Directional Statistics - The Fundamental Difference Between Linear and Circular Statistics

The Fundamental Difference Between Linear and Circular Statistics

A simple way to calculate the mean of a series of angles (in the interval [0°, 360°)) is to calculate the mean of the cosines and sines of each angle, and obtain the angle by calculating the inverse tangent. Consider the following three angles as an example: 10, 20, and 30 degrees. Intuitively, calculating the mean would involve adding these three angles together and dividing by 3, in this case indeed resulting in a correct mean angle of 20 degrees. By rotating this system anticlockwise through 15 degrees the three angles become 355 degrees, 5 degrees and 15 degrees. The naive mean is now 125 degrees, which is the wrong answer, as it should be 5 degrees. The vector mean can be calculated in the following way, using the mean sine and the mean cosine :


\bar s = \frac{1}{3} \left( \sin (355^\circ) + \sin (5^\circ) + \sin (15^\circ) \right)
= \frac{1}{3} \left( -0.087 + 0.087 + 0.259 \right)
\approx 0.086

\bar c = \frac{1}{3} \left( \cos (355^\circ) + \cos (5^\circ) + \cos (15^\circ) \right)
= \frac{1}{3} \left( 0.996 + 0.996 + 0.966 \right)
\approx 0.986

\bar \theta =
\left.
\begin{cases}
\arctan \left( \frac{\bar s}{ \bar c} \right) & \bar s > 0 ,\ \bar c > 0 \\ \arctan \left( \frac{\bar s}{ \bar c} \right) + 180^\circ & \bar c < 0 \\
\arctan \left (\frac{\bar s}{\bar c}
\right)+360^\circ & \bar s <0 ,\ \bar c >0
\end{cases}
\right\}
= \arctan \left( \frac{0.086}{0.986} \right)
= \arctan (0.087) = 5^\circ.

This may be more succinctly stated by realizing that directional data are in fact vectors of unit length. In the case of one-dimensional data, these data points can be represented conveniently as complex numbers of unit magnitude, where is the measured angle. The mean resultant vector for the sample is then:


\overline{\mathbf{\rho}}=\frac{1}{N}\sum_{n=1}^N z_n.

The sample mean angle is then the argument of the mean resultant:


\overline{\theta}=\mathrm{Arg}(\overline{\mathbf{\rho}}).

The length of the sample mean resultant vector is:


\overline{R}=|\overline{\mathbf{\rho}}|

and will have a value between 0 and 1. Thus the sample mean resultant vector can be represented as:


\overline{\mathbf{\rho}}=\overline{R}\,e^{i\overline{\theta}}.

Read more about this topic:  Directional Statistics

Famous quotes containing the words fundamental, difference, circular and/or statistics:

    Each [side in this war] looked for an easier triumph, and a result less fundamental and astounding. Both read the same Bible, and pray to the same God; and each invokes His aid against the other. It may seem strange that any men should dare to ask a just God’s assistance in wringing their bread from the sweat of other men’s faces; but let us judge not that we be not judged.
    Abraham Lincoln (1809–1865)

    I say that male and female are cast in the same mold; except for education and habits, the difference is not great.
    Michel de Montaigne (1533–1592)

    Whoso desireth to know what will be hereafter, let him think of what is past, for the world hath ever been in a circular revolution; whatsoever is now, was heretofore; and things past or present, are no other than such as shall be again: Redit orbis in orbem.
    Sir Walter Raleigh (1552–1618)

    and Olaf, too

    preponderatingly because
    unless statistics lie he was
    more brave than me: more blond than you.
    —E.E. (Edward Estlin)