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Hurst Exponent Calculation and Supporting Statistics Documentation

Hurst Exponent Calculation and Analysis Software


This C++ software based supports estimation of the Hurst exponent. For a discussion of the Hurst exponent, see the parent Web page Estimating the Hurst Exponent Value

The Hurst exponent is estimated using the classis rescaled range (R/S) technique and by using wavelet spectral analysis. The software published here was used to develop the plots which are published on the parent Web page.

Both the R/S and wavelet methods for calculating the rescaled range make use of linear regression. Supporting software calculates the linear regression slope (which is proportional to the Hurst exponent) and the error of the regression.

The application that motivated me to study the Hurst exponent and the methods used to estimate the Hurst exponent is the analysis of financial time series. Supporting software is included to read files of market data for a given stock, downloaded from Functions are also provided to calculate n-day return.

The Hurst exponent provides one way to measure long memory processes. Another statistic is autocorrelation. C++ software is also included to estimate autocorrelation.

Software to test the Hurst exponent can be tested using synthetic data sets that have a known hurst exponent value. One of the simplest is a pure random walk, which should have a Hurst exponent of 0.5. The generation of random walks with long memory is a complex topic by itself. Some methods involve "spectral" techniques, using either the Fourier or wavelet transforms. The software I developed for generating synthetics data sets was not successful. I have decided not to publish these incomplete experiments.

Hurst Exponent Estimation Classes

This code took months of nights and weekends to develop. This is not due to the size of the code for Hurst exponent estimation but rather to the time it took to read and understand the literature and develop the code. The actual hurst estimation code is in two classes:

Supporting code calculates the linear regression and regression error which is used in both the rescaled range and wavelet Hurst exponent estimators.

To create the plots and tables published on my Hurst exponent web page I wrote additional software, including support for statistical functions, reading stock market data and calculating returns. The wavelet spectrum estimation of the Hurst exponent uses the wavelet softwae base, which I wrote for other web pages. As a result of all this supporting software, the Hurst exponent code is buried in all these supporting files.

Main Programs

Sadly it looks like my mathematics adventures may have to be put aside in favor more practical software projects. But if I were to continue working on mathematics codes, I might start using a package like Octave or R. The code I wrote to test the Hurst estimation functions is sort of "hackish". To run various tests I would modify the code and recompile. This code contains lines which can be uncommented out to test the code with various data sets. This code is not terribly elegant, but I hope that it demonstrates how to apply the functions.

I've included the C++ code I wrote to apply and test the Hurst exponent estimation code, along with the supporting statistical functions. Makefiles are included for each main program. These Makefiles are written for Microsoft nmake. I appologize. I know that Microsoft is evil. UNIX (or Linux) - Live Free or Die. It should not be too difficult to convert these Makefiles for UNIX.

The classes developed to test the Hurst estimation code are:

Doxygen Documentation

To regenerate the documentation install Doxygen, modify the file Makefile and type nmake doxygen.

Copyright and Use

This copyright declaration and license applies to all of the Hurst exponent estimation software published on It also applied to any supporting software (e.g., histogram calculation, statistics functions and wavelet functions). (I did not feel like copying this copyright into every file).

You may use this source code without limitation and without fee as long as you include:

This software was written and is copyrighted by Ian Kaplan, Bear Products International,, 2003.

This software is provided "as is", without any warranty or claim as to its usefulness. Anyone who uses this source code uses it at their own risk. Nor is any support provided by Ian Kaplan and Bear Products International.

Please send any bug fixes or suggested source changes to:

Generated at Thu May 22 21:12:35 2003 for Hurst Exponent Calculation and Supporting Statistics by doxygen1.2.8.1 written by Dimitri van Heesch, © 1997-2001