February 7, 2007

Chaos Theory

When David Johnson's player rankings came out I initially posted that "goals occur randomly" and as a result, using information from a small sample hurts his analysis significantly. David Johnson replied simply that "Goals don’t occur randomly". While I can argue that they do occur randomly, I have more trouble proving that goals are not random, which got me thinking: are goals random? Statistically goals appear to be a random variable, however this does not make them random. In fact I would argue that with all possible information a goal is completely predictable, the problem is that this information is not available or not possible to get. For example the dents and shape of the ice could have an effect on the direction of a shot or the ability to pass. Consider the variables in the shooter: heart rate, blood pressure, emotional variables, physical muscle strength at any given time, oxygen concentration in the blood etc. The number of possible variables is essentially infinite. If you Knew all the variables you should be able to say if the situation will produce a goals, save, a blocked shot or a missed shot.

Why then do goals appear so random?
In simple terms, Chaos Theory can be summarized by the butterfly effect: "The phrase refers to the idea that a butterfly's wings might create tiny changes in the atmosphere that ultimately cause a tornado to appear (or, for that matter, prevent a tornado from appearing). The flapping wing represents a small change in the initial condition of the system, which causes a chain of events leading to large-scale phenomena. Had the butterfly not flapped its wings, the trajectory of the system might have been vastly different." (Wikipedia). The main point of chaos is there is a strong dependence on initial conditions as you can see in the graph to the right a difference 0.00127 resulted in the curves to eventually be on opposite sides of the graph. A chaotic system's dynamics are fully understood, so given any initial condition the result at any moment in time is known, however a small change has unpredictable effects. There are a number of real systems that are chaotic including the most classic of all examples: The Lorenzian Waterwheel. A number of other examples can be found here. A coin toss could be argued is a chaotic as could a roll of a dice, although most would agree that they are in fact random. I don't have an article: but a few years back I heard of a couple guys who could predict the numbers for the lottery as the lottery corporation used a random number generator and the individuals discovered all the initial conditions and knew the generating function and were therefor able to predict the winning numbers.

"So, is flipping a coin a random event? It really depends on the definition of the word 'random'. If we take it to mean that the outcome is theoretically unpredictable then no, the coin flip is not random. But in the real world it is practically impossible to predict the outcome of a good flip, so in that respect we can call it a random event." (Random Number Generator).

Alan Ryder concludes after going through great analysis that: "Goal scoring in a hockey game is basically a Poisson process." (Poisson Toolbox [PDF]). It is the result of hockey being chaotic with unknown variables that makes a completely determined system appear random. The more information we have on the dynamics of hockey and the more variables we know the less random the game will become. That being said, the chaotic properties would still exist and unless we had exact (infinite decimal places) values for all (infinite amount) of the information the game would still have significant random properties. And this in fact makes the game exciting. If we could predict with 100% certainty every event in hockey it wouldn't be too excite, how many people actually watch sports re-runs?

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