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Fractal Dimension by Box Counting

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Last Edited: July 21, 2009, 9:59 pm

Further expanded chaos toolkit by adding new process to calculate Fractal Dimension in a Time Series.

Below are sample runs using RPS game model (graph 1 and graph 2), scaling property is clearly visible, (scaling property is  common to fractals and an indicator that time series under study is most likely chaotic)

Calculations were performed with the help of software FD3 (see ref. 1) C language was ported to C++ and several tests were conducted to verify validity of the implementation from other sources (including successful calculation of the Fractal Dimension of the coast of Great Britain)

Fractal Dimension, might offer yet another way for entry/exit game strategy, we will at some point take time to model P3 and P4 games to translate individual number probability trajectories and apply to them exactly the same techniques as we are doing with RPS game (RPS is a stochastic game already, but with key important difference being that there is a learning algorithm that recalculates opposite player probability, nevertheless player's choice is still determined by the RNG)

It is highly interesting if: entropy/ fractal dimension/hurst exponent (not yet implemented and verified)/ lyapunov exponent and yet to be modeled: nsb-entropy (hardest but with highest potential) can be made applicable in lottery games. this question is still very much open.

It is also interesting that RPS game exhibits similar Fractal Dimension as Spectrum of Fibonacci Hamiltonian (0.88137) (see ref. 2 and 3)

Graph 1 (500 Iterations)

 

Graph 2 (1000 Iterations):

 

 

References:

1. FD3 Software - written by John Sarraille and Peter DiFalco, using ideas from "A FAST ALGORITHM TO DETERMINE FRACTAL DIMENSION BY BOX COUNTING", by Liebovitch and Toth, Physics Letters A, 141, 386-390 (1989).

2. From Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Fractal_dimension

3. List of fractals by Hausdorff Dimension (Fractal Dimension)

http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension

Entry #26

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