In probability theory and directional statistics, a wrapped Cauchy distribution is a wrapped probability distribution that results from the "wrapping" of the Cauchy distribution around the unit circle. The Cauchy distribution is sometimes known as a Lorentzian distribution, and the wrapped Cauchy distribution may sometimes be referred to as a wrapped Lorentzian distribution. The wrapped Cauchy distribution is often found in the field of spectroscopy where it is used to analyze diffraction patterns (e.g. see Fabry–Pérot interferometer).
Description The probability density function of the wrapped Cauchy distribution is:
f WC ( θ ; μ , γ ) = ∑ n = − ∞ ∞ γ π ( γ 2 + ( θ − μ + 2 π n ) 2 ) − π < θ < π {\displaystyle f_{\text{WC}}(\theta ;\mu ,\gamma )=\sum _{n=-\infty }^{\infty }{\frac {\gamma }{\pi (\gamma ^{2}+(\theta -\mu +2\pi n)^{2})}}\qquad -\pi <\theta <\pi }
where γ {\displaystyle \gamma } is the scale factor and μ {\displaystyle \mu } is the peak position of the "unwrapped" distribution. Expressing the above pdf in terms of the characteristic function of the Cauchy distribution yields:
f WC ( θ ; μ , γ ) = 1 2 π ∑ n = − ∞ ∞ e i n ( θ − μ ) − | n | γ = 1 2 π sinh γ cosh γ − cos ( θ − μ ) {\displaystyle f_{\text{WC}}(\theta ;\mu ,\gamma )={\frac {1}{2\pi }}\sum _{n=-\infty }^{\infty }e^{in(\theta -\mu )-|n|\gamma }={\frac {1}{2\pi }}\,\,{\frac {\sinh \gamma }{\cosh \gamma -\cos(\theta -\mu )}}}
The PDF may also be expressed in terms of the circular variable z = eiθ and the complex parameter ζ = ei(μ+iγ)
f WC ( z ; ζ ) = 1 2 π 1 − | ζ | 2 | z − ζ | 2 {\displaystyle f_{\text{WC}}(z;\zeta )={\frac {1}{2\pi }}\,\,{\frac {1-|\zeta |^{2}}{|z-\zeta |^{2}}}}
where, as shown below, ζ = ⟨z⟩. In terms of the circular variable z = e i θ {\displaystyle z=e^{i\theta }} the circular moments of the wrapped Cauchy distribution are the characteristic function of the Cauchy distribution evaluated at integer arguments:
⟨ z n ⟩ = ∫ Γ e i n θ f WC ( θ ; μ , γ ) d θ = e i n μ − | n | γ . {\displaystyle \langle z^{n}\rangle =\int _{\Gamma }e^{in\theta }\,f_{\text{WC}}(\theta ;\mu ,\gamma )\,d\theta =e^{in\mu -|n|\gamma }.}
where Γ {\displaystyle \Gamma \,} is some interval of length 2 π {\displaystyle 2\pi } . The first moment is then the average value of z, also known as the mean resultant, or mean resultant vector:
⟨ z ⟩ = e i μ − γ {\displaystyle \langle z\rangle =e^{i\mu -\gamma }}
The mean angle is
⟨ θ ⟩ = A r g ⟨ z ⟩ = μ {\displaystyle \langle \theta \rangle =\mathrm {Arg} \langle z\rangle =\mu }
and the length of the mean resultant is
… excerpt ends here. Continue reading the full article.
