Tuesday, March 24, 2009

The Road to Chaos

At this point we need to take a brief detour from our charge into the market to get some very helpful conceptual background. While the concepts are rooted in mathematical rigor, some nice drawings have been prepared to present them in a more elegant form. What you need to take home from this post is that chaos is anything but random. The limited determinism that exists for any chaotic system is a very meaningful and powerful tool, one we'll be using extensively and should be studied closely.

There are numerous analogies
for chaotic systems. Paul Ormerod liked to use ant models in his book, Butterfly Economics. However, a more telling example in the context of chaos in the stock market is the cigarette smoke diagram. It is the canonical example given to undergraduate students of fluid dynamics to describe the change between smooth, predictable laminar flow to irregular, chaotic turbulent flow. Cigarette smoke is an analogy that is quite extensible into the study of the stock market and therefore will be used throughout this blog to describe chaotic phenomenon in hands-on language.
Click to Enlarge Image Cigarette smoke rising off of the burning tip will rise because it is hotter than the surrounding air. As it speeds up, it grows unstable and begins to break down into turbulent flow. The initial region of instability is periodic flow, meaning it just wobbles back and forth over and over in a very repetitive, predictable fashion. The actual point where the turbulent flow begins is distinct. You can try watching a few cigarettes or incense etc for yourself to gain firsthand familiarity. (Must observe in very still air)

The key things to note here are that both the laminar region and the periodic region are very easy to model and predict, and the model will be able to predict tiny features at any point in the smoke with a relatively high degree of accuracy. However, the turbulent region is different. It will begin with very subtle perturbations that suddenly cause the smoke to change remarkably in its flow characteristics and behavior. The resulting pattern is actually non-random. All of the physics of the fluid are still intact. Everything behaves exactly as we understand it, it's just that the solution is based on so much data that we can't possibly collect enough to model it.

To illustrate some other key aspects of turbulence, the diagram includes two other illustrative (and quite suppositious) features. Imagining we could identify the region where the turbulence would begin to develop two seconds from now, we would perplexingly find nothing of great interest. In fact, it would be probably impossible to distinguish the smoke that would cause turbulence from the smoke that wouldn't. This is because chaotic behavior is the culmination of tiny, tiny features getting conserved, propagated, and amplified so that when the tipping point is reached, it will be highly dependent on nearly everything. The second illustrative feature is the arrow that says "chaos starts here." While the cigarette undoubtedly plays a role in the shape of the final turbulent smoke swirling pattern, it's again impossible in practice to tell what exactly about the cigarette lead to the smoke transitioning to turbulent when and how it eventually did.

One of the easy mistakes to make here is to believe that the smoke turning to turbulence is the result of one tiny disturbance, such as a tiny perturbation in the ambiant air or an imperfection in the cigarette paper. The truth is that every single part of the picture contributes to the final result, and no single part is dominant. No matter which part of the picture you change, be it a tiny tear in the cigarette paper or a slight twitch in the surrounding air a foot away, the resulting shape of the smoke will be different and could even be unrecognizable.

If you're asking yourself what isn't random about this extreme sensitivity to initial conditions, think of observing a large number of cigarettes. Although Phillip Morris may have you believe that every single cigarette will produce an equally unique and interesting smoke pattern, the truth is that they will get less and less unique as you keep making observations. Each smoke plume will fall within roughly the same area, and if you could measure this over a large enough cigarettes, you could predict with a high degree of certainty the maximum growth rate of the turbulent smoke.
It turns out that this characteristic of turbulent smoke is a very well understood concept known as a boundary value problem. Although every cigarette will be unique, they will all tend to fall within the same range over time, and what this essentially says is that there is a lack of uniqueness with respect to the range that the turbulent smoke will occupy. Lack of uniqueness is another way of saying determinism, which is another word for a sure thing. Limited-determinism refers to the idea that each smoke plume will be in many ways unique and in at least one way very non-unique, with larger exceptions becoming more and more rare. It's not certain what particular shape the smoke will take. Only the range that it will occupy can be defined or measured with some certainty. The applicability of this concept to the stock market should be self-foreshadowing at this point. Welcome to the chaos highway.
This illustration is simply to make clear that boundaries only tell us where the chaos should go, not where it should go within the boundaries. We'll see in the next chapter that, for the stock market, there are reasons why these boundaries are soft, but that they exist and can be used to make relatively definite predictions about the price a stock will trade at in the future in certain situations. Stay tuned.

*note there is a typo in the first cigarette smoke diagram. will get corrected at an indefinite time in the future™.

1 comment:

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