Scale Invariance & Chaos Theory

R.N. Elliott observed that financial series appear to have the same “look” on many different time scales. Elliott published what he called the “Wave Principle” in 1938. His modeled financial series is comprised of zigzags that are nested within each other at smaller and smaller scales. Market technicians still make wide use of his work.

Scientists now understand this phenomenon as scale invariance. Scale invariance (also known as self-similarity) is defined as a feature of objects or laws that do not change if length scales are multiplied by a common factor. Benoit Mandelbrot called these objects fractals because they are partially characterized by a real-numbered (decimal number) fractional dimension. The presence of scale invariance means that the object has been produced by feedback process and is governed by the science of feedback systems known as Chaos theory (which in turn is a subset of Complexity theory).

We now know that markets are not these perfect real-numbered fractals of Mandelbrot or Elliott. Financial price series often appear to have different degrees of smoothness, or fractal dimension, on different scales at the same time. This inconsistency led to a startling discovery. Our research, as well as that of fellow geophysicist Didier Sornette, confirms that financial series are complex-numbered fractals, that by definition exhibit discrete scale-invariance and converging log-periodic cycles on approach to critical trend-change points.

Our research has also shown that markets repeatedly self-organize to these critical points. By detecting log-periodic cycling at extreme high or low Hursts on multiple scales, these predictive points can be found with a high degree of accuracy. This result has the side benefit of explaining the somewhat mysterious use of Fibonacci (1, 1, 2, 3, 5, 8, 13, 21, …) and Gann (1, 2, 4, 8, 16, 32, …) number sequences by market technicians. These number series are both log periodic.

Log Periodicity in Action

The Effect of Cycle Magnitude on the Critical Point Model
The Effect of Phase on the Critical Point Model
The Effect of Wavelength on the Critical Point Model
The Effect of the Post-Critical Power-Law Exponent on the Critical Point Model