Wrapped Exponential Distribution

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

:$f_(\theta;\lambda)=\sum_^\infty \lambda e^=\frac ,$

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:

:$\varphi_n(\lambda)=\frac$

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
f_(z;\lambda)
& =\frac\sum_^\infty \frac\\ 0pt& = \begin
\frac\,\textrm(\Phi(z,1,-i\lambda))-\frac
& \textz \neq 1
\\ 2pt \frac
& \textz=1
\end
\end

where $\Phi()$ is the Lerch transcendent function.

Circular moments

In terms of the circular variable $z=e^$ the circular moments of the wrapped exponential distribution are the characteristic function of the exponential distribution evaluated at integer arguments:

:$\langle z^n\rangle=\int_\Gamma e^\,f_(\theta;\lambda)\,d\theta = \frac ,$

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:

:$\langle z \rangle=\frac .$

The mean angle is

:$\langle \theta \rangle=\mathrm\langle z \rangle = \arctan(1/\lambda) ,$

and the length of the mean resultant is

:$R=, \langle z \rangle, = \frac .$

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 $\operatorname(\theta)$.

See also

* Wrapped distribution
* Directional statistics

References

{{ProbDistributions, directional

Continuous distributions
Directional statistics