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Fractal and chaotic behaviour in discrete sequence of data

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Last Edited: June 18, 2009, 12:05 am

Found an excellent paper (ref. 1) that demonstrates methods to analyze discrete set of data (such as lottery data) without any reference to differential equations.

Usually chaos and fractal behavior is analyzed using both DE and/or PDE (differential equations and partial differential equations).

From the the time series point of view where one is seeking to locate fractal properties (or even to determine if data points exhibit fractal properties at all)  there is a need to find ways to approximate non-linear equations with set of discrete points. Such methods are clearly outlined in this paper.

Author uses well known non-linear Henon map function, transforms it to 2 dimensional representation of discretized data points and then builds so called Henon attractor.

Subsequently, via re-parametrization, a one dimensional (1 coordinate) approximate Hennon attractor in a two dimensional phase space is reconstructed.

Next step is an embedding dimension method is introduced and equations form a method to study phase space approximations (there is no limit how many embedding dimensions one can use).

The whole idea is to take any discrete data points and via above methods to approximate chaotic orbits present in the Henon phase space via estimation of Hurst (Fractal Dimension) and Lyapunov exponent for predictive power of the data points generated thru iterative process of various re-parametrization values and embedding dimensions.


This is quite exciting from our point of view as this is exactly type of applied chaos analysis thats needed for a discrete data series in form of past lottery draw, if carefully executed it will provide knowledge (to the best degree of approximation) if the methods of fractal and chaos theory can be applied at all in this context.

There is a strong possibility that this want be the case, ie Hurst exponent defaults to value as present in random walk (0 correlations) and or Lyapunov exponent yields 0 predictability, but in any case this needs to be explored and evaluated.

One drawback of the paper also is that calculations are performed using Mathematica software, and what I really need is set of c++ libraries.

In any case paper is one of best around on the subject of pragmatic approach to time series analysis using methods in chaos and fractal theory.

References:

1. Testing Chaos and Fractal Properties in Economic Time Series

http://www.internationalmathematicasymposium.org/IMS99/paper25/ims99paper25.pdf

Entry #14

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