Some of the math can be quite simple like the formulas used to create self-similar fractals like Sierpinski triangles (D=log N/Log S). Or there is Fibonacci (Fn=F(n-1) + F(n-2))) that is largely related to spiral fractals.
But more complex fractals, like those considered to be quasi-periodic, require many more formulas and infinitely more data. These fractals evolve over time while also infinitely repeating (we know… that sounds contradictory). They are the formulas used to map the expansion of the universe. They are used to model particle physics. And they’re used by us to find patterns in the financial markets.
One of the most frequent inquires we receive has to do with the science behind our programs. We discuss this in a limited way on our science page, but many of you have asked us to go much deeper. So… here you go.
Quasi-periodic fractals are identified with three areas of math and physics: electromagnetic theory, gauge theory and chaos theory. Each plays an important role in first identifying the underlying pattern while also projecting that pattern forward.
Electromagnetic theory results in the identification of sin waves between oscillating particles. Some particles oscillate North and South while others oscillate East and West. The push and pull between the two oscillations results in a sin wave with peaks and troughs that vary over time. Some waves can take a long time to develop while others appear and disappear rather quickly.
Quite simply, this relationship can be analogized to the market with buyers and sellers. Sometimes the buyers pull markets higher over time while other times the sellers pull markets lower over time. The oscillations back and forth can be modeled with electromagnetics.
Gauge theory is used to create a reference level from known data in order to find the value of behavior or unknown data as it relates to that reference level. For example: You can fit 3 green tennis balls in a bag. If you take that statement and change the color of the balls to orange, the statement remains true. This makes the color gauge invariant… meaning it doesn’t have any impact on the result.
Gauge theory is used in the mathematical identification of a fractal by looking at price and volume changes to determine whether those changes have any impact on the length of a wave going forward. In other words, is price and volume gauge variant or invariant?
Chaos theory, quite simply, is used to strip out noise that does not belong. Fed reports, elections, flash crashes, news and other noise exist in the market on a daily basis. Some of them move the market, others don’t. But none of their impacts belong on the underlying sin wave and none of them have any impact beyond the initial shock.
Fourier Transforms are used to determine wave frequency and oscillation density. They’re used to decode the universe of market data while breaking it down into individual pieces. Those individual pieces then have no reference to time, but are now only singular dots of data with a particular frequency (a frequency of one). For example, “if a recording of music were put through a Fourier transform, the result would be the musical notes that constituted it, as well as all the harmonics of the instruments that played it. The output loses the time domain; instead, it has the amplitudes of all the frequencies that composed it, as well as the phases of the sine-waves at those frequencies.”
Wave / Group Velocity Functions are used to project patterns forward after they have been identified. These formulas consider the market is a collection of millions of different harmonic inputs rather than one singular entity. For example, a bird flying in the sky does so very differently when flying alone rather than when it’s flying with a flock. The market is a flock… and wave / group velocity functions allow us to project a fractal forward with that consideration.
Over the next few weeks, we will discuss each of these topics in depth. We’ll break them down individually, while offering examples, images and analogies. Prepare for headaches.