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:

    “I believe that the fundamental proposition is that we must recognize that the hostilities in Europe, in Africa, and in Asia are all parts of a single world conflict. We must, consequently, recognize that our interests are menaced both in Europe and in the Far East.”
    —Franklin D. Roosevelt (1882–1945)

    “Belief is with them mechanical, voluntary: they believe what they are paid for—they swear to that which turns to account. Do you suppose, that after years spent in this manner, they have any feeling left answering to the difference between truth and falsehood?”
    —William Hazlitt (1778–1830)

    “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)

    “We ask for no statistics of the killed,
    For nothing political impinges on
    This single casualty, or all those gone,
    Missing or healing, sinking or dispersed,
    Hundreds of thousands counted, millions lost.”
    —Karl Shapiro (b. 1913)