What Would Galileo Do?

Scientists and people who use statistics and decision theory to make decisions are often looking for a pattern in data. By finding and characterizing patterns, the data can be organized in such a way that knowledge of how the system behaves can be transmitted from one person to another in a convenient way. Finding patterns also greatly facilitates understanding a system.

For instance, think of the tremendous bulk of empirical data that can be summarized by the observation that F = MA. Once Kepler realized that planets travel in ellipses (almost), lengthy tables of observations could be compressed into a more easily accessible form. Deviations from a pure ellipse could then be used to predict new things such as the perturbing force of a previously unseen planet.

Of course we now know that Newton’s laws are approximations that break down at high speeds, but what about those ellipses? In fact, the planets do not describe perfect ellipses. If you plot the orbit of Mars for many thousands of revolutions, you would trace out a band, not a constant ellipse. If you only look at an orbit or two, the results are very close to an ellipse, and that is what Kepler found. He saw a pattern and it is extremely useful. To this day we say planets travel in ellipses, but then we qualify it by admitting their orbits are slightly perturbed by the gravitational pull of other planets. That formalism is a more useful way of looking at the dynamics than insisting on an exact representation-for most purposes.

But what about other patterns? Sticking with the celestial examples, the constellations that look like nearby stars and really unrelated (mostly), and appear to be organized into patterns because of our position in space at this time. Generally the patterns of stars in an arbitrary constellation carry little useful information.

Examples like this cause us to pause and ask, “What constitutes a useful pattern in data?” My dictionary lists 13 definitions for pattern, and none of them apply directly to answering this question.

Assume a collection of data. Define a measure of randomness in that data. Then try to fit a function to the data such that the residual left when you subtract the fitted function from the data leaves a smaller measure of randomness. For example, you could roll balls down an inclined plane and time when they get to various places. From that collection of data, you could try to fit a curve to the results and then subtract the fitted curve from the original data points. If the scatter in the remainders is smaller than the original, then you probably have found a pattern, but is it meaningful? Is it useful? If we permit the fitting curve to have as many terms as there are data points, then we can fit them exactly, but to what goal? On the other hand, if a simple curve such as a second-order (distance down plane = a +bt + ct2) greatly reduces the variation, then we might be on the way to understanding better the mechanism of rolling down an inclined plane. That is essentially what Galileo did. Of course, he would have computed an even better fit if he had used a higher order equation, but would that have been any better at understanding the mechanism?

In response to the interest my original tutorial generated, I have completely rewritten and expanded it. Check out the tutorial availability through Lockergnome. The new version is over 100 pages long with chapters that alternate between discussion of the theoretical aspects and puzzles just for the fun of it. Puzzle lovers will be glad to know that I included an answers section that includes discussions as to why the answer is correct and how it was obtained. Most of the material has appeared in these columns, but some is new. Most of the discussions are expanded compared to what they were in the original column format.

[tags]galileo, patter, newton, kepler, ellipse, data, randomness, chaos, decision theory[/tags]