Skip to main content
SpringerLink
Log in

Wavelet Leaders in Multifractal Analysis

  • Conference paper
Book cover Wavelet Analysis and Applications

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

The properties of several multifractal formalisms based on wavelet coefficients are compared from both mathematical and numerical points of view. When it is based directly on wavelet coefficients, the multifractal formalism is shown to yield, at best, the increasing part of the weak scaling exponent spectrum. The formalism has to be based on new multiresolution quantities, the wavelet leaders, in order to yield the entire and correct spectrum of Hölder singularities. The properties of this new multifractal formalism and of the alternative weak scaling exponent multifractal formalism are investigated. Examples based on known synthetic multifractal processes are illustrating its numerical implementation and abilities.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. P. Abry, P. Goncalves and J. Lévy-Véhel Lois d’échelle, Fractales et ondelettes, Vol. 1 and 2, Hermes (2002).

    Google Scholar 

  2. P. Abry and F. Sellan, The wavelet-based synthesis for fractional Brownian motion proposed by F. Sellan and Y. Meyer: Remarks and fast implementation, Appl. Comput. Harmon. Anal. 3, No.4, pp. 377–383 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  3. P. Abry, S. Jaffard, and B. Lashermes. Revisiting scaling, multifractal, and multiplicative cascades with the wavelet leader lens. In Optic East, Wavelet Applications in Industrial Applications II, Vol. 5607, pages 103–117, Philadelphia, USA, 2004.

    Google Scholar 

  4. A. Arneodo, B. Audit, N. Decoster, J.-F. Muzy, C. Vaillant, Wavelet-based multifractal formalism: applications to DNA sequences, satellite images of the cloud structure and stock market data, in: The Science of Disasters; A. Bunde, J. Kropp, H. J. Schellnhuber eds., Springer pp. 27–102 (2002).

    Google Scholar 

  5. A. Arneodo, E. Bacry, S. Jaffard, J-F. Muzy, Oscillating singularities on Cantor sets: A grandcanonical multifractal formalism, J. Statist. Phys., vol. 87 pp. 179–209 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  6. A. Arneodo, E. Bacry, J-F. Muzy, The thermodynamics of fractals revisited with wavelets, Physica A Vol.213 p.232–275 (1995).

    Article  Google Scholar 

  7. A. Arneodo, E. Bacry, and J.-F. Muzy. Random cascades on wavelet dyadic trees. J. Math Phys., 39(8):4142–4164, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  8. J.-M. Bardet, G. Lang, G. Oppenheim, A. Philippe, and M. S. Taqqu. Generators of long-range dependent processes: a survey. In P. Doukhan, G. Oppenheim, and M. S. Taqqu, editors, Long-Range Dependence: Theory and Applications, pages 579–623, Boston, 2003. Birkhäuser.

    Google Scholar 

  9. J.-M. Aubry and S. Jaffard Random wavelet series Comm. Math. Phys., vol. 227, pp. 483–514 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  10. J. Barral and S. Seuret, From multifractal measures to multifractal wavelet series, preprint (2005).

    Google Scholar 

  11. H. Brezis, Analyse fonctionelle, Masson (1983).

    Google Scholar 

  12. G. Brown, G. Michon et J. Peyrière, On the multifractal analysis of measures, J. Statist. Phys., vol. 66 pp. 775–790 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  13. A. B. Chhabra and C. Meneveau and R. V. Jensen and K. R. Sreenivasan, Direct determination of the f(α) singularity spectrum and its application to fully developed turbulence, Physical Review A, 40(9), 1989, 5284–5294.

    Article  Google Scholar 

  14. E. Chassande-Mottin and P. Flandrin, On the time-frequency detection of chirps, Appl. Comput. Harmon. Anal., Vol. 6 pp. 252–281 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  15. A. Cohen and R. Ryan, Wavelets and multiscale signal processing, Chapman & Hall (1995).

    Google Scholar 

  16. D. Dacunha-Castelle and M. Duflo Probabilités et Statistiques Vol. 2 Masson (1993)

    Google Scholar 

  17. I. Daubechies. Orthonormal bases of compactly supported wavelets, Comm. Pure and App. Math. Vol. 41, pp. 909–996 (1988).

    MATH  MathSciNet  Google Scholar 

  18. P. Deheuvels and D. M. Mason Functional laws of the iterated logarithm for the increments of empirical and quantile processes, Ann. of Proba., Vol. 20 pp. 1248–1287 (1992).

    MATH  MathSciNet  Google Scholar 

  19. A. Durand Random wavelet series with correlated coefficients, preprint (2005).

    Google Scholar 

  20. K. Falconer Fractal geometry, John Wiley and sons (1990).

    Google Scholar 

  21. A. Fraysse, Résultats de généricité en analyse multifractale, Thèse de l’Université Paris XII (2005).

    Google Scholar 

  22. A. Grossmann, J. Morlet and T. Paul Transforms associated to square integrable group representations. I. General results, J. Math. Phys Vol 26(10) pp. 2473–2479 (1985).

    Article  MATH  MathSciNet  Google Scholar 

  23. T. Halsey, M. Jensen, L. Kadanoff, I. Procaccia and B. Shraiman, Fractal measures and their singularities: The characterization of strange sets, Phys. Rev. A, vol. 33 pp. 1141–1151 (1986)

    Article  MathSciNet  Google Scholar 

  24. G.H. Hardy, J. E. Littlewood, and G. Pólya Inequalities, Cambridge University Press (1952).

    Google Scholar 

  25. S. Jaffard, Exposants de Hölder en des points donnés et coefficients d’ondelettes, C. R. Acad. Sci. Sér. I Math., vol. 308 pp. 79–81 (1989).

    MATH  MathSciNet  Google Scholar 

  26. S. Jaffard Pointwise smoothness, two-microlocalization and wavelet coefficients, Publ. Mat. 35 (1991), 155–168.

    MATH  MathSciNet  Google Scholar 

  27. S. Jaffard, Multifractal formalism for functions. Part 1: Results valid for all functions and Part 2: Selfsimilar functions, SIAM J. Math. Anal. 28 (1997), 944–998.

    Article  MATH  MathSciNet  Google Scholar 

  28. S. Jaffard, Oscillation spaces: Properties and applications to fractal and multifractal functions, J. Math. Phys., vol. 39 pp. 4129–4141 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  29. S. Jaffard, Wavelet techniques in multifractal analysis, Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, M. Lapidus et M. van Frankenhuijsen Eds., Proceedings of Symposia in Pure Mathematics, AMS, Vol. 72 Part 2 pp. 91–152 (2004).

    Google Scholar 

  30. S. Jaffard, On Davenport expansions, Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, M. Lapidus et M. van Frankenhuijsen Eds., Proceedings of Symposia in Pure Mathematics, AMS, Vol. 72 Part 1 pp. 273–303 (2004).

    Google Scholar 

  31. S. Jaffard, Pointwise regularity associated with function spaces and multifractal analysis, Approximation and Probability, T. Figiel and A. Kamont Eds., Banach Center Pub. Vol. 72 pp. 93–110 (2006).

    Google Scholar 

  32. S. Jaffard and C. Melot, Wavelet analysis of fractal Boundaries, Part 1: Local regularity and Part 2: Multifractal formalism, Comm. Math. Phys., Vol. 258 n. 3, pp. 513–565 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  33. S. Jaffard and Y. Meyer. Wavelet methods for pointwise regularity and local oscillation of functions, Mem. Am. Math. Soc. 123 (1996), 587.

    MathSciNet  Google Scholar 

  34. S. Jaffard, Y. Meyer and R. Ryan Wavelets: Tools for Science and Technology, S.I.A.M., (2001)

    Google Scholar 

  35. J.-P. Kahane, Some Random Series of Functions, Cambridge University Press, 1985.

    Google Scholar 

  36. J.-P. Kahane, and J. Peyrière, Sur certaines martingales de Benoit Mandelbrot, Adv. Math., vol. 22, pp. 131–145 (1976).

    Article  MATH  Google Scholar 

  37. P. Kestener and A. Arneodo, Three-dimensional wavelet-based multifractal method: The need for revisiting the multifractal description of turbulence dissipation data, Physical Review Letters Vol 91 p. 194501, (2003).

    Article  Google Scholar 

  38. K.-S. Lau, Ka-Sing and S.-M. Ngai, Multifractal measures and a weak separation condition, Adv. Math. 141, No.1, 45–96 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  39. B. Lashermes, P. Abry, and P. Chainais. New insights into the es timation of scaling exponents. Int. J. of Wavelets, Multiresolution and Information Processing, 2(4):497–523, 2004.

    Article  MATH  Google Scholar 

  40. B. Lashermes, S. Jaffard, and P Abry. Wavelet Leader based Multifractal Analysis. In Proc. of the Int. Conf. on Acoust. Speech and Sig. Proc., 2005.

    Google Scholar 

  41. W. Li, Small ball probabilities for Gaussian Markov processes under the L p-norm, Stoch. Proc. Appl. Vol. 92, pp. 87–102 (2001).

    Article  MATH  Google Scholar 

  42. W. Li and Q.-M. Shao, Gaussian processes: Inequalities, small ball probabilities and applications, D. N. Shanbhag et al. ed., Stochastic processes: Theory and methods. Amsterdam: North-Holland/ Elsevier. Handb. Stat. Vol. 19, pp. 533–597 (2001).

    Google Scholar 

  43. S. Mallat A Wavelet tour of signal processing, Academic Press (1998)

    Google Scholar 

  44. S. Mallat and W. L. Hwang Singularity detection and processing with wavelets, IEEE Trans. Info. Theo. Vol. 38 pp. 617–643 (1992)

    Article  MathSciNet  Google Scholar 

  45. B. Mandelbrot, Intermittent turbulence in selfsimilar cascades: divergence of high moments and dimension of the carrier, J. Fluid Mech., vol. 62 pp. 331–358 (1974).

    Article  MATH  Google Scholar 

  46. C. Melot, Oscillating singularities in Besov spaces, J. Math. Pures Appl., Vol. 9,. Ser. 83, pp. 367–416 (2004).

    MathSciNet  Google Scholar 

  47. C. Meneveau, Analysis of turbulence in the orthonormal wavelet representation, J. Fluid Mech., 232, 469–520, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  48. C. Meneveau and K. R. Sreenivasan, The multifractal spectrum of th e dissipation field in turbulent flows, Nuclear Physics B, 2, 49–76, 1987.

    Article  Google Scholar 

  49. Y. Meyer, Ondelettes et opérateurs. Hermann, 1990.

    Google Scholar 

  50. Y. Meyer, Wavelets, Vibrations and Scalings, CRM Ser. AMS Vol. 9, Presses de l’Universitéde Montréal (1998).

    Google Scholar 

  51. Y. Meyer, F. Sellan and M. Taqqu, Wavelets, generalized white noise and fractional integration: The synthesis of fractional Brownian motion, J. Fourier Anal. Appl. 5, No.5, pp. 465–494 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  52. Y. Meyer and H. Xu. Wavelet analysis and chirps, Appl. Comput. Harmon. Anal. Vol. 4, pp. 366–379 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  53. D. Monrad and H. Rootzén Small values of Gaussian processes and functional laws of the iterated logarithm, Proba. Theo. Rel. Fields Vol. 101 p. 173–192 (1995).

    Article  MATH  Google Scholar 

  54. L. Olsen, A multifractal formalism, Adv. Math., vol. 116 pp. 92–195 (1995).

    Article  MathSciNet  Google Scholar 

  55. G. Parisi and U. Frisch, On the singularity structure of fully developed turbulence; appendix to Fully developed turbulence and intermittency, by U. Frisch; Proc. Int. Summer school Phys. Enrico Fermi, 84–88 North Holland (1985).

    Google Scholar 

  56. R. Riedi, An improved multifractal formalism and self-similar measures, J. Math. Anal. Appl. Vol. 189, No.2, pp. 462–490 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  57. S. Roux, B. Lashermes, P. Abry and S. Jaffard, Contributions l’étude des performances statistiques des estimateurs multifractals, XXIème colloque GRETSI, Louvain-la-Neuve, belgium, 2005.

    Google Scholar 

  58. K.R. Sreenivasan, Fractals and multifractals in turbulence, Ann. Rev. Fluid Mech., 23, 1991, 539–600.

    Article  MathSciNet  Google Scholar 

  59. W. Stute, The oscillation behavior of empir ical processes, Ann. Proba. Vol. 10, pp. 86–107 (1982).

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire d’Analyse et de Mathématiques Appliquées, Université Paris XII, 61 Avenue du Général de Gaulle, 94010, Créteil Cedex, France

    Stéphane Jaffard

  2. Institut Universitaire de France, France

    Stéphane Jaffard

  3. Laboratoire de Physique, École Normale Supérieure de Lyon and CNRS UMR 5672, 46, allée d’Italie, 69364, Lyon cedex, France

    Bruno Lashermes & Patrice Abry

Authors
  1. Stéphane Jaffard
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bruno Lashermes
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Patrice Abry
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Faculty of Science and Technology, University of Macau, P.O. Box 3001, Macau

    Tao Qian  & Mang I Vai  & 

  2. Department of Mathematics, Syracuse University, 215 Carnegie Hall, Syracuse, NY, 13244-1150, USA

    Yuesheng Xu

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Birkhäuser Verlag Basel/Switzerland

About this paper

Cite this paper

Jaffard, S., Lashermes, B., Abry, P. (2006). Wavelet Leaders in Multifractal Analysis. In: Qian, T., Vai, M.I., Xu, Y. (eds) Wavelet Analysis and Applications. Applied and Numerical Harmonic Analysis. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-7778-6_17

Download citation

Publish with us

Policies and ethics

Search

Navigation