Wrapped Lévy distribution explained

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

f _{WL} (\theta;\mu,c) = \sum ^{inf ty} _{n =-inf ty} \sqrt{ (c)/(2 \pi) }\,\frac

where the value of the summand is taken to be zero when

\theta +2 \pi n - \mu \le 0 , c is the scale factor and \mu is the location parameter. Expressing the above pdf in terms of the characteristic function of the Lévy distribution yields:f _{WL} (\theta;\mu,c) = (1)/(2 \pi) \sum ^{inf ty} _{n =-inf ty} e ^{-in(\theta-\mu)-\sqrt{c|n|} ( 1- i sgn {n})} = (1)/(2 \pi) \left( 1 + 2 \sum ^{inf ty} _{n =1} e ^{-\sqrt{cn} }\cos\left(n(\theta-\mu) - \sqrt\,\right)\right)

In terms of the circular variable

z = e ^{i\theta} the circular moments of the wrapped Lévy distribution are the characteristic function of the Lévy distribution evaluated at integer arguments:\langle z ^{n\rangle = \int} _{\Gamma} e ^{in\theta} f _{WL} (\theta;\mu,c)d\theta = e ^{i} ( 1- i sgn (n))}.

where

\Gamma is some interval of length 2 \pi . The first moment is then the expectation value of z, also known as the mean resultant, or mean resultant vector:\langle z \rangle = e ^{i\mu-\sqrt{c} ( 1- i)}

The mean angle is

\theta _{\mu=Arg\langle} z \rangle = \mu + \sqrt{c}

and the length of the mean resultant is

R = |\langle z \rangle| = e ^{-\sqrt{c} }

See also

• Wrapped distribution
• Directional statistics

References

• Book: Fisher, N. I. . Statistical Analysis of Circular Data . 1996 . Cambridge University Press . 978-0-521-56890-6 . 2010-02-09.