Wrapped Lévy distribution
In probability theory and directional statistics, a wrapped Lévy distribution is a wrapped probability distribution that results from the "wrapping" of the Lévy distribution around the unit circle.
Description
The pdf of the wrapped Lévy distribution is
- [math]\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}} }[/math]
where the value of the summand is taken to be zero when [math]\displaystyle{ \theta+2\pi n-\mu \le 0 }[/math], [math]\displaystyle{ c }[/math] is the scale factor and [math]\displaystyle{ \mu }[/math] is the location parameter. Expressing the above pdf in terms of the characteristic function of the Lévy distribution yields:
- [math]\displaystyle{ f_{WL}(\theta;\mu,c)=\frac{1}{2\pi}\sum_{n=-\infty}^\infty e^{-in(\theta-\mu)-\sqrt{c|n|}\,(1-i\sgn{n})}=\frac{1}{2\pi}\left(1 + 2\sum_{n=1}^\infty e^{-\sqrt{cn}}\cos\left(n(\theta-\mu) - \sqrt{cn}\,\right)\right) }[/math]
In terms of the circular variable [math]\displaystyle{ z=e^{i\theta} }[/math] the circular moments of the wrapped Lévy distribution are the characteristic function of the Lévy distribution evaluated at integer arguments:
- [math]\displaystyle{ \langle z^n\rangle=\int_\Gamma e^{in\theta}\,f_{WL}(\theta;\mu,c)\,d\theta = e^{i n \mu-\sqrt{c|n|}\,(1-i\sgn(n))}. }[/math]
where [math]\displaystyle{ \Gamma\, }[/math] is some interval of length [math]\displaystyle{ 2\pi }[/math]. The first moment is then the expectation value of z, also known as the mean resultant, or mean resultant vector:
- [math]\displaystyle{ \langle z \rangle=e^{i\mu-\sqrt{c}(1-i)} }[/math]
The mean angle is
- [math]\displaystyle{ \theta_\mu=\mathrm{Arg}\langle z \rangle = \mu+\sqrt{c} }[/math]
and the length of the mean resultant is
- [math]\displaystyle{ R=|\langle z \rangle| = e^{-\sqrt{c}} }[/math]
See also
References
- Fisher, N. I. (1996). Statistical Analysis of Circular Data. Cambridge University Press. ISBN 978-0-521-56890-6. https://books.google.com/books?id=IIpeevaNH88C&q=%22circular+variance%22+fisher. Retrieved 2010-02-09.
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