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{{Probability distribution|
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
  name      =Wrapped Normal|
  type      =density|
  pdf_image =[[File:WrappedNormalPDF.png|325px|Plot of the von Mises PMF]]<br /><small>The support is chosen to be [-π,π] with μ=0</small>|
  cdf_image  =[[File:WrappedNormalCDF.png|325px|Plot of the von Mises CMF]]<br /><small>The support is chosen to be [-π,π] with μ=0</small>|
  parameters =<math>\mu</math> real<br><math>\sigma>0</math>|
  support    =<math>\theta \in</math> any interval of length 2π|
  pdf        =<math>\frac{1}{2\pi}\vartheta\left(\frac{\theta-\mu}{2\pi},\frac{i\sigma^2}{2\pi}\right)</math>|
  cdf        =|
  mean      =<math>\mu</math>|
  median    =<math>\mu</math>|
  mode      =<math>\mu</math>|
  variance  =<math>1-e^{-\sigma^2/2}</math> (circular)|
  skewness  =|
  kurtosis  =|
  entropy    =(see text)|
  mgf        =|
  cf        =<math>e^{-\sigma^2n^2/2+in\mu}</math>|
}}
In [[probability theory]] and [[directional statistics]], a '''wrapped normal distribution''' is a [[wrapped distribution|wrapped probability distribution]] that results from the "wrapping" of the [[normal distribution]] around the [[unit circle]]. It finds application in the theory of [[Brownian motion]] and is a solution to the [[Theta function#A solution to heat equation|heat equation]] for [[periodic boundary conditions]]. It is closely approximated by the [[von Mises distribution]], which, due to its mathematical simplicity and tractability, is the most commonly used distribution in directional statistics.


==Definition==
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
The [[probability density function]] of the wrapped normal distribution is<ref name="Mardia99">{{cite book |title=Directional Statistics |last=Mardia |first=Kantilal |authorlink=Kantilal Mardia |author2=Jupp, Peter E.  |year=1999|publisher=Wiley |location= |isbn=978-0-471-95333-3 |url=http://www.amazon.com/Directional-Statistics-Kanti-V-Mardia/dp/0471953334/ref=sr_1_1?s=books&ie=UTF8&qid=1311003484&sr=1-1#reader_0471953334 |accessdate=2011-07-19}}</ref>
* Only registered users will be able to execute this rendering mode.
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:<math>
Registered users will be able to choose between the following three rendering modes:  
f_{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]
</math>


where ''μ'' and ''σ'' are the mean and standard deviation of the unwrapped distribution, respectively. [[Wrapped distribution|Expressing]] the above density function in terms of the [[characteristic function (probability theory)|characteristic function]] of the normal distribution yields:<ref name="Mardia99"/>
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


:<math>
<!--'''PNG'''  (currently default in production)
f_{WN}(\theta;\mu,\sigma)=\frac{1}{2\pi}\sum_{n=-\infty}^\infty e^{-\sigma^2n^2/2+in(\theta-\mu)} =\frac{1}{2\pi}\vartheta\left(\frac{\theta-\mu}{2\pi},\frac{i\sigma^2}{2\pi}\right) ,
:<math forcemathmode="png">E=mc^2</math>
</math>


where <math>\vartheta(\theta,\tau)</math> is the [[Theta function|Jacobi theta function]], given by
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


:<math>
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].
\vartheta(\theta,\tau)=\sum_{n=-\infty}^\infty (w^2)^n q^{n^2}
\text{ where } w \equiv e^{i\pi \theta}</math> and <math>q \equiv e^{i\pi\tau} .</math>


The wrapped normal distribution may also be expressed in terms of the [[Jacobi triple product]]:<ref name="W&W">{{cite book |title=A Course of Modern Analysis |last=Whittaker |first=E. T. |authorlink= |author2=Watson, G. N.  |year=2009 |publisher=Book Jungle |location= |isbn=978-1-4385-2815-1 |page= |pages= |url= |accessdate=}}</ref>
==Demos==


:<math>f_{WN}(\theta;\mu,\sigma)=\frac{1}{2\pi}\prod_{n=1}^\infty (1-q^n)(1+q^{n-1/2}z)(1+q^{n-1/2}/z) .</math>
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


where <math>z=e^{i(\theta-\mu)}\,</math> and <math>q=e^{-\sigma^2}.</math>


== Moments ==
* accessibility:
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


In terms of the circular variable <math>z=e^{i\theta}</math> the circular moments of the wrapped Normal distribution are the characteristic function of the Normal distribution evaluated at integer arguments:
==Test pages ==


:<math>\langle z^n\rangle=\int_\Gamma e^{in\theta}\,f_{WN}(\theta;\mu,\sigma)\,d\theta = e^{i n \mu-n^2\sigma^2/2}.</math>
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


where <math>\Gamma\,</math> is some interval of length <math>2\pi</math>. The first moment is then the average value of ''z'', also known as the mean resultant, or mean resultant vector:
*[[Inputtypes|Inputtypes (private Wikis only)]]
 
*[[Url2Image|Url2Image (private Wikis only)]]
:<math>
==Bug reporting==
\langle z \rangle=e^{i\mu-\sigma^2/2}
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
</math>
 
The mean angle is
 
:<math>
\theta_\mu=\mathrm{Arg}\langle z \rangle = \mu
</math>
 
and the length of the mean resultant is
 
:<math>
R=|\langle z \rangle| = e^{-\sigma^2/2}
</math>
 
The circular standard deviation, which is a useful measure of dispersion for the wrapped Normal distribution and its close relative, the [[von Mises distribution]] is given by:
 
:<math>
s=\sqrt{\ln(1/R^2)} = \sigma
</math>
 
== Estimation of parameters ==
 
A series of ''N'' measurements ''z''<sub>''n''</sub>&nbsp;=&nbsp;''e''<sup>&nbsp;''i&theta;''<sub>''n''</sub></sup> drawn from a wrapped normal distribution may be used to estimate certain parameters of the distribution. The average of the series {{overbar|''z''}} is defined as
 
:<math>\overline{z}=\frac{1}{N}\sum_{n=1}^N z_n</math>
 
and its expectation value will be just the first moment:
 
:<math>\langle\overline{z}\rangle=e^{i\mu-\sigma^2/2}. \,</math>
 
In other words, {{overbar|''z''}} is an unbiased estimator of the first moment. If we assume that the mean ''&mu;'' lies in the interval <nowiki>[</nowiki>&minus;''&pi;'',&nbsp;''&pi;''<nowiki>)</nowiki>, then Arg&nbsp;{{overbar|''z''}} will be a (biased) estimator of the mean&nbsp;''&mu;''.
 
Viewing the ''z''<sub>''n''</sub> as a set of vectors in the complex plane, the {{overbar|''R''}}<sup>2</sup> statistic is the square of the length of the averaged vector:
 
:<math>\overline{R}^2=\overline{z}\,\overline{z^*}=\left(\frac{1}{N}\sum_{n=1}^N \cos\theta_n\right)^2+\left(\frac{1}{N}\sum_{n=1}^N \sin\theta_n\right)^2 \, </math>
 
and its expected value is:
 
:<math>\left\langle \overline{R}^2\right\rangle = \frac{1}{N}+\frac{N-1}{N}\,e^{-\sigma^2}\,</math>
 
In other words, the statistic
 
:<math>R_e^2=\frac{N}{N-1}\left(\overline{R}^2-\frac{1}{N}\right)</math>
 
will be an unbiased estimator of ''e''<sup>&minus;''&sigma;''<sup>2</sup></sup>, and ln(1/''R''<sub>''e''</sub><sup>2</sup>) will be a (biased) estimator of&nbsp;''&sigma;''<sup>2</sup>
 
== Entropy ==
 
The [[Entropy (information theory)|information entropy]] of the wrapped normal distribution is defined as:<ref name="Mardia99"/>
 
:<math>H = -\int_\Gamma f_{WN}(\theta;\mu,\sigma)\,\ln(f_{WN}(\theta;\mu,\sigma))\,d\theta</math>
 
where <math>\Gamma</math> is any interval of length <math>2\pi</math>. Defining <math>z=e^{i(\theta-\mu)}</math> and <math>q=e^{-\sigma^2}</math>, the [[Jacobi triple product]] representation for the wrapped normal is:
 
:<math>f_{WN}(\theta;\mu,\sigma) = \frac{\phi(q)}{2\pi}\prod_{m=1}^\infty (1+q^{m-1/2}z)(1+q^{m-1/2}z^{-1})</math>
 
where <math>\phi(q)\,</math> is the [[Euler function]]. The logarithm of the density of the wrapped normal distribution may be written:
 
:<math>\ln(f_{WN}(\theta;\mu,\sigma))=  \ln\left(\frac{\phi(q)}{2\pi}\right)+\sum_{m=1}^\infty\ln(1+q^{m-1/2}z)+\sum_{m=1}^\infty\ln(1+q^{m-1/2}z^{-1})</math>
 
Using the series expansion for the logarithm:
 
:<math>\ln(1+x)=-\sum_{k=1}^\infty \frac{(-1)^k}{k}\,x^k</math>
 
the logarithmic sums may be written as:
 
:<math>\sum_{m=1}^\infty\ln(1+q^{m-1/2}z^{\pm 1})=-\sum_{m=1}^\infty \sum_{k=1}^\infty \frac{(-1)^k}{k}\,q^{mk-k/2}z^{\pm k} = -\sum_{k=1}^\infty \frac{(-1)^k}{k}\,\frac{q^{k/2}}{1-q^k}\,z^{\pm k}</math>
 
so that the logarithm of density of the wrapped normal distribution may be written as:
 
:<math>\ln(f_{WN}(\theta;\mu,\sigma))=\ln\left(\frac{\phi(q)}{2\pi}\right)-\sum_{k=1}^\infty \frac{(-1)^k}{k} \frac{q^{k/2}}{1-q^k}\,(z^k+z^{-k}) </math>
 
which is essentially a [[Fourier series]] in <math>\theta\,</math>. Using the characteristic function representation for the wrapped normal distribution in the left side of the integral:
 
:<math>f_{WN}(\theta;\mu,\sigma) =\frac{1}{2\pi}\sum_{n=-\infty}^\infty q^{n^2/2}\,z^n</math>
 
the entropy may be written:
 
:<math>H = -\ln\left(\frac{\phi(q)}{2\pi}\right)+\frac{1}{2\pi}\int_\Gamma \left( \sum_{n=-\infty}^\infty\sum_{k=1}^\infty \frac{(-1)^k}{k} \frac{q^{(n^2+k)/2}}{1-q^k}\left(z^{n+k}+z^{n-k}\right) \right)\,d\theta</math>
 
which may be integrated to yield:
 
:<math>H = -\ln\left(\frac{\phi(q)}{2\pi}\right)+2\sum_{k=1}^\infty \frac{(-1)^k}{k}\, \frac{q^{(k^2+k)/2}}{1-q^k}</math>
 
== See also ==
 
* [[Wrapped distribution]]
* [[Dirac comb]]
* [[Wrapped Cauchy distribution]]
 
== References ==
{{More footnotes|date=June 2014}}
<references/>
* {{cite book |title=Statistics of Earth Science Data |last=Borradaile |first=Graham |year=2003 |publisher=Springer |isbn=978-3-540-43603-4 |url=http://books.google.com/books?id=R3GpDglVOSEC&printsec=frontcover&source=gbs_navlinks_s#v=onepage&q=&f=false |accessdate=31 Dec 2009}}
* {{cite book |title=Statistical Analysis of Circular Data |last=Fisher |first=N. I. |year=1996 |publisher=Cambridge University Press |location= |isbn=978-0-521-56890-6
|url=http://books.google.com/books?id=IIpeevaNH88C&dq=%22circular+variance%22+fisher&source=gbs_navlinks_s |accessdate=2010-02-09}}
* {{cite journal |last1=Breitenberger |first1=Ernst |year=1963 |title=Analogues of the normal distribution on the circle and the sphere |journal=Biometrika |volume=50 |pages=81 |url=http://biomet.oxfordjournals.org/cgi/pdf_extract/50/1-2/81 |doi=10.2307/2333749}}
 
==External links==
* [http://www.codeproject.com/Articles/190833/Circular-Values-Math-and-Statistics-with-Cplusplus Circular Values Math and Statistics with C++11], A C++11 infrastructure for circular values (angles, time-of-day, etc.) mathematics and statistics
 
{{ProbDistributions|directional}}
 
[[Category:Continuous distributions]]
[[Category:Directional statistics]]
[[Category:Normal distribution]]
[[Category:Probability distributions]]

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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