Editing Directional statistics (section)

==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]].