Directional statistics

Directional statistics (also circular statistics or spherical statistics) is the subdiscipline of statistics that deals with directions (unit vectors in Rⁿ), axes (lines through the origin in Rⁿ) or rotations in Rⁿ. More generally, directional statistics deals with observations on compact Riemannian manifolds.

The fact that 0 degrees and 360 degrees are identical angles, so that for example 180 degrees is not a sensible mean of 2 degrees and 358 degrees, provides one illustration that special statistical methods are required for the analysis of some types of data (in this case, angular data). Other examples of data that may be regarded as directional include statistics involving temporal periods (e.g. time of day, week, month, year, etc.), compass directions, dihedral angles in molecules, orientations, rotations and so on.

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

is

This concept can be extended to the multivariate context by an extension of the simple sum to a number of ${\displaystyle F}$ sums that cover all dimensions in the feature space:

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