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:

    “POLITICIAN, n. An eel in the fundamental mud upon which the superstructure of organized society is reared. When he wriggles he mistakes the agitation of his tail for the trembling of the edifice. As compared with the statesman, he suffers the disadvantage of being alive.”
    —Ambrose Bierce (1842–1914?)

    “I see less difference between a city and a swamp than formerly.”
    —Henry David Thoreau (1817–1862)

    “Loving a baby is a circular business, a kind of feedback loop. The more you give the more you get and the more you get the more you feel like giving.”
    —Penelope Leach (20th century)

    “July 4. Statistics show that we lose more fools on this day than in all the other days of the year put together. This proves, by the number left in stock, that one Fourth of July per year is now inadequate, the country has grown so.”
    —Mark Twain [Samuel Langhorne Clemens] (1835–1910)