Directional Statistics - Examples of Circular Distributions

Examples of Circular Distributions

  • The von Mises distribution is a circular distribution which, like any other circular distribution, may be thought of as a wrapping of a certain linear probability distribution around the circle. The underlying linear probability distribution for the von Mises distribution is mathematically intractable, however, for statistical purposes, there is no need to deal with the underlying linear distribution. The usefulness of the von Mises distribution is twofold: it is the most mathematically tractable of all circular distributions, allowing simpler statistical analysis, and it is a close approximation to the wrapped normal distribution, which, analogously the linear normal distribution, is important because it is the limiting case for the sum of a large number of small angular deviations. In fact, the von Mises distribution is often known as the "circular normal" distribution because of its ease of use and its close relationship to the wrapped normal distribution (Fisher, 1993).
The pdf of the von Mises distribution is:
where is the modified Bessel function of order 0.
  • The pdf of the circular uniform distribution is given by
  • The pdf of the wrapped normal distribution (WN) is:

WN(\theta;\mu,\sigma)=\frac{1}{\sigma \sqrt{2\pi}} \sum^{\infty}_{k=-\infty} \exp \left=\frac{1}{2\pi}\zeta\left(\frac{\theta-\mu}{2\pi},\frac{i\sigma^2}{2\pi}\right)
where μ and σ are the mean and standard deviation of the unwrapped distribution, respectively and is the Jacobi theta function:

\zeta(\theta,\tau)=\sum_{n=-\infty}^\infty (w^2)^n q^{n^2}
where and
  • The pdf of the wrapped Cauchy distribution (WC) is:

WC(\theta;\theta_0,\gamma)=\sum_{n=-\infty}^\infty \frac{\gamma}{\pi(\gamma^2+(\theta+2\pi n-\theta_0)^2)}
=\frac{1}{2\pi}\,\,\frac{\sinh\gamma}{\cosh\gamma-\cos(\theta-\theta_0)}
where is the scale factor and is the peak position.
  • The pdf of the Wrapped Lévy distribution (WL) is:

f_{WL}(\theta;\mu,c)=\sum_{n=-\infty}^\infty \sqrt{\frac{c}{2\pi}}\,\frac{e^{-c/2(\theta+2\pi n-\mu)}}{(\theta+2\pi n-\mu)^{3/2}}
where the value of the summand is taken to be zero when, is the scale factor and is the location parameter.

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