The battlefield is a scene of constant chaos.
The winner will be the one who controls that chaos,
both his own and the enemies.
-Napoleon
The winner will be the one who controls that chaos,
both his own and the enemies.
-Napoleon
The book Random Walks and Random Environments: Volume 1: Random Walks by Barry D. Hughes details the logic and mathematics behind chaotic motion patterns. He opens his text with a quote.
The truth is that the limit of human faculties often imposes upon us, as a condition of advance, temporary departure from the standard of strict method. The work of the discoverer may thus precede that of the systematizer; and the division of labour will have its advantage here as well as in other fields.
Lord Rayleigh
This zealous insight on development may be only roughly and hastily applied to stochastic processes, but its truth lies in the nature of the study. On one hand, Rayleigh's vague criticism of the "systematizer" is ironic because of the clearly instructive nature of Random Walks. It is true, however, that "departure from the standard of strict method" is necessary when analysing shapes such as the Cantor set; exceptions must be made when dealing with exceptional figures.
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| The Cantor set, a shape warranting extreme measures, creative measures. |
Throughout the book, Hughes provides tools and insight to help quantify such tricky objects. He links chemistry, physics, math, and the aftermath of partying to one another in their randomness, demonstrating how connected separate fields of study are. To help ease this fusion, Hughes provides a helpful table that can translate the woes of engineers to the cries of physicists. The context of this Rosetta Stone is the use of balls and sticks in illustrated models.
Discipline
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Name for 'points'
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Name for 'connections'
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physics
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site or atom
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bond
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mathematics
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vertex
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edge
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engineering
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node
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link
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Deterministic Chaos
One complex shape that Hughes demonstrates the chaotic properties of is the Lorenz attractor. Hughes provides these parameters for the function, a simple model for atmosphere hydrodynamics.
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| Parameters used in the definitive differential equations of the Lorenz attractor, a dynamical system. |
This simple set of rules, Hughes points out, are enough to generate an incredible pattern. The pattern shown below is classified as deterministic chaos because it generates disorder as it operates by a set of rules.
Hughes makes the interesting point that non-deterministic chaos is easier to analyse and study. I find this odd because deterministic chaos grows from simple equations, in most cases, while non-deterministic chaos like stock price patterns are devoid of process. The fact that processes are involved in deterministic chaos makes them seem a lot more approachable to me.
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| An animated view of the collection of information generated by the Lorenz equations. Note the qualities of the pattern listed below. |
The attractor operates by the simple set of three rules shown above. Based on these laws, I make observations or predictions about the pattern produced.
- The velocity at each point on the traced path is based on the position of the particle.
- The traced path never crosses itself. If it did, it would begin to follow the path it made last time it came to that point because of point [1], going in a loop.
- The attractor is labeled a "strange" attractor because its orbit about one of the two central nodes is unpredictable; there are no means by which to tell which of the two a point will circulate around at a given time.
- One starting point will determine either a path of infinite length that folds upon itself infinitely, or a loop.
Hughes mentions that the evolution of the system from any initial position is uniquely determined. This goes along with point [4] because .
The Lorenz attractor is not a random walk. Hughes presents it as a counterexample. The attractor is a fantastic example of deterministic chaos, not random chaos. Its elegant curves and calculated trajectory will not be seen reflected in any of the later examples that Hughes provides for us. From here, it gets rough. You'll see what I mean.
Random Chaos and Research Possibilities
The characteristic that most concludes the examination of stock market prices is roughness. Aside from brutal competition and abrasive attitudes, the patterns followed by stock prices are rough.
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| A stock time series. This plots the price of shares of Chipotle Corporation stock over time. |
As we can see from records of the Chipotle Corporation stock, "roughness" is present. The specific characteristics that are central to this type of chaos are that it is:
- Continuous
- Nowhere-differentiable
Because of these qualities, it is difficult to predict the future of such a time sequence. I'd like to research ways that you can make predictions even on a function that is nowhere-differentiable.
Below are some other cool examples of functions that are nowhere differentiable.
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| The Weierstrass continuous and nowhere-differentiable function. |
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| The Darboux continuous and nowhere-differentiable function and the equation determining it. |
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| A list of other discovered continuous nowhere-differentiable functions. |
when you say, "from here it gets rough..." and I was thinking "rougher?" OK. So I think my husband will have to tutor me on some of this language I do not speak. What, exactly, will you be examining? It's not all stock predictions, right? I think one thing you will have to be aware of when you write your expository essay is your audience. How do you explain complex mathematical to someone like me who feels like a foreigner to this "language"?
ReplyDeleteBeing a math guy, I find this stuff interesting. In all the math classes I’ve taken up to this point, I haven’t spent a lot of time on continuous, nowhere-differentiable functions, although they can serve as models for many real-life phenomena. The Lorenz equations look really cool when you graph them--I would be interested to know if there are any real-world things that follow them. There might not be, since--as you point out--real life tends to be a little more “rough.”
ReplyDeleteSince many real-world phenomena don’t follow neat equations, it would be very useful to be able to predict nowhere-differentiable functions. You say you would like to research ways of doing that. I assume there are ways, although I don’t know what they are, and I would be interested in reading a paper about them. But I think (and Ms. Romano would likely agree with me) that I would enjoy reading your paper more if you described some specific real-life things. You mentioned stock prices, but I’m sure there are plenty of others out there.
I’m going off on a tangent here, but I think it could be interesting to study predictions in general. We encounter predictions every day--about the weather, the baseball season, the economy, et cetera--and so many of them turn out to be wrong. I wonder if there are any patterns to bad predictions--logical traps that people keep falling into. If there are such patterns, they might reveal something interesting about human nature, as well as help us make better predictions in the future. I haven’t read it, but I know that Nate Silver, who’s pretty well known for his statistical analysis of baseball and political elections, has a book called The Signal and the Noise about the science (the art?) of predictions. This paragraph might be totally off base as far as what you’re interested in researching--I’m just throwing out an idea.
Although it might not prove useful for your project, it’s also just fun to read about terrible predictions that people have made in the past. For example, a New York Times reporter wrote in 2006, “Everyone’s always asking me when Apple will come out with a cell phone. My answer is, ‘Probably never.’”
To be entirely honest, I understood the poem by Napoleon, and very little afterwords. I feel like the definition of chaos that this book utilizes is different than what I'd consider chaos. For instance, if the Lorenz attractor can be defined by a series of equations, can it be considered chaotic? This is just me talking out of ignorance, but shouldn't something that's chaotic defy rational explanation?
ReplyDeleteLike Ms. Romano said, I think that It'll be really difficult for you to communicate what you mean to your audience when you write your paper. Also, you have to do four other things that you are going to have to present. We went over a possible list today in class (you weren't here) but I on't see how they lend themselves easily to your topic.
You with all of the stock charts that you have, is there a way that you can accurately (or semi-accurately) begin to predict the stock market or something like it?