Probability 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[ 1]
f
W
E
(
θ θ -->
;
λ λ -->
)
=
∑ ∑ -->
k
=
0
∞ ∞ -->
λ λ -->
e
− − -->
λ λ -->
(
θ θ -->
+
2
π π -->
k
)
=
λ λ -->
e
− − -->
λ λ -->
θ θ -->
1
− − -->
e
− − -->
2
π π -->
λ λ -->
,
{\displaystyle 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
≤ ≤ -->
θ θ -->
<
2
π π -->
{\displaystyle 0\leq \theta <2\pi }
λ λ -->
>
0
{\displaystyle \lambda >0}
truncated distribution obtained by restricting observed values X from the exponential distribution with rate parameter λ to the range
0
≤ ≤ -->
X
<
2
π π -->
{\displaystyle 0\leq 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:
φ φ -->
n
(
λ λ -->
)
=
1
1
− − -->
i
n
/
λ λ -->
{\displaystyle \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:
f
W
E
(
z
;
λ λ -->
)
=
1
2
π π -->
∑ ∑ -->
n
=
− − -->
∞ ∞ -->
∞ ∞ -->
z
− − -->
n
1
− − -->
i
n
/
λ λ -->
=
{
λ λ -->
π π -->
Im
(
Φ Φ -->
(
z
,
1
,
− − -->
i
λ λ -->
)
)
− − -->
1
2
π π -->
if
z
≠ ≠ -->
1
λ λ -->
1
− − -->
e
− − -->
2
π π -->
λ λ -->
if
z
=
1
{\displaystyle {\begin{aligned}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{aligned}}}
where
Φ Φ -->
(
)
{\displaystyle \Phi ()}
Lerch transcendent function.
Circular moments
In terms of the circular variable
z
=
e
i
θ θ -->
{\displaystyle z=e^{i\theta }}
⟨ ⟨ -->
z
n
⟩ ⟩ -->
=
∫ ∫ -->
Γ Γ -->
e
i
n
θ θ -->
f
W
E
(
θ θ -->
;
λ λ -->
)
d
θ θ -->
=
1
1
− − -->
i
n
/
λ λ -->
,
{\displaystyle \langle z^{n}\rangle =\int _{\Gamma }e^{in\theta }\,f_{WE}(\theta ;\lambda )\,d\theta ={\frac {1}{1-in/\lambda }},}
where
Γ Γ -->
{\displaystyle \Gamma \,}
2
π π -->
{\displaystyle 2\pi }
z , also known as the mean resultant, or mean resultant vector:
⟨ ⟨ -->
z
⟩ ⟩ -->
=
1
1
− − -->
i
/
λ λ -->
.
{\displaystyle \langle z\rangle ={\frac {1}{1-i/\lambda }}.}
The mean angle is
⟨ ⟨ -->
θ θ -->
⟩ ⟩ -->
=
A
r
g
⟨ ⟨ -->
z
⟩ ⟩ -->
=
arctan
-->
(
1
/
λ λ -->
)
,
{\displaystyle \langle \theta \rangle =\mathrm {Arg} \langle z\rangle =\arctan(1/\lambda ),}
and the length of the mean resultant is
R
=
|
⟨ ⟨ -->
z
⟩ ⟩ -->
|
=
λ λ -->
1
+
λ λ -->
2
.
{\displaystyle 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
≤ ≤ -->
θ θ -->
<
2
π π -->
{\displaystyle 0\leq \theta <2\pi }
E
-->
(
θ θ -->
)
{\displaystyle \operatorname {E} (\theta )}
[ 1]
See also
References
Discrete
with finite with infinite
Continuous
supported on a supported on a supported with support
Mixed
Multivariate Directional Degenerate singular Families
![{\displaystyle {\begin{aligned}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{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95e9059afd3d0d8d75dd6cb3cbe6e0acf8cb11b5)
