Fractal Analysis Programs of the National Simulation Resource
Introduction
Our long range purpose is to provide a set of analytical tools for fractal analysis. We currently have programs available for: (1) the generation of synthetic 1-dimensional signals that are simple fractional Brownian noise, and (2) analysis programs for determining the fractal dimension D (or the Hurst coefficient H, H = E + 1 - D, where E is the Euclidean dimension) from a simple fractal time series, i.e. a 1-dimensional signal. The analysis methods are described briefly by Schepers, van Beek, and Bassingthwaighte (1992) and in more detail by Bassingthwaighte, Liebovitch, and West (1994), which also review applications. Specific publications on the methods are listed with the programs.
A short description of the programs currently available for distribution is given below. Each package includes the source code for the product, its test program, and all subprograms upon which they depend. Also included are a README file with notes about the files, a manual page (plain text and UNIX troff source versions), a Makefile to create and run the test program, and the auxiliary files required by the test program.
These software packages can be transferred using anonymous ftp by clicking on the name of the package desired. Transferred files are compressed tar archives. (Some browsers will uncompress the file automatically). Extracting files from the archive will place the source files in a new subdirectory with the same name as the program.
(NOTE: Beyond NSR, Francesco Potortì has made available, under the Gnu Public License, some small Octave functions for measuring and generating the Hurst parameters of unidimensional fractional Brownian noise. These functions can be obtained via his Software Page.)
Available Programs
Signal generating programs
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Description: Generate a fractional (fractal) Gaussian process 1-dimensional series at evenly spaced intervals.
Notes: This is the recommended generating algorithm. It has the correct falloff in power spectral density and the correct autocorrelation.
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Description: Generate a fractional (fractal) Brownian noise or motion 1-dimensional series at evenly spaced intervals using the spectral synthesis method.
Notes: While the spectral synthesis method is a standard method that has been widely reported and used, it is not correct. It has the correct power spectral density but not the correct autocorrelation. The correlation between the first and seconds points is the same as between the first and last. If N data points are desired, the error is reduced by generating a signal of at least 2N, preferably 8N, and then taking a segment of length N out of the results.
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Description: Generate a fractional (fractal) Brownian noise or motion 1-dimensional series at evenly spaced intervals using the successive random addition method.
Notes: Like the SSM method, the successive random addition method is also inaccurate, but we have not yet well characterized its inadequacies While it is a ``standard method,'' beware.
Signal analysis programs
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Description: Perform dispersion analysis on a fractal time series.
Notes: The dispersion analysis program is designed for the analysis of fractional Brownian noise and is not suitable for analyzing fractal Brownian motion. (Given a Brownian motion signal, the noise signal can be obtained by taking the differences between adjacent points.) Disp requires that the signal to be analyzed be at uniform intervals. Dispersion analysis is the recommended method for time series analysis.
Publication: Bassingthwaighte and Raymond (1995)
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Description: Perform Hurst rescaled range (R/S) analysis on a fractal time series.
Notes: The rescaled range analysis is included for historical perspective and in recognition of the marvellous insights of Harold Edwin Hurst into the analysis of natural time series. The method is, however, seriously flawed and give biased results even if trend correction is used. The notes about disp related to analysis of noise vs. motion and uniform intervals also apply to hurst.
Publication: Bassingthwaighte and Raymond (1994)
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Description: Perform relative dispersion (fractal) and correlation analysis of flow distributions.
Notes: flowrect reads coordinates and flow values for voxels of a 3-dimensional organ, calculates the relative dispersion (RD) of flow for ever-larger groupings of adjacent voxels, and estimates the slope of the graph of log(RD) versus log(volume). For the same data, the correlation statistic (r) for pairs of flows as a function of distance is also calculated.
Publication: NONE
Beta test programs
Some beta test programs are available. These programs have not been extensively tested and are offered "as is." Use them at your own risk.
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Description: Estimate the Hurst coefficient of a time series using scaled windowed variance analysis.
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Description: Estimate the Hurst coefficient of a time series using bridge detrended scaled windowed variance analysis.
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Description: Estimate the Hurst coefficient of a time series using linearly detrended scaled windowed variance analysis.
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Description: Calculate the autocorrelation function (ACF) and Hurst coefficient.
Problems and Questions
To report any problems or obtain further information, send e-mail to: librarian@bioeng.washington.edu
References
Bassingthwaighte, J. B., and G. M. Raymond. Evaluating rescaled range analysis for time series. Ann. Biomed. Eng. 22:432-444, 1994.
Bassingthwaighte, J. B., L. S. Liebovitch, and B. J. West. Fractal Physiology. New York, London: Oxford University Press, 1994.
Bassingthwaighte, J. B., and G. M. Raymond. Evaluation of the dispersional analysis method for fractal time series. Ann. Biomed. Eng. 23:491-505, 1995.
Schepers, H. E., J. H. G. M. van Beek, and J. B. Bassingthwaighte. Comparison of four methods to estimate the fractal dimension from self-affine signals. IEEE Eng. Med. Biol. 11:57-64x71, 1992.
Last modified 03Jun03, 1:31 pm.