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:

    “In this choice of inheritance we have given to our frame of polity the image of a relation in blood; binding up the constitution of our country with our dearest domestic ties; adopting our fundamental laws into the bosom of our family affections; keeping inseparable and cherishing with the warmth of all their combined and mutually reflected charities, our state, our hearths, our sepulchres, and our altars.”
    —Edmund Burke (1729–1797)

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

    “If one doubts whether Grecian valor and patriotism are not a fiction of the poets, he may go to Athens and see still upon the walls of the temple of Minerva the circular marks made by the shields taken from the enemy in the Persian war, which were suspended there. We have not far to seek for living and unquestionable evidence. The very dust takes shape and confirms some story which we had read.”
    —Henry David Thoreau (1817–1862)

    “He uses statistics as a drunken man uses lamp-posts—for support rather than illumination.”
    —Andrew Lang (1844–1912)