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.

Web References

Ian Kaplan
May 2003

Pseudo-random Numbers

Calculating the Hurst Exponent using the Wavelet Transform

Wavelets and Signal Processing