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Wavelet p-Leader and Bootstrap based MultiFractal analysis (PLBMF) toolbox

This toolbox for MATLAB enables the (discrete) wavelet domain based multifractal analysis of (1d) signals and (2d) images, with bootstrap confidence intervals and significance tests, using wavelet p-leaders and wavelet leaders.
The toolbox is a generalization (to p-leaders based analysis) and significant update (to a more efficient code) of the wavelet leader based WLBMF toolbox.

The following functionalities are implemented:
Wavelet p-leader and dyadic wavelet coefficient based multifractal analysis
- multifractal attributes (scaling exponents, log-cumulants, multifractal spectrum)
- stucture functions
- minimal regularity exponent
- preliminary wavelet domain fractional integration
Wavelet-domain block bootstrap for multifractal analysis
- time- (space-) block, and time-scale- (space-scale-) block bootstrap
- ordinary and double bootstrap methods
- several confidence intervals for structure functions and multifractal attributes
- tests for null hypotheses on the precise value of multifractal attributes
- test for the null hypothesis of time constancy of multifractal attributes (1d only)
- automatic selection of scaling range (see this reference)

The toolbox is self-contained and entirely written in MATLAB (with the exception of one function that can be replaced with a mex C file that is provided in the release). Several demo files are included and use of the toolbox should be straightforward.

The theoretical and practical aspects of the renewing of multifractal analysis through the use of p-leaders are discussed in reference [JI.16] and reference [JI.15], respectively (which also establishes connections with the wavelet leaders and MFDFA). Further details may be found in reference [BC.5],

You can find a brief overview on (discrete) wavelet transform and leaders based multifractal analysis in this reference [JI.2], and in this tutorial article on wavelet-based multifractal analysis (with main focus on continuous wavelet transform). Specific aspects and particularities in the multifractal analysis of images are treated in this reference [JI.5]. Another useful reference can be found in this review article [BC.4].
You may also want to have a look at my thesis, which includes a synthetic overview of wavelet-based multifractal analysis, and collects material on a large number of theoretical and practical issues in empirical multifractal analysis.

If you use the code in your work, please cite the following references:
"Bootstrap for empirical multifractal analysis" [JI.2 .bib]
"p-exponent and p-leaders, Part II: Multifractal Analysis. Relations to Detrended Fluctuation Analysis" [JI.16 .bib]
"Wavelet leaders and bootstrap for multifractal analysis of images" [JI.5 .bib].

download: PLBMF toolbox

Release notes
01/10/2016 Initial release of the toolbox.
03/02/2017 Major fix for analysis of (2D) images and minor bug fixes. If you downloaded an earlier release of the toolbox, please replace it.

Testing fractal connectivity in multivariate long memory processes

This MATLAB function enables the (pairwise) test for fractal connectivity in multivariate time series. Fractal connectivity constitutes a particular model within which the low frequencies of the inter-spectrum of each pair of process components are determined by the auto-spectra of the components, the underlying intuition being that the long memory of each component results from the same single mechanism (see e.g. S. Achard et al., "Fractal connectivity of long-memory networks," Phys. Rev. E, vol. 77, no. 3, pp. 036104, 2008. ).
The test is based on the discrete wavelet coefficient coherence function, Fisher's z-transform and Pearson correlation coefficient. For details, please refer to this reference [CI.7].

If you use the code in your work, please cite the following reference:
"Testing fractal connectivity in multivariate long memory processes" [CI.7 .bib].

FractalConnectTest



For a collection of MATLAB routines for the synthesis of multivariate stationary Gaussian and non-Gaussian series, see the HERMIR toolbox by Hannes Helgason, Vladas Pipiras, and Patrice Abry.

Bayesian estimation for the multifractality parameter

Bayesian estimation for the multifractality parameter for Signals (1D) and Images (2D)

This collection of MATLAB function enables the wavelet leader based Bayesian estimation for the multifractality parameter (i.e., the intermittency parameter or second log-cumumant) for (1D) time series and (2D) images.
The estimators are based on a generic Bayesian model for the multivariate statistics of log-wavelet leaders. This model is motivated by the asymptotic covariance of multiscale quantities associated with multifractal multiplicative cascade based processes.
The Bayesian estimators are approximated using a Gibbs sampler. A Whittle likelihood is used in the Bayesian model in order to enable estimation for large sample size. For details, please refer to this reference [JI.12] and reference [CI.17].
Furthermore, for (2D) images, a suitable Hankel transformation is employed, effectively casting most 2D computations into more efficient 1D computations and significantly reducing computation times with respect to those indicated in [JI.12]. Please refer to this reference [CI.31] for details.
The toolbox includes a demo file that illustrates the usage of the estimation procedures.

If you use the code in your work, please cite the following references:
"Bayesian Estimation of the Multifractality Parameter for Image Texture Using a Whittle Approximation" [JI.12 .bib]
"Bayesian estimation of the multifractality parameter for images via a closed-form Whittle likelihood" [CI.31 .bib]
"Bayesian estimation for the multifractality parameter" [CI.17 .bib].

Bayesian c2 Toolbox



Bayesian estimation for the multifractality parameter and integral scale for signals (1D)

This MATLAB function complements the above "Bayesian c2 Toolbox" and enables the wavelet leader based Bayesian joint estimation for the multifractality parameter and the integral scale of (1D) time series.
The estimator is based on a modified generic Bayesian model that enables to also assess the integral scale of time series, jointly with the multifractality parameter. For details, please refer to this reference [CI.17] and reference [CI.28].

If you use the code in your work, please cite the following references:
"A Bayesian approach for the joint estimation of the multifractality parameter and integral scale" [CI.28 .bib]
"Bayesian estimation for the multifractality parameter" [CI.17 .bib].

Bayesian c2 and Integral Scale Code



Wavelet Leader and Bootstrap based MultiFractal analysis (WLBMF) toolbox

This toolbox for MATLAB enables the (discrete) wavelet domain based multifractal analysis of (1d) signals and (2d) images, with bootstrap confidence intervals and significance tests, using wavelet leaders. The WLBMF toolbox has been developped between 2008 and 2015 and is superseded by the PLBMF toolbox.

It includes and extends the following functionalities:
Wavelet leader and dyadic wavelet coefficient based multifractal analysis
- multifractal attributes (scaling exponents, log-cumulants, multifractal spectrum)
- stucture functions
- minimal regularity exponent
- preliminary wavelet domain fractional integration
Wavelet-domain block bootstrap for multifractal analysis
- time- (space-) block, and time-scale- (space-scale-) block bootstrap
- ordinary and double bootstrap methods
- several confidence intervals for structure functions and multifractal attributes
- tests for null hypotheses on the precise value of multifractal attributes
- test for the null hypothesis of time constancy of multifractal attributes (1d only)

The toolbox is self-contained and entirely written in MATLAB.

You can find a brief overview on (discrete) wavelet transform based empirical multifractal analysis in this reference [JI.2], and in this tutorial article on wavelet-based multifractal analysis (with main focus on continuous wavelet transform). Specific aspects and particularities in the multifractal analysis of images are treated in this reference [JI.5]. Another useful reference can be found in this review article [BC.4].
You may also want to have a look at my thesis, which includes a synthetic overview of wavelet-based multifractal analysis, and collects material on a large number of theoretical and practical issues in empirical multifractal analysis.

If you use the code in your work, please cite the following references:
"Bootstrap for empirical multifractal analysis" [JI.2 .bib]
"Multifractality tests using bootstrapped wavelet leaders" [JI.1 .bib]
"Wavelet leaders and bootstrap for multifractal analysis of images" [JI.5 .bib].

WLBMF toolbox
WLBMF example data


The codes are made freely available for non-commercial use only, specifically for research and teaching purposes, provided that the copyright headers at the head of each file are not removed, and suitable citation is made in published work using the routine(s). No responsibility is taken for any errors in the code. The headers in the files contain copyright dates and authors. In addition there is a copyright on the collection as made available here (September 2008).