The Index of Clumpiness, Part Three: One Dimension

In the last two posts, we talked about clumpiness in two-dimensional “star fields.”

• In the first, we discussed the problem in general and used a measure of clumpiness created by taking the mean of the distances from the stars to their nearest neighbors. The smaller this number, the clumpier the field.
• In the second, we divided the field up into bins (“cells”) and found the variance of the counts in the bins. The larger this number, the clumpier the field.

Both of these schemes worked, but the second seemed to work a little better, at least the way we had it set up.

We also saw that this was pretty complicated, and we didn’t even touch the details of how to compute these numbers. So this time we’ll look at a version of the same problem that’s easier to wrap our heads around, by reducing its dimension from 2 to 1.  This is often a good strategy for making things more understandable.

Where do we see one-dimensional clumpiness? Here’s an example:

One day, a few years ago, I had some time to kill at George Bush Intercontinental, IAH, the big Houston airport. If you’ve been to big airports, you know that the geometry of how to fit airplanes next to buildings often creates vast, sprawling concourses. In one part of IAH (I think in Terminal C) there’s a long, wide corridor connecting the rest of the airport to a hub with a slew of gates. But this corridor, many yards long, had no gates, no restaurants, no shoe-shine stands, no rest rooms. It was just a corridor. But it did have seats along the side, so I sat down to rest and people-watch.

The Index of Clumpiness, Part Two

Last time, we discussed random and not-so-random star fields, and saw how we could use the mean of the minimum distances between stars as a measure of clumpiness. The smaller the mean minimum distance, the more clumpy.

Star fields of different clumpiness, from K = 0.0 (no stars are in the clump; they’re all random) to K = 0.5 to K = 1.0 (all stars are in the big clump)

What other measures could we use?

It turns out that the Professionals have some. I bet there are a lot of them, but the one I dimly remembered from my undergraduate days was the “index of clumpiness,” made popular—at least among astronomy students—by Neyman (that Neyman), Scott, and Shane in the mid-50s. They were studying Shane (& Wirtanen)’s catalog of galaxies and studying the galaxies’ clustering. We are simply asking, is there clustering? They went much further, and asked, how much clustering is there, and what are its characteristics?

They are the Big Dogs in this park, so we will take lessons from them. They began with a lovely idea: instead of looking at the galaxies (or stars) as individuals, divide up the sky into smaller regions, and count how many fall in each region.

The Index of Clumpiness, Part One

1000 points. All random. The colors indicate how close the nearest neighbor is.

There really is such a thing. Some background: The illustration shows a random collection of 1000 dots. Each coordinate (x and y) is a (pseudo-)random number in the range [0, 1) — multiplied by 300 to get a reasonable number of pixels.

The point is that we can all see patterns in it. Me, I see curves and channels and little clumps. If they were stars, I’d think the clumps were star clusters, gravitationally bound to each other.

But they’re not. They’re random. The patterns we see are self-deception. This is related to an activity many stats teachers have used, in which the students are to secretly record a set of 100 coin flips, in order, and also make up a set of 100 random coin flips. The teacher returns to the room and can instantly tell which is the real one and which is the fake. It’s a nice trick, but easy: students usually make the coin flips too uniform. There aren’t enough streaks. Real randomness tends to have things that look non-random.

Here is a snap from a classroom activity: