Wrapped distribution

In probability theory and directional statistics, a wrapped probability distribution is a continuous probability distribution that describes data points that lie on a unit n-sphere. In one dimension, a wrapped distribution will consist of points on the unit circle.

Any probability density function $p(\phi )$ on the line can be "wrapped" around the circumference of a circle of unit radius. That is, the pdf of the wrapped variable

$\theta =\phi \mod 2\pi$ in some interval of length $2\pi$

is

$p_{w}(\theta )=\sum _{k=-\infty }^{\infty }{p(\theta +2\pi k)}.$

Theory

In most situations, a process involving circular statistics produces angles ($\phi$) which lie in the interval from negative infinity to positive infinity, and are described by an "unwrapped" probability density function $p(\phi )$. However, a measurement will yield a "measured" angle $\theta$ which lies in some interval of length $2\pi$ (for example $[0,2\pi )$). In other words, a measurement cannot tell if the "true" angle $\phi$ has been measured or whether a "wrapped" angle $\phi +2\pi a$ has been measured where a is some unknown integer. That is:

$\theta =\phi +2\pi a.$

If we wish to calculate the expected value of some function of the measured angle it will be:

$\langle f(\theta )\rangle =\int _{-\infty }^{\infty }p(\phi )f(\phi +2\pi a)d\phi .$

We can express the integral as a sum of integrals over periods of $2\pi$ (e.g. 0 to $2\pi$):

$\langle f(\theta )\rangle =\sum _{k=-\infty }^{\infty }\int _{2\pi k}^{2\pi (k+1)}p(\phi )f(\phi +2\pi a)d\phi .$

Changing the variable of integration to $\theta '=\phi -2\pi k$ and exchanging the order of integration and summation, we have

$\langle f(\theta )\rangle =\int _{0}^{2\pi }p_{w}(\theta ')f(\theta '+2\pi a')d\theta '$

where $p_{w}(\theta ')$ is the pdf of the "wrapped" distribution and a' is another unknown integer (a'=a+k). It can be seen that the unknown integer a' introduces an ambiguity into the expectation value of $f(\theta )$. A particular instance of this problem is encountered when attempting to take the mean of a set of measured angles. If, instead of the measured angles, we introduce the parameter $z=e^{i\theta }$ it is seen that z has an unambiguous relationship to the "true" angle $\phi$ since:

$z=e^{i\theta }=e^{i\phi }.$

Calculating the expectation value of a function of z will yield unambiguous answers:

$\langle f(z)\rangle =\int _{0}^{2\pi }p_{w}(\theta ')f(e^{i\theta '})d\theta '$

and it is for this reason that the z parameter is the preferred statistical variable to use in circular statistical analysis rather than the measured angles $\theta$. This suggests, and it is shown below, that the wrapped distribution function may itself be expressed as a function of z so that:

$\langle f(z)\rangle =\oint p_{w}(z)f(z)\,dz$

where $p_{w}(z)$ is defined such that $p_{w}(\theta )\,d\theta =p_{w}(z)\,dz$. This concept can be extended to the multivariate context by an extension of the simple sum to a number of $F$ sums that cover all dimensions in the feature space:

$p_{w}({\vec {\theta }})=\sum _{k_{1}=-\infty }^{\infty }{p({\vec {\theta }}+2\pi k_{1}\mathbf {e} _{1}+\dots +2\pi k_{F}\mathbf {e} _{F})}$

where $\mathbf {e} _{k}=(0,\dots ,0,1,0,\dots ,0)^{\mathsf {T}}$ is the $k$th Euclidean basis vector.

Expression in terms of characteristic functions

A fundamental wrapped distribution is the Dirac comb which is a wrapped delta function:

$\Delta _{2\pi }(\theta )=\sum _{k=-\infty }^{\infty }{\delta (\theta +2\pi k)}.$

Using the delta function, a general wrapped distribution can be written

$p_{w}(\theta )=\sum _{k=-\infty }^{\infty }\int _{-\infty }^{\infty }p(\theta ')\delta (\theta -\theta '+2\pi k)\,d\theta '.$

Exchanging the order of summation and integration, any wrapped distribution can be written as the convolution of the "unwrapped" distribution and a Dirac comb:

$p_{w}(\theta )=\int _{-\infty }^{\infty }p(\theta ')\Delta _{2\pi }(\theta -\theta ')\,d\theta '.$

The Dirac comb may also be expressed as a sum of exponentials, so we may write:

$p_{w}(\theta )={\frac {1}{2\pi }}\,\int _{-\infty }^{\infty }p(\theta ')\sum _{n=-\infty }^{\infty }e^{in(\theta -\theta ')}\,d\theta '$

again exchanging the order of summation and integration,

$p_{w}(\theta )={\frac {1}{2\pi }}\,\sum _{n=-\infty }^{\infty }\int _{-\infty }^{\infty }p(\theta ')e^{in(\theta -\theta ')}\,d\theta '$

using the definition of $\phi (s)$, the characteristic function of $p(\theta )$ yields a Laurent series about zero for the wrapped distribution in terms of the characteristic function of the unwrapped distribution:

$p_{w}(\theta )={\frac {1}{2\pi }}\,\sum _{n=-\infty }^{\infty }\phi (-n)\,e^{in\theta }={\frac {1}{2\pi }}\,\sum _{n=-\infty }^{\infty }\phi (-n)\,z^{n}$

or

$p_{w}(z)={\frac {1}{2\pi i}}\,\sum _{n=-\infty }^{\infty }\phi (-n)\,z^{n-1}.$

By analogy with linear distributions, the $\phi (m)$ are referred to as the characteristic function of the wrapped distribution (or perhaps more accurately, the characteristic sequence).

Moments

The moments of the wrapped distribution $p_{w}(z)$ are defined as:

$\langle z^{m}\rangle =\oint p_{w}(z)z^{m}\,dz.$

Expressing $p_{w}(z)$ in terms of the characteristic function and exchanging the order of integration and summation yields:

$\langle z^{m}\rangle ={\frac {1}{2\pi i}}\sum _{n=-\infty }^{\infty }\phi (-n)\oint z^{m+n-1}\,dz.$

From the theory of residues we have

$\oint z^{m+n-1}\,dz=2\pi i\delta _{m+n}$

where $\delta _{k}$ is the Kronecker delta function. It follows that the moments are simply equal to the characteristic function of the unwrapped distribution for integer arguments:

$\langle z^{m}\rangle =\phi (m).$

See also

• Wrapped normal distribution
• Wrapped Cauchy distribution

References

• Borradaile, Graham (2003). Statistics of Earth Science Data. Springer. ISBN 978-3540436034. Retrieved 31 December 2009.
• Fisher, N. I. (1996). Statistical Analysis of Circular Data. Cambridge University Press. ISBN 9780521568906. Retrieved 9 February 2010.