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Simulating Fractional Brownian Motion

Fractional brownian motion is a random walk that has a defined Hurst
exponent. A random walk, with no long memory dependence, has a hurst
exponent of 0.5. Fractional brownian motion data sets where 0 < H <
0.5 or 0.5 < H < 1 simulate long memory processes. Fractional
brownian motion data sets are critical for testing the quality of
algorithms that estimate the Hurst exponent. Generating high quality
fractional brownian motion data sets is surprisingly complex. There
are a variety of techniques, some of which are based on the wavelet
transform.

I implemented the so called "random mid-point displacement" algorithm
which is used to generate fractal landscapes (rock and mountains) in
computer graphics. Unfortunately this is not an effective way to
generate a data set with a defined Hurst exponent. This code is not
published here since I regard it as experimental.

Wavelet techniques can also be used to generate fractional Brownian
motion (fBm). Sadly I had to move on to other topics with a less
esoteric nature and more commercial potential before I could finish
this work. So this web page is incomplete. I tested my Hurst
exponent estimation code with data sets I got from other people.

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Web References

Ian Kaplan

May 2003

Revised:

*Pseudo-random Numbers*

*Calculating the Hurst Exponent using the Wavelet Transform*

*Wavelets and Signal Processing*