A Distribution on a Cylinder

Here's a utility pole at a traffic intersection in Aspen Hill, Maryland. The pole has served as display for the many yard sales, community meetings, and businesses that have had their advertising flyers posted on the pole. The flyers have long since been removed. Only their staples and nails remain. These accumulated staples show a distribution of the heights of flyer postings. 

The close-up view below shows the distribution of individual staples and nails on the cylinder of the pole. The staples are distributed both around the pole and vertically up and down the pole. Vertically, it's too difficult to put flyers high on the pole and few staples can be found there. Flyers very low on the pole wouldn't be easily seen by those passing by, so few staples are also found there. Most staples and nails are at a comfortable shoulder and viewing height. If we imagine the height of a staple above the ground is our random variable, we find few staples with small height, few with large height, and many more with a medium height. This is a bell-shaped pattern up and down the pole that we have seen often.

Horizontally, the staples are distributed circularly around the pole. They would also have greatest frequency towards the traffic and lesser frequency on the backside of the pole. This is likely also a bell-shaped distribution, wrapped around a circle. We have seen such a distribution before connected with the characteristic function of a random variable. We've also seen a distribution on a pole at the Rodin museum in Paris.

Wearing of the Sphere

This is a kids' playground item at Wheaton Regional Park in Maryland. It's a toy/chair/apparatus that is basically a painted sphere on a pedestal. Children sit, slip off, or perhaps spin around, balancing on top of the sphere. This produces the wear pattern contours shown here. The central circular region has the most paint rubbed off, down to the metal interior. Around this is a ring of lesser wear. The circular nature of the contours indicates that the children don't seem favor one approach to sit, one method of sitting, or one direction of dismounting the sphere. No one direction of wear seems overly favored, or avoided, even though the light ring of lesser wear may not have exactly equal width around the contour. What remains are the circular contours of the frequency of use of a distribution defined on a sphere.

Pine Needles in a Circle

A view of the cover of a 'sanitary sewer,' (and why would you want any other kind?!). And as promised last week, notice the accumulation of pine needles around the edge of the cover. On this day the wind was quite gusty, blowing along this sidewalk from the bottom of the image to the top. Due to this wind direction, many more pine needles have accumulated and piled up around the cover at the bottom than at the top. This forms a histogram of the frequency distribution of wind action distributed around the circumference of the sewer cover.

If we had data situated around the circumference of a circle we could display it as a circular dot plot as shown below from the book "Circular Statistics" by Fisher. These are arrival times, on a 24hr clock, are for 254 patients at an intensive care unit. Few arrive in the morning, many more arrive in late afternoon and early evening.

An estimate of the density of circular sample can be computed using something like the code for a circular density curve in the programming language R, as shown below.

 

Such graphical tools are the beginnings of modeling on spheres and other manifolds studied under the general heading of directional statistics.

A probability density function defined on the real line is sometimes wrapped around a circle. We have earlier seen that the results of such wrapping give rise to the characteristic function, a fundamental tool of probability modeling.

Wine consumption for various countries proportional to the area of each circle. Strangemaps.wordpress.com has an interesting discussion of trying to make sense of the numbers in this map that has no legend to tell you what the numbers mean. Why is Luxembourg so big?