Directional statistics

Definition

Let $p \ge 2$ and let $S^{p-1} = \set{x \in \R^p : \abs{x} = 1}$ be the unit sphere. Directional statistics is the statistics of data $x_1, \ldots, x_n$ with values in $S^{p-1}$ . More generally the data may lie in a compact manifold such as a rotation group. For $p = 2$ it is called circular statistics. Its basic statistics come from the resultant vector $R = x_1 + \cdots + x_n$ .

The mean resultant length is $\bar R = \abs{R}/n$ , a number in $[0, 1]$ .
If $R \ne 0$ , the mean direction is $R / \abs{R} \in S^{p-1}$ .
For $p = 2$ , the circular variance is $1 - \bar R$ .

Remarks

Example: the angles $350°$ and $10°$ give the unit vectors $(\cos 10°, -\sin 10°)$ and $(\cos 10°, \sin 10°)$ . Their sum is $R = (2 \cos 10°, 0)$ , so the mean direction is $0°$ and $\bar R = \cos 10° \approx 0.985$ . Non-example: for the angles $0°$ and $180°$ , $R = 0$ , so the mean direction is not defined; then $\bar R = 0$ and the circular variance is $1$ . The value $\bar R$ is close to $1$ when the directions are concentrated, and close to $0$ when they are spread out or balanced. The mean direction is the point $m \in S^{p-1}$ that maximises $\sum_i m^T x_i$ , so it does not depend on the choice of a zero angle.

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Statistic concept
n-sphere object
Directional statistics definition

Tag	`A6BW0KPXNC`
URI	`m2://A6BW0KPXNC`
Kind	definition
Writing	W0 stub
MSC	62H11
Defines	Directional statistics
Uses	n-sphere, Statistic

Directional statistics

Definition

Remarks

Reading path

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