Bayesian multifractal analysis toolboxes
The following toolboxes are mostly based on my publications (detailed here) and my PhD work (manuscript). They are entirely written in MATLAB. Each toolbox folder contains a demo file that illustrates the usage of the corresponding estimation procedure. Bayesian estimation for the multifractality parameter for univariate time series and images

PRESENTATION:
This first toolbox enables the wavelet leader based Bayesian estimation for the multifractality parameter (i.e., the intermittency parameter or second logcumumant, denoted c_{2} in all my publications) for univariate time series and images.
The proposed Bayesian model is based on a Gaussian model for the logarithm of recently introduced multiresolution quantities, the wavelet leaders. This model builds on a parametric covariance structure motivated by the asymptotic covariance of multiresolution quantities associated with multifractal multiplicative cascade based processes. The Bayesian estimators are approximated using a MetropoliswithinGibbs algorithm. During the inference, the Gaussian distribution is numerically evaluated using a spectral approximation, called Whittle approximation, which enables the use of the estimation procedure for large sample sizes.

RELATED REFERENCES:
If you use the code in your work, please cite the following references:
 S. Combrexelle, H. Wendt, N. Dobigeon, J.Y. Tourneret, S. McLaughlin, P. Abry, "Bayesian estimation of the multifractality parameter for image texture using a Whittle approximation," IEEE T. Image Proces., vol. 24, no. 8, pp. 25402551, Aug. 2015.
 S. Combrexelle, H. Wendt, J.Y. Tourneret, P. Abry, S. McLaughlin, "Bayesian estimation of the multifractality parameter for images via a closedform Whittle likelihood," 23rd European Signal Proces. Conf. (EUSIPCO), Nice, France, August 2015.
 H. Wendt, N. Dobigeon, J.Y. Tourneret, P. Abry, "Bayesian estimation for the multifractality parameter," IEEE Int. Conf. Acoust., Speech, and Signal Proces. (ICASSP), Vancouver, Canada, May 2013.
Details on theoretical and practical aspects of multifractal analysis along with a definition of wavelet leaders can be found in the following references: S. Jaffard. "Wavelet techniques in multifractal analysis." In M. Lapidus and M. van Frankenhuijsen, editors, Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot, Proc. Symp. Pure Math., volume 72(2), pages 91152. AMS, 2004.
 S. Jaffard, B. Lashermes, and P. Abry. "Wavelet leaders in multifractal analysis" . In T. Qian, M. I. Vai, and X. Yuesheng, editors, Wavelet Analysis and Applications, page 219264. Cambridge, 2006.
 H. Wendt, P. Abry, S. Jaffard, H. Ji, and Z. Shen. "Wavelet leader multifractal analysis for texture classification" . In Proc. IEEE Int. Conf. Image Proces. (ICIP), Cairo, Egypt, Nov. 2009.
 DOWNLOADS:
Bayesian estimation for the multifractality parameter and integral scale for univariate time series

PRESENTATION:
This second toolbox complements the above "Bayesian c2 Toolbox" and enables the wavelet leader based Bayesian joint estimation for the multifractality parameter and the integral scale of time series.
This approach is based on a modified generic statistical model that enables to also assess the integral scale of time series, jointly with the multifractality parameter.
Note: only implemented for time series.

RELATED REFERENCES:
If you use the code in your work, please cite the following references:
 S. Combrexelle, H. Wendt, P. Abry, N. Dobigeon, S. McLaughlin, J.Y. Tourneret, "A Bayesian approach for the joint estimation of the multifractality parameter and integral scale based on the Whittle approximation," IEEE Int. Conf. Acoust., Speech, and Signal Proces. (ICASSP), Brisbane, Australia, April 2015.
 H. Wendt, N. Dobigeon, J.Y. Tourneret, P. Abry, "Bayesian estimation for the multifractality parameter," IEEE Int. Conf. Acoust., Speech, and Signal Proces. (ICASSP), Vancouver, Canada, May 2013.
 DOWNLOADS:
Bayesian multifractal analysis for multivariate images and time series

PRESENTATION:
Multifractal analysis has been successfully used in various fields of signal and image processing. Yet, its application remains so far conceptually limited to the independent analysis of one single component. This toolbox enables a Bayesian estimation for the multifractal analysis of multivariate time series/images (implemented cases: ndimensional sets of timeseries/images for n=1,2,3).
The proposed Bayesian approach starts from the univariate statistical model for logleaders considered in the previous toolboxes, on which we build multivariate Bayesian models jointly describing the collection of multifractal parameters via the design of suitable multivariate prior distributions enforcing smoothness. For the collection of logcumulants c_{2}, we consider a gamma Markov random field prior. Unlike in the previous toolboxes, the first logcumulant c_{1} is here considered, and we assign to the collection of logcumulants c_{1} a simultaneously autoregressive prior.
Finally, the combination of these priors with a data augmented version of our statistical model leads to a posterior distribution which is associated with standard conditional distributions. Bayesian estimators are computed using an efficient Gibbs sampler.
Note: This toolbox also works for the univariate processing of a single time series or image. It can be used in place of the first toolbox "Bayesian c2 Toolbox", with a reduced computational cost thanks to a data augmentation approach.

RELATED REFERENCES:
Technical details can be found directly in my PhD manuscript and a journal paper to be submitted:
 S. Combrexelle, H. Wendt, Y. Altmann, J.Y. Tourneret, S. McLaughlin, P. Abry, "Multifractal analysis of multivariate images using gamma Markov random field priors"
Illustrations of the methodology can be found in the following conference papers: S. Combrexelle, H. Wendt, Y. Altmann, J.Y. Tourneret, S. McLaughlin, P. Abry, "Bayesian joint estimation of the multifractality parameter of image patches using gamma Markov random fied priors," IEEE Int. Conf. on Image Proces. (ICIP), Phoenix, Arizona, USA, September 2016.
 S. Combrexelle, H. Wendt, Y. Altmann, J.Y. Tourneret, S. McLaughlin, P. Abry, "Bayesian estimation for the local assessment of the multifractality parameter of multivariate time series," 24th European Signal Proces. Conf. (EUSIPCO), Budapest, Hungary, August 2016.
 S. Combrexelle, H. Wendt, J.Y. Tourneret, S. McLaughlin, P. Abry, "Bayesian multifractal analysis of multitemporal images using smooth priors," IEEE Workshop on Statistical Signal Proces. (SSP), Palma de Mallorca, Spain, June 2016.
 S. Combrexelle, H. Wendt, J.Y. Tourneret, Y. Altmann, S. McLaughlin, P. Abry, "A Bayesian Approach for the Multifractal Analysis of SpatioTemporal Data," Int. Conf. Systems, Signals and Image Proces. (IWSSIP), Bratislava, Slovakia, May 2016.
Details on gamma Markov random field priors, simultaneously autoregressive priors and data augmentation can be found in: O. Dikmen and A.T. Cemgil. "Gamma Markov random elds for audio source modeling." IEEE Trans. Audio, Speech, and Language Proces., 18(3):589601, March 2010.
 N. Cressie. "Statistics for spatial data." John Wiley & Sons, 2015.
 S. Combrexelle, H. Wendt, Y. Altmann, J.Y. Tourneret, S. McLaughlin, P. Abry, "A Bayesian framework for the multifractal analysis of images using data augmentation and a Whittle approximation," IEEE Int. Conf. Acoust., Speech, and Signal Proces. (ICASSP), Shanghai, China, March 2016.
 DOWNLOADS: