Wavelet Leaders: A new method to estimate the multifractal singularity spectra

Wavelet Leaders is a novel alternative based on wavelet analysis for estimating the Multifractal Spectrum. It was proposed by Jaffard and co-workers improving the usual wavelet methods. In this work, we analyze and compare it with the well known Multifractal Detrended Fluctuation Analysis. The latter is a comprehensible and well adapted method for natural and weakly stationary signals. Alternatively, Wavelet Leaders exploits the wavelet self-similarity structures combined with the Multiresolution Analysis scheme. We give a brief introduction on the multifractal formalism and the particular implementation of the above methods and we compare their effectiveness. We expose several cases: Cantor measures, Binomial Multiplicative Cascades and also natural series from a tonic–clonic epileptic seizure. We analyze the results and extract the conclusions.

[1]  R. Jensen,et al.  Direct determination of the f(α) singularity spectrum , 1989 .

[2]  E. Bacry,et al.  Wavelets and multifractal formalism for singular signals: Application to turbulence data. , 1991, Physical review letters.

[3]  U. Frisch.  FULLY DEVELOPED TURBULENCE AND INTERMITTENCY , 1980 .

[4]  H. Stanley,et al.  On growth and form : fractal and non-fractal patterns in physics , 1986 .

[5]  E. Bacry,et al.  BEYOND CLASSICAL MULTIFRACTAL ANALYSIS USING WAVELETS: UNCOVERING A MULTIPLICATIVE PROCESS HIDDEN IN THE GEOMETRICAL COMPLEXITY OF DIFFUSION LIMITED AGGREGATES , 1993 .

[6]  Multifractal fluctuations in seismic interspike series , 2005, cond-mat/0502168.

[7]  V. Plerou,et al.  Scale invariance and universality: organizing principles in complex systems , 2000 .

[8]  Patrice Abry,et al.  Comprehensive multifractal analysis of turbulent velocity using the wavelet leaders , 2008 .

[9]  P. Grassberger,et al.  14. Estimating the fractal dimensions and entropies of strange attractors , 1986 .

[10]  H. Stanley,et al.  On Growth and Form , 1985 .

[11]  H. Stanley,et al.  Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series. , 1995, Chaos.

[12]  H. Stanley,et al.  Multifractal Detrended Fluctuation Analysis of Nonstationary Time Series , 2002, physics/0202070.

[13]  Jensen,et al.  Direct determination of the f( alpha ) singularity spectrum and its application to fully developed turbulence. , 1989, Physical review. A, General physics.

[14]  Stéphane Jaffard,et al.  Multifractal formalism for functions part I: results valid for all functions , 1997 .

[15]  P. Grassberger.  On the Hausdorff dimension of fractal attractors , 1981 .

[16]  C. Chui,et al.  Wavelets : theory, algorithms, and applications , 1994 .

[17]  E. Bacry,et al.  Singularity spectrum of fractal signals from wavelet analysis: Exact results , 1993 .

[18]  Yuechao Wu,et al.  Detrended fluctuation analysis of human brain electroencephalogram , 2004 .

[19]  H E Stanley,et al.  Statistical properties of DNA sequences. , 1995, Physica A.

[20]  S. Jaffard,et al.  Wavelet Leaders in Multifractal Analysis , 2006 .

[21]  P. A. Prince,et al.  Anomalous fluctuations in the dynamics of complex systems: from DNA and physiology to econophysics , 1996 .

[22]  O. Rosso,et al.  Study of EEG Brain Maturation Signals with Multifractal Detrended Fluctuation Analysis , 2007 .

[23]  Kenneth Falconer,et al.  Fractal Geometry: Mathematical Foundations and Applications , 1990 .

[24]  H. Stanley,et al.  Magnitude and sign correlations in heartbeat fluctuations. , 2000, Physical review letters.

[25]  H. Preissl,et al.  Detrended fluctuation analysis of short datasets : An application to fetal cardiac data , 2007 .

[26]  S. Mallat.  A wavelet tour of signal processing , 1998 .

[27]  J. Moreira,et al.  Roughness exponents to calculate multi-affine fractal exponents , 1997 .

[28]  J. Kwapień,et al.  Wavelet versus detrended fluctuation analysis of multifractal structures. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[29]  P. Grassberger,et al.  Measuring the Strangeness of Strange Attractors , 1983 .

[30]  K. Falconer.  Techniques in fractal geometry , 1997 .

[31]  Alejandra Figliola,et al.  A multifractal approach for stock market inefficiency , 2008 .

[32]  Luciano Pietronero,et al.  FRACTALS IN PHYSICS , 1990 .

[33]  Stéphane Jaffard,et al.  Some Mathematical Results about the Multifractal Formalism for Functions , 1994 .

[34]  B. Mandelbrot.  A Multifractal Walk down Wall Street , 1999 .