Skip to main content
SpringerLink
Log in

Scaling and Wavelets: An Introductory Walk

  • Chapter
  • First Online:
Book cover Processes with Long-Range Correlations

Part of the book series: Lecture Notes in Physics ((LNP,volume 621))

Abstract

This chapter offers, first, an introductory walk through the notions related to scaling phenomena and intuitions behind are gathered to formulate a tentative definition. Second, it introduces the mathematical model of self-similar processes with stationary increments, understood as the canonical reference to describe scaling. Then, it shows how and why the wavelet transform constitutes a powerful and relevant tool for the analysis (detection, identification, estimation) of self-similarity. It is finally explained why self-similarity is too restrictive a model to account for the large variety of scaling encountered in empirical data and a review of the various models related to scaling— long range dependence, local Hölder regularity, fractal and multifractal processes, multiplicative or cascade processes— is proposed. Their interrelations and differences, as well as estimation issues, are discussed. A set of Matlab routines has beendev eloped to implement the wavelet-based analysis for scaling described here. It is available at www.ens-lyon.fr/~pabry.

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 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
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. Flandrin: ‘Point processes, long-range dependence and wavelets’. In: Wavelets in Medicine and Biology, ed. by A. Aldroubi and M. Unser (CRC Press, Boca Raton(FL) 1996) pp. 413–438

    Google Scholar 

  2. P. Abry, D. Veitch: IEEE Trans. on Info. Theory, 44, 2 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  3. P. Abry, L. Delbeke and P. Flandrin: IEEE-ICASSP-99, Phoenix (AZ) (1999)

    Google Scholar 

  4. P. Abry, P. Flandrin, M.S. Taqqu and D. Veitch: Wavelets for the analysis, estimation andsynthesis of scaling data A chapter in

    Google Scholar 

  5. P. Abry, B. Pesquet-Popescu, M. S. Taqqu: ‘Wavelet Based Estimators for Self Similar α-Stable Processes’. In: Int. Conf. on Signal Proc., 16th WorldComputer Congress, Beijing, China, 2000

    Google Scholar 

  6. P. Abry, P. Gonçcalvès and J. Lévy Véhel: Traité Information, Commandes, Contrôle, Hermès Editions, Paris, France, April (2002)

    Google Scholar 

  7. A. Arnéodo, J.F. Muzy, S.G. Roux: J. Phys. II France, 7, 363 (1997)

    Article  Google Scholar 

  8. R. Averkamp and C. Houdré: IEEE Trans. on Info. Theory, Vol. 44, 1111 (1998)

    Article  MATH  Google Scholar 

  9. V. Bareikis and R. Katilius: Noise in Physical Systems and 1/f Fluctuations (World Scientific 1995)

    Google Scholar 

  10. J. Beran: Statistics for Long-Memory Processes (Chapman and Hall, New York 1994)

    MATH  Google Scholar 

  11. S. Cambanis and C. Houdré: IEEE Trans. on Info. Theory, 41, 628 (1995)

    Article  MATH  Google Scholar 

  12. B. Castaing, Y. Gagne, E. Hopfinger: Physica D, 46, 177 (1990)

    Article  MATH  ADS  Google Scholar 

  13. B. Castaing: J. Phys. II France, 6, 105 (1996)

    Article  Google Scholar 

  14. P. Chainais, P. Abry, and J. F. Pinton: Phys. Fluids, 11, 3524 (1999)

    Article  ADS  MATH  Google Scholar 

  15. P. Chainais: Cascades log-infiniment divisibles et analyse multirésolution. Application à l’étude des intermittences en turbulence. PhD Thesis, Ecole Normale Supérieure de Lyon(2001)

    Google Scholar 

  16. P. Chainais, R. Riedi and P. Abry: ‘Infinitely Divisible Processes’. In: Stochastic processes and Their Applications (May 2002)

    Google Scholar 

  17. I. Daubechies: Ten Lectures on Wavelets (SIAM, Philadelphia 1992)

    Google Scholar 

  18. L. Delbeke and P. Abry: Stochastic Processes and their Applications, 86, 177 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  19. P. Frankhauser: Population, 4, 1005 (1997)

    Article  Google Scholar 

  20. U. Frisch: Turbulence. The legacy of A. Kolmogorov (Cambridge University Press, UK 1995)

    Google Scholar 

  21. A. C. Gilbert, W. Willinger and A. Feldmann: IEEE Trans. on Info. Theory 45, 971 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  22. P. Gonçcalvès and R. H. Riedi: Proc. 17ème Colloque GRETSI, Vannes, France (1999)

    Google Scholar 

  23. M. Keshner: proc. of the IEEE, 70, 212 (1982)

    Article  ADS  Google Scholar 

  24. A. N. Kolmogorov: ‘a) Dissipationof energy inthe locally isotropic turbulence, b) The local structure of turbulence in incompressible viscous fluid for very large Reynolds number, c) Ondegen erationof isotropic turbulence inanin compressible viscous liquid’. In: Turbulence, Classic papers on statistical theory, ed.by S.K. Friedlander and L. Topper (Interscience publishers 1941) pp. 151–161

    Google Scholar 

  25. W. E. Leland, M. S. Taqqu, W. Willinger, and D. V. Wilson: IEEE/ACM Trans. on Networking, 2, 1 (1994)

    Article  Google Scholar 

  26. L. Liebovitch and A. Todorov: The American Physiological Society, 1446 (1996)

    Google Scholar 

  27. B. B. Mandelbrot and J. W. Van Ness: SIAM Rev, 10, 422 (1968)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  28. B. B. Mandelbrot: J. of Fluid Mech., 62, 331 (1974)

    Article  MATH  ADS  Google Scholar 

  29. B. B. Mandelbrot: Fractales, Hasardet Finance (Flammarion 1997)

    Google Scholar 

  30. S. Mallat: A Wavelet Tour of Signal Processing (Academic Press, Boston 1997)

    Google Scholar 

  31. E. Masry: IEEE Trans. on Info. Theory, 39, 260 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  32. P. Flandrin: IEEE Trans. on Info. Theory, 38, 910 (1992)

    Article  MathSciNet  Google Scholar 

  33. P. Flandrin: ‘Fractional Brownian motion and wavelets’. In: Wavelets, Fractals, and Fourier Transforms ed. by M. Farge, J.C.R. Hunt and J.C. Vassilicos (Clarendon Press, Oxford 1993) pp. 109–122

    Google Scholar 

  34. B. Pesquet-Popescu: Signal Processing 75 (1999)

    Google Scholar 

  35. K. Park and W. Willinger: Self-Similar Network Traffic and Performance Evaluation (Wiley,Interscience Division 2000)

    Google Scholar 

  36. R. Riedi, M.S. Crouse, V.J. Ribeiro and R.G. Baraniuk: IEEE Trans. on Info. Theory, 45, 992 April (1999)

    Google Scholar 

  37. R. H. Riedi: ‘Multifractal processes’. In: Long range dependence: theory and applications ed. by Doukhan, Oppenheim and Taqqu (2002)

    Google Scholar 

  38. L. C. G. Rogers and D. Williams: Diffusions, Markov processes andmartingales, 2nd edition, vol 1 (Foundations, Wiley 1994)

    Google Scholar 

  39. M. Roughan, D. Veitch, and P. Abry: IEEE Trans. on Networking, 8, 467 (2000)

    Article  Google Scholar 

  40. M. Roughan, D. Veitch, J. Yates, M. Ashberg, H. Elgelid, M. Castro, M. Dwyer and P. Abry: ‘Real-Time Measurement of Long-Range Dependence in ATM Networks’. In: Passive andA ctive Measurement Workshop, Hamilton, New-Zealand 2000

    Google Scholar 

  41. G. Samorodnitsky and M. S. Taqqu: Stable Non-Gaussian Processes: Stochastic Models with Infinite Variance (Chapman and Hall, New York, London 1994)

    MATH  Google Scholar 

  42. A. Saucier: ‘Méthodes multifractales pour l’analyse d’images et de signaux’. In

    Google Scholar 

  43. D. Sornette: Critical Phenomena in Natural Sciences (Springer 2000)

    Google Scholar 

  44. A. Torcini and M. Antoni: Physical Review E, 59, 2746 (1999)

    Article  ADS  Google Scholar 

  45. M. Teich, S. Lowen, B. Jost and K. Vibe-Rheymer: Heart rate Variability: Measures end Models (preprint 2000)

    Google Scholar 

  46. A. H. Tewfik and M. Kim: IEEE Trans. Info. Theory, 38, 904 (1992)

    Article  MathSciNet  Google Scholar 

  47. D. Veitch and P. Abry: IEEE Trans. on Info. Theory, 45, 878, April (1999)

    Google Scholar 

  48. D. Veitch, P. Abry, P. Flandrin and P. Chainais: ‘Infinitely Divisible Cascade Analysis of Network Traffic Data’. In: Proceedings of ICASSP, Istanbul, June 2000

    Google Scholar 

  49. D. Veitch and P. Abry: ‘A statistical test for the constancy of scaling exponents’. In: itshape IEEE Trans. on Sig. proc. 2001

    Google Scholar 

  50. D. Veitch, P. Abry and M. S. Taqqu: On the automatic selection of the onset of scaling (Preprint 2002)

    Google Scholar 

  51. C. Walter: Lois d’échelle en finance in

    Google Scholar 

  52. B. West and A. Goldberger: American Scientist, 75, 354 (1987)

    ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CNRS, UMR 5672, Physics Lab., Ecole Normale Supérieure de Lyon, 46, allée d’Italie, F-69364, Lyon cedex 7, France

    Patrice Abry

Authors
  1. Patrice Abry
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Dept. of Mathematics, Indian Institute of Science, 560 012, Bangalore, India

    Govindan Rangarajan

  2. Center for Complex Systems and Brain Sciences, Florida Atlantic University, 33431, Boca Raton, FL, USA

    Mingzhou Ding

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Abry, P. (2003). Scaling and Wavelets: An Introductory Walk. In: Rangarajan, G., Ding, M. (eds) Processes with Long-Range Correlations. Lecture Notes in Physics, vol 621. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44832-2_3

Download citation

  • DOI: https://doi.org/10.1007/3-540-44832-2_3

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-40129-2

  • Online ISBN: 978-3-540-44832-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation