Wrapped exponential distribution

Wrapped Exponential

Probability density function The support is chosen to be [0,2π]
Cumulative distribution function The support is chosen to be [0,2π]
Parameters	$\lambda>0$
Support	$0\le\theta<2\pi$
PDF	$\frac{\lambda e^{-\lambda \theta}}{1-e^{-2\pi \lambda}}$
CDF	$\frac{1-e^{-\lambda \theta}}{1-e^{-2\pi \lambda}}$
Mean	$\arctan(1/\lambda)$ (circular)
Variance	$1-\frac{\lambda}{\sqrt{1+\lambda^2}}$ (circular)
Entropy	$1+\ln\left(\frac{\beta-1}{\lambda}\right)-\frac{\beta}{\beta-1}\ln(\beta)$ where $\beta=e^{2\pi\lambda}$ (differential)
CF	$\frac{1}{1-in/\lambda}$

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

$f_{WE}(\theta;\lambda)=\sum_{k=0}^\infty \lambda e^{-\lambda (\theta+2 \pi k)}=\frac{\lambda e^{-\lambda \theta}}{1-e^{-2\pi \lambda}} ,$

for $0 \le \theta < 2\pi$ where $\lambda > 0$ 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 λ to the range $0\le X < 2\pi$.

Characteristic function

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

$\varphi_n(\lambda)=\frac{1}{1-in/\lambda}$

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:

$\begin{align} f_{WE}(z;\lambda) & =\frac{1}{2\pi}\sum_{n=-\infty}^\infty \frac{z^{-n}}{1-in/\lambda}\\[10pt] & = \begin{cases} \frac{\lambda}{\pi}\,\textrm{Im}(\Phi(z,1,-i\lambda))-\frac{1}{2\pi} & \text{if }z \neq 1 \\[12pt] \frac{\lambda}{1-e^{-2\pi\lambda}} & \text{if }z=1 \end{cases} \end{align}$

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:

$\langle z^n\rangle=\int_\Gamma e^{in\theta}\,f_{WE}(\theta;\lambda)\,d\theta = \frac{1}{1-in/\lambda} ,$

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:

$\langle z \rangle=\frac{1}{1-i/\lambda} .$

The mean angle is

$\langle \theta \rangle=\mathrm{Arg}\langle z \rangle = \arctan(1/\lambda) ,$

and the length of the mean resultant is

$R=|\langle z \rangle| = \frac{\lambda}{\sqrt{1+\lambda^2}} .$

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 $\operatorname{E}(\theta)$.

See also

• Wrapped distribution
• Directional statistics