Description
The code source files will enable you to reproduce the experiments on multifractal singularity analysis detailed in Turiel et al.'s article on numerical methods for the estimation of singularity spectra on sampled data: a comparative study referenced below (cite this source code or the reference's doi: 10.1016/j.jcp.2005.12.004).
See also multifractal project for multifractal analysis of 1D (time-series) and 2D (images) signals.
Usage
settings
You need to create your own hierarchy of directories, as follows:
EXP -------- benchmark ------ A
| |--- B
| |--- C
|
|---- wtmm ----------- A
| |--- B
| |--- C
|
|---- lastwave ------- A
| |--- B
| |--- C
|
|---- wavelab -------- A
| |--- B
| |--- C
| |--- wavelib
| |--- test
|
|---- fraclab -------- A
|--- B
|--- C
|--- fraclib
|--- test
Note that the name of the main directory (EXP in the example above) doesn't matter whereas the others do.
benchmark
The directory benchmark/ will contain all the data used in the various experiments.
In this directory, there is a script:
buildbench.sh
that enables to generate all the multifractal signals and to install them in the corresponding sub-directories A/, B/ and C/ once multifractal_generator has been copied in bin/ (note : in this file, change the name multifractal_generator.exe into multifractal_generator).
wtmm
The directory wtmm/ will contain all the results of the WTMM analysis with our code.
In this directory, there is a script:
procwtmm.sh
that enables to compute the singularity spectra of the signals in benchmark and to install the results of these estimations in the corresponding sub-directories A/, B/ and C/ once wtmm_processor has been copied in bin/.
(note : in procwtmm.sh,
i/ first change the name wtmm_processor.exe into wtmm_processor,
ii/ then read carefully the first lines of comments to have info about the processing,
iii/ the program needs approx. 20 mn to run with our data)
lastwave
The directory lastwave/ stores the main program for analyzing the signals with the WTMM method implemented in the LastWave software.
In this directory, the file analysisLastWTMM1d.def contains the routines to process the analysis. To use the functions defined in this file, launch the software LastWave and type:
a> source analysisLastWTMM1d.def
("a>" is the prompt of lastwave). With the command:
a> help AnalyzeSeriesLastWTMM
you will have some information about the main function used to analyze our data.
If you want to analyze for instance the 10 signals Log-Poisson of size 4096 and with parameters h_infty=-0.5 and D_\infty=0, type the command:
a> AnalyzeSeriesLastWTMM '../benchmark/B' 'B' 'Log-Poisson.h-0.50-coD1.00_1D' \
10 1.5 8. 10. 'g2' {-200} -200 -200
See the help of function AnalyzeSeriesLastWTMM and the code for a complete description.
wavelab
The directory wavelab/ contains the main program for analyzing the signals with the WTMM method implemented in the WaveLab802 software.
- either you install the library WaveLab in your environment; then, you add the path of the directory WaveLab to the
PATHvariable of Matlab, e.g. thanks to the command addpath (See the help of functionprocWaveWTMM1d, below). - you use the 'stand-alone' files provided in the sub-directory
wavelib/of the directorywavelab/. They are copies of the Wavelab
necessary files.
In Matlab, move to the directory wavelab/. Then, in order to perform the estimation for all the series, you should type:
> procWaveWTMM1d('all','all');
(">" is the prompt of Matlab). The processing of all the series takes about 15mn. However, you may want to try first over small dataset to begin, so type for example:
> procWaveWTMM1d('A',1);
to perform the analysis only for the multifractal serie stored in file Log-Poisson.h-0.50-coD1.00_1D-N000.txt of set A. For more details, get the help of function procWaveWTMM1d by typing:
> help procWaveWTMM1d
The function performing the main computation is stored in the file myAnalyzeSeriesWaveWTMM; type
> help myAnalyzeSeriesWaveWTMM
for help and have a look on the code inside this file. Note that this method uses only the classical method to compute the Legendre transform (not the canonical one, like in Lastwave).
You can have a look in the test file, whose purpose is to use different implementations (but all using the main WaveLab functions) for retrieving similar results.
fraclab
The directory fraclab/ contains the main program for analyzing the signals with the WTMM method implemented in the FracLab software.
First of all, you should be sure that the fraclab functions are known in the Matlab environment. For this purpose, you have two solutions:
- either you install the library FracLab in your environment; then,
you need to add the path of the directory FracLab to the
PATHvariable of Matlab, e.g. thanks to the command addpath (See the help of functionprocFracWTMM1d, below). - you use the 'stand-alone' files provided in the sub-directory
fraclib/of the directoryfraclab/.
In Matlab, move to the directory fraclab/. Then, in order to perform the estimation for all the series, you should type:
> procFracWTMM1d('all','all');
The processing of all the series takes about 25mn. However, you may want to try first over small dataset to begin, so type for example:
> procFracWTMM1d('A',1);
to perform the analysis only for the multifractal stored in file Log-Poisson.h-0.50-coD1.00_1D-N000.txt of set A. For more details, get the help of function procFracWTMM1d by typing:
> help procFracWTMM1d
The function performing the main computation is stored in the file myAnalyzeSeriesFracWTMM; type
> help myAnalyzeSeriesFracWTMM
for help and have a look on the code. Note that this method uses only the classical method to compute the Legendre transform.
We however recommand first to have a look on the README of the sub-directory test/ and to perform the operations described therein. The aim of the described operations is to help you to understand how FracLab works, to see the results it provides and eventually see some of its limitations (not the canonical one, like in lastwave).
- Pont A., Turiel A., and Pérez-Vicente C.J. (2009): Empirical evidences of a common multifractal signature in economic, biological and physical systems, Physica A, 388(10):2025-2035, doi: 10.1016/j.physa.2009.01.041.
- Turiel A., Pérez-Vicente C.J., and Grazzini J. (2006): Numerical methods for the estimation of singularity spectra on sampled data: a comparative study, Journal of Computational Physics, 216(1):362-390, doi: 10.1016/j.jcp.2005.12.004.
- Lévy Véhel J. and Tricot, C. (2004): On various multifractal spectra, Progress in Probability, Fractal Geometry and Stochastics III, Birkhauser Verlag, 57, pp.23-42.
- Arneodo A., Audit B., Decoster N., Muzy J.-F. and Vaillant C. (2002): Wavelet based multifractal formalism: Application to DNA sequences, satellite images of the cloud structure and stock market data, in Bunde A. et al. eds.: The Science of Disasters: Climate Disruptions, Heart Attacks, and Market Crashes, Springer Verlag, pp. 26-102.
- Abry P., Gonçalves P., Lévy Véhel J. (2002): Lois d’Echelle, Fractales et Ondelettes, Hermès Sciences Publications.
- Arneodo A., Bacry E., Jaffard S. and Muzy J.-F. (1998): Singularity spectrum of multifractal functions involving oscillating singularities, Journal of Fourier Analysis & Applications, 4(2):159-174. doi:10.1007/BF02475987.
- Bacry E., Muzy J.F. & Arneodo A. (1993): Singularity spectrum of fractal signals from wavelet analysis: Exact results, Journal of Statistical Physics, 70(3):635–674, doi:10.1007/BF01053588.
- Donoho D.L. (1993): Nonlinear wavelet methods for recovery of signals, images, and densities from noisy and incomplete data, in Daubechies I. (ed.): Different Perspectives on Wavelets, American Mathematical Society, pp.173-205.