Wrapped exponential distribution explained

In probability theory and directional statistics, a wrapped exponential distribution is a wrapped probability distribution that results from the "wrapping" of the exponential distribution around the unit circle.

Definition

The probability density function of the wrapped exponential distribution is^[1]

f _{WE} (\theta; λ ) = \sum ^{inf ty} _{k =0} λ e ^{-λ} = (λ e ^{-λ})/(1- e ^{-2\pi}) ,

for

0 \le \theta < 2 \pi where λ > 0 is the rate parameter of the unwrapped distribution. This is identical to the truncated distribution obtained by restricting observed values X from the exponential distribution with rate parameter &lambda; to the range 0 \le X < 2 \pi . Note that this distribution is not periodic.

Characteristic function

The characteristic function of the wrapped exponential is just the characteristic function of the exponential function evaluated at integer arguments:

\varphi _{n( λ ) = (1)/(1- in/ λ)}

which yields an alternate expression for the wrapped exponential PDF in terms of the circular variable z=e ^{i (θ-m)} valid for all real θ and m:

\begin{align} f _{WE} (z; λ ) & = (1)/(2 \pi) \sum ^{inf ty} _{n =-inf ty} (z ^{-n})/(1- in/ λ) \\[ 10 pt] & = \begin{cases} (λ)/(\pi) rm{Im}(\Phi(z,1,-iλ))- (1)/(2\pi) & if z ≠ 1 \\[ 12 pt] (λ)/(1- e ^{-2\piλ}) & if z =1 \end{cases} \end{align}

where

\Phi is the Lerch transcendent function.

Circular moments

In terms of the circular variable

z = e ^{i\theta} the circular moments of the wrapped exponential distribution are the characteristic function of the exponential distribution evaluated at integer arguments:\langle z ^{n\rangle = \int} _{\Gamma} e ^{in\theta} f _{WE} (\theta; λ )d\theta = (1)/(1- in/ λ) ,

where

\Gamma is some interval of length 2 \pi . The first moment is then the average value of z, also known as the mean resultant, or mean resultant vector:\langle z \rangle = (1)/(1- i/ λ) .

The mean angle is

\langle \theta \rangle = Arg \langle z \rangle = \arctan( 1 / λ ) ,

and the length of the mean resultant is

R = |\langle z \rangle| = (λ)/(\sqrt{ 1+λ ^{2}) } .

and the variance is then 1-R.

Characterisation

The wrapped exponential distribution is the maximum entropy probability distribution for distributions restricted to the range

0 \le \theta < 2 \pi for a fixed value of the expectation \operatorname{E}(\theta) .^[1]

See also

• Wrapped distribution
• Directional statistics

References

1. Jammalamadaka . S. Rao . Kozubowski . Tomasz J. . 2004 . New Families of Wrapped Distributions for Modeling Skew Circular Data . Communications in Statistics - Theory and Methods . 33 . 9 . 2059–2074 . 10.1081/STA-200026570. 2011-06-13 .