Mean of circular quantities

In mathematics, a mean of circular quantities is a mean which is suited for quantities like angles, daytimes, and fractional parts of real numbers. This is necessary since most of the usual means fail on circular quantities. For example, the arithmetic mean of 0° and 360° is 180°, although 0° would be clearly the better choice. ^[1] This is one of the simplest examples of statistics of non-Euclidean spaces.

Mean of angles

Since the arithmetic mean is not effective for angles, the following method can be used to obtain both a mean value and measure for the variance of the angles:

Convert all angles to corresponding points on the unit circle, e.g., α to (cos α,sin α). That is convert polar coordinates to Cartesian coordinates. Then compute the arithmetic mean of these points. The resulting point will lie on the unit disk. Convert that point back to polar coordinates. The angle is a reasonable mean of the input angles. The resulting radius will be 1 if all angles are equal. If the angles are uniformly distributed on the circle, then the resulting radius will be 0, and there is no circular mean. In other words, the radius measures the concentration of the angles.

Given the angles $\alpha_1,\dots,\alpha_n$ the mean is computed by

$\bar{\alpha} = \operatorname{atan2}\left(\frac{1}{n}\cdot\sum_{j=1}^n \sin\alpha_j, \frac{1}{n}\cdot\sum_{j=1}^n \cos\alpha_j\right)$

using the atan2 variant of the arctangent function, or

$\bar{\alpha} = \arg\left(\frac{1}{n}\cdot\sum_{j=1}^n \exp(i\cdot\alpha_j)\right)$

using complex numbers.

Properties

The mean $\bar{\alpha}$

• maximizes the likelihood of the mean parameter of the von Mises distribution and
• minimizes the sum of a certain distance on the circle, more precisely

$\bar{\alpha} = \underset{\beta}{\operatorname{argmin}} \sum_{j=1}^n d(\alpha_j,\beta)$, where d(φ,β) = 1 − cos(φ − β).

The distance d(φ,β) is equal to half the squared Euclidean distance between the two points on the unit circle associated with φ and β.

See also

• Directional statistics
• Circular distribution

References

1. ^ Christopher M. Bishop: Pattern Recognition and Machine Learning (Information Science and Statistics), ISBN 0387310738