Information theory and coordinates

The previous post explains how the Maidenhead location system works. The first character in a location code specifies the longitude in 20 degree increments and the second character specifies the latitude in 10 degree increments. Both are a letter from A through R that breaks the possible range down into 18 parts. (The longitude range is wider because longitude ranges over 360 degrees but latitude ranges over 180 degrees.)

If you divide a range into 16 equally probable parts, you get four bits of information because 24 = 16. Since we’re dividing longitude and latitude into 18 parts, we get around 4 bits of information from each. But the longitude component carries more information than the latitude component. This post will explain why.

Defining information

The Shannon entropy of a random variable with N possible values, each with probability pi, is defined to be

-\sum_{i=1}^N p_i \log_b \, p_i

where the base b is usually 2. Occasionally b might be another base, such as 2 or 10. (More on that here.) The statement above about 16 equal parts giving 4 bits of information assumes b = 2.

The expected information gain from observing the value of a random variable is measured by the random variable’s Shannon entropy.

Even vs uneven allocation

Suppose we pick a point at random on a sphere. The first component of the point’s Maidenhead location narrows the location of the point to one of 18 equally probable sectors. So each of the ps in the equation above is 1/18. That means the information content in the first component is log2(1/18) = 4.17 bits.

The second component of the location also divides the sphere into 18 parts, but not evenly. A 10 degree band of latitude near the equator covers more area than a 10 degree band of latitude near the polls.

Shannon entropy is maximized when all the ps are equal. Said another way, the amount of surprise from observing a random variable is greatest when all possibilities are equally likely.

Knowing that a random point on a sphere is within 10 degrees of the equator is not as surprising as knowing that it is within 10 degrees of the North Pole because there’s less land within 10 degrees of the North Pole.

Archimedes and Shannon

To calculate the information content in the latitude band, we need to know the area of each band.

Archimedes (c. 287 – 212 BC) discovered that if you have a sphere sitting inside a cylinder, the area of a band on the sphere is proportional to the area of its projection onto the cylinder. That means the area of a band of latitude is proportional to the difference in the z coordinates of the band.

The 18 bands of latitude run from -90° to 90° in increments of 10°. If we number the bands starting with 1 at the South Pole, the probability of landing in the nth band is

(sin(−90° + 10n°) − sin(-90° + 10(n − 1)°)/2

and you can calculate from this that the Shannon entropy is 3.97.

So the first component of the location, the longitude, carries 4.17 bits of information, and the second component, the latitude, carries 3.97 bits.