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Multifractal Estimators of Long Range Dependences

Danuta Makowiec 

Gdansk University, Institute of Theoretical Physics and Astrophysics, (IFTiA UG), Wita Stwosza 57, Gdańsk 80-952, Poland

Abstract
Multifractal approach is a  generalization of fractal techniques. It offers an attractive and compact frame to qualify and quantify multiscaling aspects immersed in any system represented by time series.  A lot of  simulations  were performed to validate numerical procedures (Abry et al., Stoev et al., Oświęcimka et al., Lashermes et al., Makowiec et al. 2009, Makowiec et al. 2010). However  available tools still seem to be  not powerful enough to discern reliably all specificity of the real data organization. One can find  a lot of advice how to overcome  possible traps. For example, it is advised to investigate series that are long enough (Oświęcimka et al.),  to use  parallelly  tools which are of different numerical origin (Kantelhardt). As a good rule of thumb is to question  results  if a multifractal spectrum is  concentrated at  h=1 or  h=0 (Gao et al.). Finally, it is said that specific knowledge about the phenomenon from which a signal is analyzed should be carefully taken into account (Veitch et al., Stoev et al.). Therefore multifractal estimates demand special protocols describing how to proceed with systematic errors of numerical methods used, and how to extract reliable fractal characterization of a given data.

We want to present the  method which consists in working together on two multifractal spectra received numerically from a given signal. The first spectrum describes the  signal itself while the second spectrum is calculated for the integrated signal. By such approach  each  signal is analyzed twice: if it is a stochastic walk (a direct signal), and if it is a noise (when integrated). Moreover,  the scaling options for the partition functions are kept firmly within the particular time interval to detect phenomena related to that time interval. Our proposition bases ob wide simulation experiments  in which the software of Physionet (i.e.,  multifractal.c  and  dfa.c modified  to receive the multifractal scaling)  was applied to  synthetic data.

Abry B, Jaffard S and Lashermes B 2004, Revisiting scaling, multifractal, and multiplicative cascades with wavelet leader lens  Optic East, Wavelet Applications in Industrial Applications II vol.  5607 103, Philadelphia, USA, 2004.

Gao J, Cao Y, Tung W-W, Hu J 2007,  Multiscale analysis of complex time series. Integration of Chaos and random Fractal Theory, and Beyond  (John Wiley\& Sons, Inc).

Kantelhardt J W  2009, Fractal and multifractal time series in  Encyclopedia of Complexity and Systems Science Meyers R A(Ed.), Springer.

Lashermes B,  Roux S G,  Abry P  and Jaffard J 2008, Comprehensive multifractal analysis of turbulent velocity using the wavelet leaders  Eur.Phys.J B  61 201.
Makowiec D, Dudkowska A, Gałąska R and Rynkiewicz A 2009, Multifractal estimates of monofractality in RR-heart series in power spectrum ranges  Physica A 388 3486.
Makowiec D and Fuliński A 2010, Multifractal Detrended Fluctuation Analysis as the Estimator of Long-Range Dependence.  Acta Phys. Pol. B  41 1025.
Oświęcimka P, Kwapień J and Drożdż Z 2006, Wavelet versus detrended fluctuation analysis of multifractal structures  Phys. Rev. E  74 016103.
Stoev S, Taqqu M S, Park Ch and Marron J S 2005, On the wavelet spectrum diagnostic for Hurst parameter estimation in the analysis of Internet traffic  Comp.Networks  48 423.
Veitch D, Hohn N and Abry P 2005, Multifractality in TCP\ IP traffic: the case against
 Comp. Networks  48 293.
 

Auxiliary resources (full texts, presentations, posters, etc.)
  1. PRESENTATION: Multifractal Estimators of Long Range Dependences, PDF document, version 1.4, 9.2MB
 

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Presentation: Oral at 5 Ogólnopolskie Sympozjum "Fizyka w Ekonomii i Naukach Społecznych", by Danuta Makowiec
See On-line Journal of 5 Ogólnopolskie Sympozjum "Fizyka w Ekonomii i Naukach Społecznych"

Submitted: 2010-10-15 16:24
Revised:   2010-10-15 16:50