Statistical Methods & Applications

, Volume 21, Issue 3, pp 251–277 | Cite as

On wavelet analysis of the nth order fractional Brownian motion

  • Hedi Kortas
  • Zouhaier Dhifaoui
  • Samir Ben Ammou


In this paper, we investigate the use of wavelet techniques in the study of the nth order fractional Brownian motion (n-fBm). First, we exploit the continuous wavelet transform’s capabilities in derivative calculation to construct a two-step estimator of the scaling exponent of the n-fBm process. We show, via simulation, that the proposed method improves the estimation performance of the n-fBm signals contaminated by large-scale noise. Second, we analyze the statistical properties of the n-fBm process in the time-scale plan. We demonstrate that, for a convenient choice of the wavelet basis, the discrete wavelet detail coefficients of the n-fBm process are stationary at each resolution level whereas their variance exhibits a power-law behavior. Using the latter property, we discuss a weighted least squares regression based-estimator for this class of stochastic process. Experiments carried out on simulated and real-world datasets prove the relevance of the proposed method.


nth order fBm Scaling exponent Wavelet transform Derivative operator Signal-to-noise ratio Weighted least squares estimator 


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  1. Abry P, Flandrin P, Taqqu MS, Veitch D (2000) Wavelets for the analysis, estimation, and synthesis of scaling data. In: Park K, Willinger W (eds) Self-similar network traffic and performance evaluation. Wiley, New York, pp 39–88CrossRefGoogle Scholar
  2. Abry P, Flandrin P, Taqqu MS, Veitch D (2003) Self-similarity and long-range dependence through the wavelet lens. In: Doukhan P, Oppenheim G, Taqqu MS (eds) Long-Range Dependence: Theory and Applications, Birkhäuser, Boston, pp 527–556Google Scholar
  3. Bel L, Oppenheim G, Robbiano L, Viano MC (1998) Distribution processes with stationary fractional increments. In: Proceedings of the colloquium FDS’98: fractional differential systems: models, methods and applications ESAIM: Proceedings 5, pp 43–54Google Scholar
  4. Beran J (1994) Statistics for long-memory processes, monographs on statistics and applied probability, vol 61. Chapman & Hall, New YorkGoogle Scholar
  5. Biermé H (2005) Champs Aléatoires, Autosimilarité, Anisotropie et Etude Directionnelle. Ph.D. thesis, Université d’OrléansGoogle Scholar
  6. Brockwell PJ, Davis RA (1991) Time series: theory and methods, 2 edn. Springer, New YorkCrossRefGoogle Scholar
  7. Brouste A, Istas J, Lambert-Lacroix S (2007) On fractional gaussian random fields simulations. J Stat Softw 23: 1–23Google Scholar
  8. Coeurjolly J-F (2001) Estimating the parameters of a fractional Brownian motion by discrete variations of its sample paths. Stat Inference Stoch Process 4: 199–227MathSciNetCrossRefzbMATHGoogle Scholar
  9. Daubechies I (1992) Ten lectures on wavelets. CBMS-NSF regional conference series on applied mathematics, SIAM, Philadelphia, PA, p 61Google Scholar
  10. Dickey DA, Fuller WA (1981) The likelihood ratio statistics for autoregressive time series with a unit root. Econometrica 49: 1057–1072MathSciNetCrossRefzbMATHGoogle Scholar
  11. Dietrich CR, Newsam GN (1997) Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix. SIAM J Sci Comput 18: 1088–1107MathSciNetCrossRefzbMATHGoogle Scholar
  12. Flandrin P (1992) Wavelet analysis and synthesis of fractional Brownian motion. IEEE Trans Inf Theory 38: 910–917MathSciNetCrossRefzbMATHGoogle Scholar
  13. Gupta A, Joshi SD, Prasad S (2005) A new approach for estimation of statistically matched wavelet. IEEE Trans Signal Process 53: 1778–1793MathSciNetCrossRefGoogle Scholar
  14. Handy CR, Murenzi R (1998) Moment-wavelet quantization: a first principles analysis of quantum mechanics through continuous wavelet transform theory. Phys Lett A 248: 7–15CrossRefzbMATHGoogle Scholar
  15. Hwang W-L (1999) Estimation of fractional Brownian motion embedded in a noisy environment using nonorthogonal wavelets. IEEE Trans Signal Process 47: 2211–2219MathSciNetCrossRefzbMATHGoogle Scholar
  16. Istas J, Lang G (1997) Quadratic variations and estimation of the local Hölder index of a gaussian process. Annales de l’institut Henri Poincaré (B) Probabilités et Statistiques 33: 407–436MathSciNetCrossRefzbMATHGoogle Scholar
  17. Kwiatkowski D, Phillips P, Schmidt P, Shin Y (1992) Testing the null hypothesis of stationary against the alternative of a unit root: how sure are we that economic time series have a unit root?. J Econom 54: 159–178CrossRefzbMATHGoogle Scholar
  18. Liu H-F, Yang Y-Z, Dai Z-H, Yu Z-H (2003a) The largest Lyapunov exponent of chaotic dynamical system in scale space and its application. Chaos 13: 839–844MathSciNetCrossRefzbMATHGoogle Scholar
  19. Liu H-F, Li W-F, Dai Z-H, Yu Z-H (2003b) The dimension of chaotic dynamical system in wavelet space and its application. Phys Lett A 316: 44–54MathSciNetCrossRefzbMATHGoogle Scholar
  20. Loussot T, Harba R, Jacquet G, Benhamou CL, Lespesailles E, Julien A (1996) An oriented fractal analysis for the characterization of texture: application to bone radiographs. EUSIPCO Signal Process 1: 371–374Google Scholar
  21. Mallat S (1989) A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans Pattern Anal Machine Intell 11: 674–693CrossRefzbMATHGoogle Scholar
  22. Mallat S (1998) A wavelet tour of signal processing. Academic Press, New YorkzbMATHGoogle Scholar
  23. Mallat S, Hwang WH (1992) Singularity detection and processing with wavelets. IEEE Trans Inf Theory 38: 617–643MathSciNetCrossRefzbMATHGoogle Scholar
  24. Mandelbrot BB (1999) Multifractals and 1/f noise: wild self-affinity in physics. Springer, New YorkGoogle Scholar
  25. Mandelbrot BB, Van Ness JW (1968) Fractional Brownian motion, fractional noises and applications. SIAM Rev 10: 422–438MathSciNetCrossRefzbMATHGoogle Scholar
  26. Meyer Y (1992) Wavelets and operators (trans: DH Salinger). Cambridge University, CambridgeGoogle Scholar
  27. Mielniczuk J, Wojdyllo P (2007) Estimation of Hurst exponent revisited. Comput Stat Data Anal 51: 4510–4525MathSciNetCrossRefzbMATHGoogle Scholar
  28. Parra C, Iftekharuddin K, Rendon D (2003) Wavelet based estimation of the fractal dimension in fBm images. In: First International IEEE EMB Conference on Neural Engineering. Conference Proceedings, pp 533–536Google Scholar
  29. Pérez DG, Zunino L, Garavaglia M, Rosso OA (2006) Wavelet entropy and fractional Brownian motion time series. Phys A 365: 282–288CrossRefGoogle Scholar
  30. Perrin E, Harba R, Berzin-Joseph C, Iribarren I, Bonami A (2001) Nth order fractional Brownian motion and fractional Gaussian noises. IEEE Trans Signal Process 49: 1049–1059CrossRefGoogle Scholar
  31. Perrin E, Harba R, Jennane R, Iribarren I (2002) Fast and exact synthesis for 1-D fractional Brownian motion and fractional gaussian noises. IEEE Signal Process Lett 9: 382–384CrossRefGoogle Scholar
  32. Pesquet-Popescu B, Larzabal P (1997) Higher order and lower order properties of the Wavelet decomposition of self similar process. IEEE signal processing workshop on higher-order statistics (SPW-HOS ’97), spwhos, pp 4–58Google Scholar
  33. Phillips PCB, Perron P (1988) Testing for unit roots in time series regression. Biometrica 75: 335–346MathSciNetCrossRefzbMATHGoogle Scholar
  34. Power GJ, Turvey GC (2010) Long-range dependence in the volatility of commodity futures prices: wavelet-based evidence. Phys A 389: 79–90CrossRefGoogle Scholar
  35. Sembiring J, Soemintapoera K, Kobayachi T, Akizuki K (2003) Diffusive representation of Nth order fractional brownian motion. In: Proceedings of the 13th IFAC symposium on system identification, Rotterdam, the Netherlands, 1, pp 181–186Google Scholar
  36. Shao X, Ma C (2003) A general approach to derivative calculation using wavelet transform. Chemom Intell Lab Syst 69: 157–165CrossRefGoogle Scholar
  37. Stoev S, Taqqu MS (2005) Path properties of the linear multifractional stable motion. Fractals 13: 157–178MathSciNetCrossRefGoogle Scholar
  38. Tewfik AH, Kim M (1992) Correlation structure of the discrete wavelet coefficients of fractional Brownian motion. IEEE Trans Inf Theory 38: 904–909MathSciNetCrossRefzbMATHGoogle Scholar
  39. Trimech A, Kortas H, Benammou S, Benammou S (2009) Multiscale Fama-French model: application to the French market. J Risk Finance 10: 179–192CrossRefGoogle Scholar
  40. Veitch D. (2001) MATLAB code for estimation of scaling exponents. Available at:
  41. Veitch D, Abry P (1999) A wavelet-based joint estimator of the parameters of long-range dependence. IEEE Trans Inf Theory 45: 878–897MathSciNetCrossRefzbMATHGoogle Scholar
  42. Zhang L, Bao P, Wu X (2004) Wavelet estimation of fractional Brownian motion embedded in a noisy environment. IEEE Trans Inf Theory 50: 2194–2200MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Hedi Kortas
    • 1
  • Zouhaier Dhifaoui
    • 1
  • Samir Ben Ammou
    • 2
  1. 1.Higher Institute of ManagementSousseTunisia
  2. 2.Computational Mathematics LaboratoryFaculty of ScienceMonastirTunisia

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