Circular distribution

In probability and statistics, a circular distribution or polar distribution is a probability distribution of a random variable whose values are angles, usually taken to be in the range [0, 2π). A circular distribution is often a continuous probability distribution, and hence has a probability density, but such distributions can also be discrete, in which case they are called circular lattice distributions. Circular distributions can be used even when the variables concerned are not explicitly angles: the main consideration is that there is not usually any real distinction between events occurring at the opposite ends of the range, and the division of the range could notionally be made at any point.

Graphical representation
If a circular distribution has a density


 * $$p(\phi)  \qquad  \qquad (0\le\phi<2\pi),\,$$

it can be graphically represented as a closed curve


 * $$[x(\phi),y(\phi)] = [r(\phi)\cos\phi, \, r(\phi)\sin\phi], \, $$

where the radius $$r(\phi)\,$$ is set equal to
 * $$r(\phi) = a+b p(\phi), \, $$

and where a and b are chosen on the basis of appearance.

Examples
By computing the probability distribution of angles along a handwritten ink trace, a lobe-shaped polar distribution emerges. The main direction of the lobe in the first quadrant corresponds to the slant of handwriting (see: graphonomics).

An example of a circular lattice distribution would be the probability of being born in a given month of the year, with each calendar month being thought of as arranged round a circle, so that "January" is next to "December".