Visualizing randomness

May 19, 2009 in Math

Daniel Becker's diploma dissertation was on the visualization of randomness - finding concrete ways to map the highly abstract idea of random behaviors and patterns. The resulting portfolio is fascinating, even for someone without a statistical background, in particular for the way in which it lends a semblance of order to these inherently chaotic processes.

The first one that caught my eye was a clean visualization of Benford's law (I recently rambled on the subject). It isn't a derivation of the law, but rather an illustration of its presence in a dataset of countries' areas in square kilometers (a zoomable version is on the diploma website):

However, my favorite visualization is one of a quincunx. A quincunx (also called a Galton box, but that's not as fun to say) is a device consisting of a vertical space filled by evenly spaced horizontal pins. A ball is dropped from the top of the box and allowed to bounce off the pins on its way down. At each pin, it is impossible to say whether the ball will bounce left or right. However, the box is used to illustrate the normal distribution (strictly speaking, it really illustrates a binomial distribution, which in the limit approaches a normal distribution) because the ball's ultimate location will take the shape of a familiar bell curve - it is likely to land directly under the point at which it was dropped, and less likely to land far away.

My dad used to have a quincunx in his office when I was little. It took the form of a game (circa 1900) - there were bins along the bottom, and you dropped a penny in the top. If the penny landed in one of the extreme bins, you won -- but you lost if it landed in one of the central bins.  In retrospect, the game was obviously mispriced, since many of the bins had equal payouts despite their chances of receiving the penny. But I'm sure the mispricing was in the house's favor.

Anyway, here is Becker's visuaization of the process:

Finally, I also liked this visualization of the output of various random number generators. Needless to say, there should be no apparent pattern in uniform numbers generated by an algorithm - the presence of such a pattern indicates faulty number generators. (It is well documented that Excel's random number generator is particularly disappointing). The box in the lower right represents the "iPhone" of  random number generators - the Mersenne Twister. It is the closest to being pattern-free, though it looks a bit like a Clayton copula to me!

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