Skip to main content
SpringerLink
Log in

Statistical Applications of Wavelets

  • Reference work entry
Computational Complexity

  • 267 Accesses

Article Outline

Glossary

Definition of the Subject

Introduction

Decomposition View of Random Processes

Wavelets

Signal Estimation or Denoising

2-D Extensions of Wavelets

Covariance Estimation Using Wavelets

Future Directions

Bibliography

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 1,500.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 1,399.99
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

Abbreviations

Locally stationary process:

A stochastic process that when sampled in time, its sampled points have joint characteristics in a short time window that only depend on the difference between the sampled time points, and not the global time point of the samples within the time window.

Multiresolution analysis:

The analysis of a phenomenon of interest over different temporal or spatial scales and locations.

Shrinkage:

A method of estimation where the initialestimator is decreased by multiplication with a factor of magnitude less than or equal to one. If the object being estimated is small, then a reduced variance will arise, that sets off the increased bias in estimation.

Sparsity:

Sparsity in a vector corresponds to the support of the vector being limited. The sparsity can either be strict, i. e. the vector is perfectly supported in a small subset of all its entries, or the magnitudes of the entries decay sufficiently rapidly, if ordered in magnitude.

Stationary process:

A stochastic process that when sampled in time, its sampled points have joint characteristics that only depend on the difference between the sampled time points, and not the global time point of the samples.

Thresholding:

A method of estimation where the initialestimator is set to zero if it does not exceed a given threshold. Thresholding is a special case of shrinkage.

Wavelet and scaling functions:

The functions that the wavelet and scaling filters converge to in increasing scale, when using discrete wavelet filters. These are not for any finite scale equivalent to the wavelet and scaling filters.

Wavelet and scaling filters:

The high and low pass digital filters that are used to calculate the discrete wavelet transform of a given sequence or vector. These are a pair of quadrature mirror filters that satisfy the perfect reconstruction property.

Bibliography

  1. AntoniadisA, Paparoditis E, Sapatinas T (2006) A functional wavelet‐kernel approach for time series prediction. J Roy Stat Soc B68:837–857

    Article  MathSciNet  MATH  Google Scholar 

  2. Antoniadis A, Sapatinas T (2003) Wavelet methods for continuous‐timeprediction using Hilbert‐valued autoregressive processes. J Multivar Anal 87:133–158

    Article  MathSciNet  MATH  Google Scholar 

  3. Averkamp R, Houdre C (1998) Some distributional properties of the continuouswavelet transform of random processes. IEEE Trans Info Theory 44:1111–1124

    Article  MathSciNet  MATH  Google Scholar 

  4. Averkamp R, Houdre C (2000) A note on the discrete wavelet transform ofsecond‐order processes. IEEE Trans Info Theory 46:1673–1676

    Article  MathSciNet  MATH  Google Scholar 

  5. Bellegem SV, Fryzlewicz P, von Sachs R (2003) A wavelet‐based model forforecasting non‐stationary processes. Inst Phys Conf Ser 173:955–958

    Google Scholar 

  6. Buresti G, Lombardi G, Bellazzini J (2004) On the analysis of fluctuatingvelocity signals through methods based on the wavelet and hilbert transforms. Chaos Solitons Fractals 20:149–158

    Article  MathSciNet  MATH  Google Scholar 

  7. Cai T, Silverman BW (2001) Incorporating information on neighbouring coefficientsinto wavelet estimation. Sankhyā Ser B 63:127–148

    MathSciNet  MATH  Google Scholar 

  8. Cambanis S, Houdre C (1995) On the continuous wavelet transform ofsecond‐order random processes. IEEE Trans Info Theory 41:628–642

    Article  MathSciNet  MATH  Google Scholar 

  9. Candès E, Donoho DL (1999) Ridgelets: a key to higher‐dimensionalintermittency? Philos Trans: Math Phys Eng Sci 357:2495–2509

    Google Scholar 

  10. Daubechies I (1988) Orthonormal bases of compactly supported wavelets. CommPure Appl Math 41:909–996

    Article  MathSciNet  MATH  Google Scholar 

  11. Daubechies I (1992) Ten lectures on wavelets. SIAM,Philadelphia

    Book  MATH  Google Scholar 

  12. Daubechies I, Sweldens W (1998) Factoring wavelet transforms into liftingsteps. J Fourier Anal Appl 4:247–269

    Article  MathSciNet  MATH  Google Scholar 

  13. Delprat N, Escudié B, Guillemain P, Kronland-Martinet R, Tchamitchian P,Torresani B (1992) Asymptotic wavelet and gabor analysis: extraction of instantaneous frequencies. IEEE Trans Inform Theory38:644–64

    Google Scholar 

  14. Donoho DL (2006) Compressed sensing. IEEE Trans Inf Theory52:1289–1306

    Article  MathSciNet  Google Scholar 

  15. Donoho DL, Johnstone IM (1994) Ideal spatial adaptation by waveletshrinkage. Biometrika 81:425–455

    Article  MathSciNet  MATH  Google Scholar 

  16. Donoho DL, Johnstone IM (1995) Adapting to unknown smoothness via waveletshrinkage. J Am Stat Assoc 90:1200–1224

    Article  MathSciNet  MATH  Google Scholar 

  17. Donoho DL, Johnstone IM, Kerkyacharian G, Picard D (1995) Waveletshrinkage – asymptotia. J Roy Stat Soc B 57:301–337

    MathSciNet  MATH  Google Scholar 

  18. Downie TR, Silverman BW (1998) The discrete multiple wavelet transform andthreshold methods. IEEE Trans Signal Process 46:2558–2561

    Article  Google Scholar 

  19. Dragotti PL, Vetterli M (2003) Wavelet footprints: theory, algorithms, andapplications. IEEE Trans Signal Process 51:1306–1323

    Article  MathSciNet  Google Scholar 

  20. Fryzlewicz P, Bellegem SV, von Sachs R (2003) Forecasting non‐stationarytime series by wavelet process modelling. Ann Inst Stat Math 55:737–764

    Article  MATH  Google Scholar 

  21. Fryzlewicz P, Nason GP (2004a) A Haar–Fisz algorithm for poissonintensity estimation. J Comp Graph Stat 13:621–638

    Article  MathSciNet  Google Scholar 

  22. Fryzlewicz P, Nason GP (2004b) A Haar–Fisz algorithm for poissonintensity estimation. J Comp Graph Stat 13:621–638

    Article  MathSciNet  Google Scholar 

  23. Fryzlewicz P, Nason GP (2006) Haar–Fisz estimation of evolutionarywavelet spectra. J Roy Stats Soc B 68:611–634

    Article  MathSciNet  MATH  Google Scholar 

  24. Grossman A, Morlet J (1984) Decomposition of Hardy functions in squareintegrable wavelets of constant shape. Siam J Math Anal 15:723–736

    Article  MathSciNet  Google Scholar 

  25. Guérin C-A (2000) Wavelet analysis and covariance structure of someclasses of non‐stationary processes. J Fourier Anal Appl 6:403–425

    Google Scholar 

  26. Haar A (1910) On the theory of orthogonal function systems. Math Ann69:331–371

    Article  MathSciNet  MATH  Google Scholar 

  27. Holschneider M (1995) Wavelets: an analysis tool. Oxford University Press,Oxford

    MATH  Google Scholar 

  28. Jaffard S, Melot S (2005) Wavelet analysis of fractal boundaries. Commun MathPhys 258:513–159

    Article  MathSciNet  MATH  Google Scholar 

  29. Johnstone I, Silverman BW (2005) Empirical Bayes selection of waveletthresholds. Ann Stat 33:1700–1752

    Article  MathSciNet  MATH  Google Scholar 

  30. Johnstone IM, Silverman BW (1997) Wavelet threshold estimators for data withcorrelated noise. J Roy Stat Soc Ser B 59:319–351

    Article  MathSciNet  MATH  Google Scholar 

  31. Kim IK, Kim YY (2005) Damage size estimation by the continuous wavelet ridgeanalysis of dispersive bending waves in a beam. J Sound Vib 287:707–722

    Article  MATH  Google Scholar 

  32. Krim H, Pesquet J-C (1995) Multiresolution analysis of a class of nonstationaryprocesses. IEEE Trans Inf Theory 41:1010–1020

    Article  MATH  Google Scholar 

  33. le Pennec E, Mallat S (2005) Sparse geometric image representations. IEEETrans Image Process 14:423–438

    Article  MathSciNet  Google Scholar 

  34. Li T-H, Oh H-S (2002) Wavelet spectrum and its characterization property forrandom processes. IEEE Trans Inf Theory 48:2922–2937

    Article  MathSciNet  MATH  Google Scholar 

  35. Lilly JM, Gascard J-C (2006) Wavelet ridge diagnosis of time‐varyingelliptical signals with application to an oceanic eddy. Nonlinear Process Geophys 13:467–483

    Article  Google Scholar 

  36. Loève M (1945) Sur le fonctions aléatoire de second ordre. Rev Sci83:297–303

    Google Scholar 

  37. Mallat S (1999) A Wavelet Tour of Signal Processing, 2nd edn. Academic Press,New York

    MATH  Google Scholar 

  38. McCoy EJ, Walden AT (1996) Wavelet analysis and synthesis of stationarylong‐memory processes. J Comp Graph Stat 5:26–56

    MathSciNet  Google Scholar 

  39. Moulines E, Roueff F, Taqqu MS (2007) On the spectral density of the waveletcoefficients of long‐memory time series with application to the log‐regression estimation of the memory parameter. J Time Ser Anal28:155–187

    Article  MathSciNet  MATH  Google Scholar 

  40. Nason GP, von Sachs R (1999) Wavelets in time‐series analysis. PhilosTrans Math Phys Eng Sci 357:2511–2526

    Article  MATH  Google Scholar 

  41. Nason GP, von Sachs R, Kroisandt G (2000) Wavelet processes and adaptiveestimation of the evolutionary wavelet spectrum. J Roy Stat Soc B 62:271–292

    Article  Google Scholar 

  42. Olhede S (2007) Hyperanalytic denoising. IEEE Trans Image Process16:1522–1537

    Article  MathSciNet  Google Scholar 

  43. Olhede S, Walden AT (2004a) ‘Analytic’ denoising. Biometrika91:955–973

    Article  MathSciNet  MATH  Google Scholar 

  44. Olhede SC, Walden AT (2004b) The Hilbert spectrum via wavelet projections. ProcRoy Soc Lond A 460:955–975

    Article  MathSciNet  MATH  Google Scholar 

  45. Olhede SC, Walden AT (2005) Wavelet denoising for signals in quadrature. IntegrComputer-Aided Eng 12:109–117

    Google Scholar 

  46. Percival DB, Walden AT (2000) Wavelet methods for time seriesanalysis. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  47. Schuster A (1898) On the investigation of hidden periodicities with applicationto a supposed 26-day period of meteorological phenomena. Terr Mag Atmos Elect 3:13–41

    Article  Google Scholar 

  48. Selesnick IW, Baraniuk RG, Kingsbury NG (2005) The dual‐tree complexwavelet transform. IEEE Signal Process Mag 22(6):123–151

    Article  Google Scholar 

  49. Serroukh A, Walden AT (2000a) Wavelet scale analysis of bivariate time seriesi: motivation and estimation. J Nonparametr Stat 13:1–36

    Article  MathSciNet  MATH  Google Scholar 

  50. Serroukh A, Walden AT (2000b) Wavelet scale analysis of bivariate time seriesii: statistical properties for linear processes. J Nonparametr Stat 13:37–56

    Article  MathSciNet  MATH  Google Scholar 

  51. Serroukh A, Walden AT, Percival DB (2000) Statistical properties and uses ofthe wavelet variance estimator for the scale analysis of time series. J Am Stat Assoc 95:184–196

    Article  MathSciNet  MATH  Google Scholar 

  52. Starck JL, Candès EJ, Donoho DL (2002) The curvelet transform for imagedenoising. IEEE Trans Image Process 11:670–684

    Google Scholar 

  53. Stein C (1981) Estimation of the mean of a multivariate normaldistribution. Ann Stat 9:1135–1151

    Article  MATH  Google Scholar 

  54. Torrence C, Compo GP (1998) A practical guide to wavelet analysis. Bull AmMeteorol Soc 79:61–78

    Article  Google Scholar 

  55. Tu C-L, Hwang W-L, Ho J (2005) Analysis of singularities from modulus maxima ofcomplex wavelets. IEEE Trans Inf Theory 51:1049–1062

    Article  MathSciNet  Google Scholar 

  56. Wasserman L (2006) All of nonparametric statistics. Springer, NewYork

    MATH  Google Scholar 

  57. Whitcher B, Guttorp P, Percival DB (2000) Wavelet analysis of covariance withapplication to atmospheric time series. J Geophys Res Atmos 105(14):14941–14962

    Article  Google Scholar 

  58. Wolfe PJ, Godsill SJ, Ng WJ (2004) Bayesian variable selection andregularisation for time‐frequency surface estimation. J Roy Stat Soc Ser B 66:575–589

    Article  MathSciNet  MATH  Google Scholar 

  59. Zhang Z, Ren Z, Huang W (2003) A novel detection method of motor broken rotorbars based on wavelet ridge. IEEE Trans Energy Convers 18:417–423

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistical Science, University College London, London, UK

    Sofia Olhede

Authors
  1. Sofia Olhede
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag

About this entry

Cite this entry

Olhede, S. (2012). Statistical Applications of Wavelets. In: Meyers, R. (eds) Computational Complexity. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1800-9_189

Download citation

Publish with us

Policies and ethics

Search

Navigation