that may be regarded as directional include statistics involving temporal periods (e.g. time of day, week, month, year, etc.), compass directions, [[dihedral angle]]s in molecules, orientations, rotations and so on.

==Circular distributions==
{{main|Circular distribution}}
{{anchor|Distributions}}
Any [[probability density function]] (pdf) <math>\ p(x)</math> on the line can be [[wrapped distribution|"wrapped"]] around the circumference of a circle of unit radius.<ref>Bahlmann, C., (2006), [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.330.9384&rep=rep1&type=pdf Directional features in online handwriting recognition], Pattern Recognition, 39</ref> That is, the pdf of the wrapped variable 
<math display="block">\theta = x_w=x \bmod 2\pi\ \ \in (-\pi,\pi]</math>
is
<math display="block">p_w(\theta) = \sum_{k=-\infty}^{\infty}{p(\theta+2\pi k)}.</math>

This concept can be extended to the multivariate context by an extension of the simple sum to a number of <math>F</math> sums that cover all dimensions in the feature space:
<math display="block">p_w(\boldsymbol\theta) = \sum_{k_1=-\infty}^{\infty} \cdots \sum_{k_F=-\infty}^\infty {p(\boldsymbol\theta + 2\pi k_1\mathbf{e}_1 + \dots + 2\pi k_F\mathbf{e}_F)}</math>
where <math>\mathbf{e}_k = (0, \dots, 0, 1, 0, \dots, 0)^{\mathsf{T}}</math> is the <math>k</math>-th Euclidean basis vector.

The following sections show some relevant circular distributions.

===von Mises circular distribution===
{{main|von Mises distribution}}
The ''von Mises distribution'' is a circular distribution which, like any other circular distribution, may be thought of as a wrapping of a certain linear probability distribution around the circle. The underlying linear probability distribution for the von Mises distribution is mathematically intractable; however, for statistical purposes, there is no need to deal with the underlying linear distribution. The usefulness of the von Mises distribution is twofold: it is the most mathematically tractable of all circular distributions, allowing simpler statistical analysis, and it is a close approximation to the [[wrapped normal]] distribution, which, analogously to the linear normal distribution, is important because it is the limiting case for the sum of a large number of small angular deviations. In fact, the von Mises distribution is often known as the "circular normal" distribution because of its ease of use and its close relationship to the wrapped normal distribution.{{sfn|Fisher|1993}}

The pdf of the von Mises distribution is: <math display="block">f(\theta;\mu,\kappa) = \frac{e^{\kappa\cos(\theta-\mu)}}{2\pi I_0(\kappa)}</math> where <math>I_0</math> is the modified [[Bessel function]] of order 0.
<!-- * A fundamental wrapped distribution is the [[Dirac comb]] of period <math>2\pi\,</math> which is a wrapped delta function:

<math>\Delta_{2\pi}(\theta)=\sum_{k=-\infty}^{\infty}{\delta(\theta+2\pi k)}</math> -->

===Circular uniform distribution===
{{main|Circular uniform distribution}}
The probability density function (pdf) of the ''circular uniform distribution'' is given by
<math display="block">U(\theta) = \frac 1 {2\pi}.</math>

It can also be thought of as <math>\kappa = 0</math> of the von Mises above.

===Wrapped normal distribution===
{{main|Wrapped normal distribution}}
The pdf of the ''wrapped normal distribution'' (WN) is:
<math display="block">
WN(\theta;\mu,\sigma) = \frac{1}{\sigma \sqrt{2\pi}} \sum^{\infty}_{k=-\infty} \exp \left[\frac{-(\theta - \mu - 2\pi k)^2}{2 \sigma^2} \right] = \frac{1}{2\pi}\vartheta\left(\frac{\theta-\mu}{2\pi},\frac{i\sigma^2}{2\pi}\right)
</math>
where μ and σ are the mean and standard deviation of the unwrapped distribution, respectively and  <math>\vartheta(\theta,\tau)</math> is the [[Theta function|Jacobi theta function]]:
<math display="block">
\vartheta(\theta,\tau) = \sum_{n=-\infty}^\infty (w^2)^n q^{n^2} 
</math> where <math>w \equiv e^{i\pi \theta}</math> and <math>q \equiv e^{i\pi\tau}.</math>

===Wrapped Cauchy distribution===
{{main|Wrapped Cauchy distribution}}
The pdf of the ''wrapped Cauchy distribution'' (WC) is:
<math display="block">WC(\theta;\theta_0,\gamma) = \sum_{n=-\infty}^\infty \frac{\gamma}{\pi(\gamma^2+(\theta+2\pi n-\theta_0)^2)}
= \frac{1}{2\pi}\,\,\frac{\sinh\gamma}{\cosh\gamma-\cos(\theta-\theta_0)}</math>
where <math>\gamma</math> is the scale factor and <math>\theta_0</math> is the peak position.

===Wrapped Lévy distribution===
{{main|Wrapped Lévy distribution}}
The pdf of the ''wrapped Lévy distribution'' (WL) is:
<math display="block">f_{WL}(\theta;\mu,c) = \sum_{n=-\infty}^\infty \sqrt{\frac{c}{2\pi}}\,\frac{e^{-c/2(\theta+2\pi n-\mu)}}{(\theta+2\pi n-\mu)^{3/2}}</math>
where the value of the summand is taken to be zero when <math>\theta+2\pi n-\mu \le 0</math>, <math>c</math> is the scale factor and <math>\mu</math> is the location parameter.

===Projected normal distribution===
{{main|Projected normal distribution}}
The projected normal distribution is a circular distribution representing the direction of a random variable with multivariate normal distribution, obtained by radial projection of the variable over the unit (n-1)-sphere. Due to this, and unlike other commonly used circular distributions, it is not symmetric nor [[Unimodality|unimodal]].

==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>

== Moments ==
The raw vector (or trigonometric) moments of a circular distribution are defined as

:<math>
m_n=\operatorname E(z^n)=\int_\Gamma P(\theta) z^n \, d\theta
</math>

where <math>\Gamma</math> is any interval of length <math>2\pi</math>, <math>P(\theta)</math> is the [[Probability density function|PDF]] of the circular distribution, and <math>z=e^{i \theta}</math>. Since the integral <math>P(\theta)</math> is unity, and the integration interval is finite, it follows that the moments of any circular distribution are always finite and well defined.

Sample moments are analogously defined:

:<math>
\overline{m}_n=\frac{1}{N}\sum_{i=1}^N z_i^n.
</math>

The population resultant vector, length, and mean angle are defined in analogy with the corresponding sample parameters.

:<math>
\rho=m_1
</math>

:<math>
R=|m_1|
</math>

:<math>
\theta_n=\operatorname{Arg}(m_n).
</math>

In addition, the lengths of the higher moments are defined as:

:<math>
R_n=|m_n|
</math>

while the angular parts of the higher moments are just <math>(n \theta_n) \bmod 2\pi</math>. The lengths of all moments will lie between 0 and 1.

== Measures of location and spread ==
Various measures of [[central tendency]] and [[statistical dispersion]] may be defined for both the population and a sample drawn from that population.{{sfn|Fisher|1993}}

=== Central tendency ===
{{Further|Circular mean}}

The most common measure of location is the circular mean. The population circular mean is simply the first moment of the distribution while the sample mean is the first moment of the sample. The sample mean will serve as an unbiased estimator of the population mean.

When data is concentrated, the [[median]] and [[Mode (statistics)|mode]] may be defined by analogy to the linear case, but for more dispersed or multi-modal data, these concepts are not useful.

=== Dispersion ===
{{See also|Yamartino method}}

The most common measures of circular spread are:

* The '''{{visible anchor|circular variance|Variance}}'''. For the sample the circular variance is defined as: <math display="block">
\overline{\operatorname{Var}(z)} = 1 - \overline{R}
</math> and for the population <math display="block">
\operatorname{Var}(z) = 1 - R
</math> Both will have values between 0 and 1.
* The '''{{visible anchor|circular standard deviation|Standard deviation}}''' <math display="block">
S(z) = \sqrt{\ln(1/R^2)} = \sqrt{-2\ln(R)}
</math> <math display="block">
\overline{S}(z) = \sqrt{\ln(1/{\overline{R}}^2)} = \sqrt{-2\ln({\overline{R}})}
</math> with values between 0 and infinity. This definition of the standard deviation (rather than the square root of the variance) is useful because for a wrapped normal distribution, it is an estimator of the standard deviation of the underlying normal distribution. It will therefore allow the circular distribution to be standardized as in the linear case, for small values of the standard deviation. This also applies to the von Mises distribution which closely approximates the wrapped normal distribution. Note that for small <math>S(z)</math>, we have <math>S(z)^2 = 2 \operatorname{Var}(z)</math>.
* The '''{{visible anchor|circular dispersion|Dispersion}}''' <math display="block">\delta = \frac{1-R_2}{2R^2}</math> <math display="block">
\overline{\delta}=\frac{1-{\overline{R}_2}}{2{\overline{R}}^2}
</math> with values between 0 and infinity. This measure of spread is found useful in the statistical analysis of variance.

== Distribution of the mean ==
Given a set of ''N'' measurements <math>z_n=e^{i\theta_n}</math> the mean value of ''z'' is defined as:

:<math>
\overline{z}=\frac{1}{N}\sum_{n=1}^N z_n
</math>

which may be expressed as

:<math>
\overline{z} = \overline{C}+i\overline{S}
</math>

where

:<math>
\overline{C} = \frac{1}{N}\sum_{n=1}^N \cos(\theta_n) \text{ and } \overline{S} = \frac{1}{N}\sum_{n=1}^N \sin(\theta_n)
</math>

or, alternatively as:

:<math>
\overline{z} = \overline{R}e^{i\overline{\theta}}
</math>

where

:<math>
\overline{R} = \sqrt{{\overline{C}}^2+{\overline{S}}^2} \text{ and } \overline{\theta} = \arctan (\overline{S} / \overline{C}).
</math>

The distribution of the mean angle (<math>\overline{\theta}</math>) for a circular pdf ''P''(''&theta;'') will be given by:

:<math>
P(\overline{C},\overline{S}) \, d\overline{C} \, d\overline{S} =
P(\overline{R},\overline{\theta}) \, d\overline{R} \, d\overline{\theta} = 
\int_\Gamma \cdots \int_\Gamma \prod_{n=1}^N \left[ P(\theta_n) \, d\theta_n \right]
</math>

where <math>\Gamma</math> is over any interval of length <math>2\pi</math> and the integral is subject to the constraint that <math>\overline{S}</math> and <math>\overline{C}</math> are constant, or, alternatively, that <math>\overline{R}</math> and <math>\overline{\theta}</math> are constant.

The calculation of the distribution of the mean for most circular distributions is not analytically possible, and in order to carry out an analysis of variance, numerical or mathematical approximations are needed.{{sfn|Jammalamadaka|Sengupta|2001}}

The [[central limit theorem]] may be applied to the distribution of the sample means. (main article: [[Central limit theorem for directional statistics]]). It can be shown{{sfn|Jammalamadaka|Sengupta|2001}} that the distribution of <math>[\overline{C},\overline{S}]</math> approaches a [[bivariate normal distribution]] in the limit of large sample size.

==Goodness of fit and significance testing==
For cyclic data – (e.g., is it uniformly distributed) :
* [[Rayleigh test]] for a unimodal cluster
* [[Kuiper's test]] for possibly multimodal data.

== See also ==
* [[Circular correlation coefficient]]
* [[Complex normal distribution]]
* [[Wrapped distribution]]

==References==
<references />

==Books on directional statistics==
* {{cite book|last=Batschelet|first=E.|title=Circular statistics in biology|publisher=[[Academic Press]]|location=London|year=1981|isbn=0-12-081050-6}}
* {{cite book|authorlink=Nicholas Fisher (statistician)|last=Fisher|first=N. I.|title=Statistical Analysis of Circular Data|language=en|publisher=[[Cambridge University Press]]|year=1993|isbn=0-521-35018-2}}
* {{cite book|authorlink1=Nicholas Fisher (statistician)|last1=Fisher|first1=N. I.|last2=Lewis|first2=T.|last3=Embleton|first3=BJJ|title=Statistical Analysis of Spherical Data|publisher=[[Cambridge University Press]]|year=1993|isbn=0-521-45699-1}}
* {{cite book|last1=Jammalamadaka|first1=S. Rao|last2=Sengupta|first2=A.|title=Topics in Circular Statistics|publisher=World Scientific|location=New Jersey|year=2001|isbn=981-02-3778-2|url=https://books.google.com/books?id=sKqWMGqQXQkC&q=Jammalamadaka+Topics+in+circular|access-date=2011-05-15}}
* {{cite book|authorlink1=Kantilal Mardia|last1=Mardia|first1=K. V.|last2=Jupp|first2=P.|title=Directional Statistics|edition=2nd|publisher=John Wiley and Sons Ltd.|year=2000|isbn=0-471-95333-4}}
* {{cite book|last1=Ley|first1=C.|last2=Verdebout|first2=T.|title=Modern Directional Statistics|publisher=[[CRC Press]] [[Taylor & Francis Group]]|year=2017|isbn=978-1-4987-0664-3}}

{{ProbDistributions|directional}}

[[Category:Directional statistics| ]]
[[Category:Statistical data types]]
[[Category:Statistical theory]]
[[Category:Types of probability distributions]]