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Circular distribution
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<h2 id="graphical-representation">Graphical representation</h2> <p>If a circular distribution has a density </p> p ( ϕ ) ( 0 ≤ ϕ < 2 π ) , {\displaystyle p(\phi )\qquad \qquad (0\leq \phi <2\pi ),\,} <p>it can be graphically represented as a closed <a href="/facts/Curve/xBQ0pTBW">curve</a> </p> [ x ( ϕ ) , y ( ϕ ) ] = [ r ( ϕ ) cos ϕ , r ( ϕ ) sin ϕ ] , {\displaystyle [x(\phi ),y(\phi )]=[r(\phi )\cos \phi ,\,r(\phi )\sin \phi ],\,} <p>where the radius r ( ϕ ) {\displaystyle r(\phi )\,} is set equal to </p> r ( ϕ ) = a + b p ( ϕ ) , {\displaystyle r(\phi )=a+bp(\phi ),\,} <p>and where <i>a</i> and <i>b</i> are chosen on the basis of appearance. </p> <h2 id="examples">Examples</h2> <p>By computing the probability distribution of angles along a handwritten ink trace, a lobe-shaped polar distribution emerges. The main direction of the lobe in the first quadrant corresponds to the <a href="/facts/Slant_(handwriting)/hKpJWl5Q">slant</a> of handwriting (see: <a href="/facts/Graphonomics/gLMYTYjR">graphonomics</a>). </p><p>An example of a circular lattice distribution would be the probability of being born in a given month of the year, with each calendar month being thought of as arranged round a circle, so that "January" is next to "December". </p> <p class="note">Main article: Circular distribution</p> <p> Any <a href="/facts/Probability_density_function/zvfybna4">probability density function</a> (pdf) p ( x ) {\displaystyle \ p(x)} on the line can be <a href="/facts/Wrapped_distribution/ysA2fo8W">"wrapped"</a> around the circumference of a circle of unit radius.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> That is, the pdf of the wrapped variable θ = x w = x mod 2 π ∈ ( − π , π ] {\displaystyle \theta =x_{w}=x{\bmod {2}}\pi \ \ \in (-\pi ,\pi ]} is p w ( θ ) = ∑ k = − ∞ ∞ p ( θ + 2 π k ) . {\displaystyle p_{w}(\theta )=\sum _{k=-\infty }^{\infty }{p(\theta +2\pi k)}.} </p><p>This concept can be extended to the multivariate context by an extension of the simple sum to a number of F {\displaystyle F} sums that cover all dimensions in the feature space: p w ( θ ) = ∑ k 1 = − ∞ ∞ ⋯ ∑ k F = − ∞ ∞ p ( θ + 2 π k 1 e 1 + ⋯ + 2 π k F e F ) {\displaystyle 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})}} where e k = ( 0 , … , 0 , 1 , 0 , … , 0 ) T {\displaystyle \mathbf {e} _{k}=(0,\dots ,0,1,0,\dots ,0)^{\mathsf {T}}} is the k {\displaystyle k} -th Euclidean basis vector. </p><p>The following sections show some relevant circular distributions. </p> <h3>von Mises circular distribution</h3> <p class="note">Main article: <a href="/facts/Von_Mises_distribution/bnFma2cF">von Mises distribution</a></p> <p>The <i>von Mises distribution</i> 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 <a href="/facts/Wrapped_normal/N16Nx35y">wrapped normal</a> 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.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p><p>The pdf of the von Mises distribution is: f ( θ ; μ , κ ) = e κ cos ( θ − μ ) 2 π I 0 ( κ ) {\displaystyle f(\theta ;\mu ,\kappa )={\frac {e^{\kappa \cos(\theta -\mu )}}{2\pi I_{0}(\kappa )}}} where I 0 {\displaystyle I_{0}} is the modified <a href="/facts/Bessel_function/hSyvFPqC">Bessel function</a> of order 0. </p> <h3>Circular uniform distribution</h3> <p class="note">Main article: <a href="/facts/Circular_uniform_distribution/rNHWEAJ7">Circular uniform distribution</a></p> <p>The probability density function (pdf) of the <i>circular uniform distribution</i> is given by U ( θ ) = 1 2 π . {\displaystyle U(\theta )={\frac {1}{2\pi }}.} </p><p>It can also be thought of as κ = 0 {\displaystyle \kappa =0} of the von Mises above. </p> <h3>Wrapped normal distribution</h3> <p class="note">Main article: <a href="/facts/Wrapped_normal_distribution/N16Nx35y">Wrapped normal distribution</a></p> <p>The pdf of the <i>wrapped normal distribution</i> (WN) is: W N ( θ ; μ , σ ) = 1 σ 2 π ∑ k = − ∞ ∞ exp [ − ( θ − μ − 2 π k ) 2 2 σ 2 ] = 1 2 π ϑ ( θ − μ 2 π , i σ 2 2 π ) {\displaystyle WN(\theta ;\mu ,\sigma )={\frac {1}{\sigma {\sqrt {2\pi }}}}\sum _{k=-\infty }^{\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)} where μ and σ are the mean and standard deviation of the unwrapped distribution, respectively and ϑ ( θ , τ ) {\displaystyle \vartheta (\theta ,\tau )} is the <a href="/facts/Theta_function/g09xvoQQ">Jacobi theta function</a>: ϑ ( θ , τ ) = ∑ n = − ∞ ∞ ( w 2 ) n q n 2 {\displaystyle \vartheta (\theta ,\tau )=\sum _{n=-\infty }^{\infty }(w^{2})^{n}q^{n^{2}}} where w ≡ e i π θ {\displaystyle w\equiv e^{i\pi \theta }} and q ≡ e i π τ . {\displaystyle q\equiv e^{i\pi \tau }.} </p> <h3>Wrapped Cauchy distribution</h3> <p class="note">Main article: <a href="/facts/Wrapped_Cauchy_distribution/bhJ7Qgdq">Wrapped Cauchy distribution</a></p> <p>The pdf of the <i>wrapped Cauchy distribution</i> (WC) is: W C ( θ ; θ 0 , γ ) = ∑ n = − ∞ ∞ γ π ( γ 2 + ( θ + 2 π n − θ 0 ) 2 ) = 1 2 π sinh γ cosh γ − cos ( θ − θ 0 ) {\displaystyle 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})}}} where γ {\displaystyle \gamma } is the scale factor and θ 0 {\displaystyle \theta _{0}} is the peak position. </p> <h3>Wrapped Lévy distribution</h3> <p class="note">Main article: <a href="/facts/Wrapped_L%C3%A9vy_distribution/CFGKApQo">Wrapped Lévy distribution</a></p> <p>The pdf of the <i>wrapped Lévy distribution</i> (WL) is: f W L ( θ ; μ , c ) = ∑ n = − ∞ ∞ c 2 π e − c / 2 ( θ + 2 π n − μ ) ( θ + 2 π n − μ ) 3 / 2 {\displaystyle 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}}}} where the value of the summand is taken to be zero when θ + 2 π n − μ ≤ 0 {\displaystyle \theta +2\pi n-\mu \leq 0} , c {\displaystyle c} is the scale factor and μ {\displaystyle \mu } is the location parameter. </p> <h3>Projected normal distribution</h3> <p class="note">Main article: <a href="/facts/Projected_normal_distribution/7npTaUpr">Projected normal distribution</a></p> <p>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 <a href="/facts/Unimodality/Xo3bhgeE">unimodal</a>. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Circular_mean/eMWmRu8P">Circular mean</a></li> <li><a href="/facts/Circular_uniform_distribution/rNHWEAJ7">Circular uniform distribution</a></li> <li><a href="/facts/Von_Mises_distribution/bnFma2cF">von Mises distribution</a></li></ul> <h2 id="sources">Sources</h2> <ul><li><a href="/facts/Nicholas_Fisher_(statistician)/7WCRxrgF">Fisher, N. I.</a> (1993). <i>Statistical Analysis of Circular Data</i>. <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-35018-2.</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="http://www.codeproject.com/Articles/190833/Circular-Values-Math-and-Statistics-with-Cplusplus">Circular Values Math and Statistics with C++11</a>, A C++11 infrastructure for circular values (angles, time-of-day, etc.) mathematics and statistics</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Dodge, Y. (2006). The Oxford Dictionary of Statistical Terms. OUP. ISBN 0-19-920613-9. <a href="0-19-920613-9" target="_blank">0-19-920613-9</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Dodge, Y. (2006). The Oxford Dictionary of Statistical Terms. OUP. ISBN 0-19-920613-9. <a href="0-19-920613-9" target="_blank">0-19-920613-9</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Bahlmann, C., (2006), Directional features in online handwriting recognition, Pattern Recognition, 39 <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.330.9384&rep=rep1&type=pdf" target="_blank">http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.330.9384&rep=rep1&type=pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Fisher 1993. - Fisher, N. I. (1993). Statistical Analysis of Circular Data. Cambridge University Press. ISBN 0-521-35018-2. <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>