Comptes Rendus
Fourier and the science of today / Fourier et la science d'aujourd'hui
Multivariate scale-free temporal dynamics: From spectral (Fourier) to fractal (wavelet) analysis
[Dynamique temporelle multivariée invariante d'échelle : de l'analyse spectrale (Fourier) à l'analyse fractale (ondelette)]
Comptes Rendus. Physique, Volume 20 (2019) no. 5, pp. 489-501.

La transformée de Fourier (ou analyse spectrale) est aujourd'hui devenue un outil universel pour l'analyse de données issues de nombreuses applications réelles de natures très différentes, particulièrement pertinent pour la caractérisation de la dynamique temporelle ou spatiale. La transformée en ondelettes (ou analyse multéchelle) peut être vue comme une analyse spectrale adaptée à des classes de signaux ou fonctions dont la dynamique est invariante d'échelle. La présente contribution propose d'abord de faire un état de l'art des relations formelles entre ces deux analyses dans le cadre des processus aléatoires stationaires multivariés, puis de montrer la capacité de la transformée en ondelettes à étendre l'analyse de l'invariance d'échelle multivariée au-delà des statistiques de second ordre (fonction de covariance et spectre de Fourier), à l'auto-similarité multivariée et à la multifractalité multivariée. Quelques illustrations et éléments de discussion sur la pertinence de ces concepts et outils pour l'analyse de l'activité cérébrale macroscopique sont proposés.

The Fourier transform (or spectral analysis) has become a universal tool for data analysis in many different real-world applications, notably for the characterization of temporal/spatial dynamics in data. The wavelet transform (or multiscale analysis) can be regarded as tailoring spectral estimation to classes of signals or functions defined by scale-free dynamics. The present contribution first formally reviews these connections in the context of multivariate stationary processes, and second details the ability of the wavelet transform to extend multivariate scale-free temporal dynamics analysis beyond second-order statistics (Fourier spectrum and autocovariance function) to multivariate self-similarity and multivariate multifractality. Illustrations and qualitative discussions of the relevance of scale-free dynamics for macroscopic brain activity description using MEG data are proposed.

Publié le :
DOI : 10.1016/j.crhy.2019.08.005
Keywords: Fourier transform, wavelet transform, Multivariate signals, Scale-free dynamics, Self-similarity, Multifractality
Mot clés : Transformée de Fourier, transformée en ondelettes, Signaux multivariés, Dynamique invariante d'échelle, Auto-similarité, Multifractalité

Patrice Abry 1 ; Herwig Wendt 2 ; Stéphane Jaffard 3 ; Gustavo Didier 4

1 Université de Lyon, ENS de Lyon, CNRS, Laboratoire de physique, Lyon, France
2 IRIT, CNRS (UMR 5505), Université de Toulouse, France
3 Université Paris-Est, LAMA (UMR 8050), UPEM, UPEC, CNRS, Créteil, France
4 Department of Mathematics, Tulane University, New Orleans, LA, USA
@article{CRPHYS_2019__20_5_489_0,
     author = {Patrice Abry and Herwig Wendt and St\'ephane Jaffard and Gustavo Didier},
     title = {Multivariate scale-free temporal dynamics: {From} spectral {(Fourier)} to fractal (wavelet) analysis},
     journal = {Comptes Rendus. Physique},
     pages = {489--501},
     publisher = {Elsevier},
     volume = {20},
     number = {5},
     year = {2019},
     doi = {10.1016/j.crhy.2019.08.005},
     language = {en},
}
TY  - JOUR
AU  - Patrice Abry
AU  - Herwig Wendt
AU  - Stéphane Jaffard
AU  - Gustavo Didier
TI  - Multivariate scale-free temporal dynamics: From spectral (Fourier) to fractal (wavelet) analysis
JO  - Comptes Rendus. Physique
PY  - 2019
SP  - 489
EP  - 501
VL  - 20
IS  - 5
PB  - Elsevier
DO  - 10.1016/j.crhy.2019.08.005
LA  - en
ID  - CRPHYS_2019__20_5_489_0
ER  - 
%0 Journal Article
%A Patrice Abry
%A Herwig Wendt
%A Stéphane Jaffard
%A Gustavo Didier
%T Multivariate scale-free temporal dynamics: From spectral (Fourier) to fractal (wavelet) analysis
%J Comptes Rendus. Physique
%D 2019
%P 489-501
%V 20
%N 5
%I Elsevier
%R 10.1016/j.crhy.2019.08.005
%G en
%F CRPHYS_2019__20_5_489_0
Patrice Abry; Herwig Wendt; Stéphane Jaffard; Gustavo Didier. Multivariate scale-free temporal dynamics: From spectral (Fourier) to fractal (wavelet) analysis. Comptes Rendus. Physique, Volume 20 (2019) no. 5, pp. 489-501. doi : 10.1016/j.crhy.2019.08.005. https://comptes-rendus.academie-sciences.fr/physique/articles/10.1016/j.crhy.2019.08.005/

[1] A. Papoulis Signal Analysis, vol. 191, McGraw-Hill, New York, 1977

[2] T.W. Körner Fourier Analysis, Cambridge University Press, 1988

[3] B.G. Osgood Lectures on the Fourier Transform and Its Applications, American Mathematical Society, Providence, RI, USA, 2019

[4] P.J. Brockwell; R.A. Davis Time Series: Theory and Methods, Springer Science and Business Media, 1991

[5] J.W. Cooley; J.W. Tukey An algorithm for the machine calculation of complex Fourier series, Math. Comput., Volume 19 (1965) no. 90, pp. 297-301

[6] J.W. Cooley The re-discovery of the fast Fourier transform algorithm, Mikrochim. Acta, Volume 93 (1987) no. 1–6, pp. 33-45

[7] G. Buzsáki; A. Draguhn Neuronal oscillations in cortical networks, Science, Volume 304 (2004) no. 5679, pp. 1926-1929

[8] D. Veitch; P. Abry A wavelet-based joint estimator of the parameters of long-range dependence, IEEE Trans. Inf. Theory, Volume 45 (1999) no. 3, pp. 878-897

[9] K. Park; W. Willinger Self-similar network traffic: an overview (K. Park; W. Willinger, eds.), Self-Similar Network Traffic and Performance Evaluation, Wiley, 2000, pp. 1-38

[10] P. Abry; R. Baraniuk; P. Flandrin; R. Riedi; D. Veitch Multiscale nature of network traffic, IEEE Signal Process. Mag., Volume 19 (2002) no. 3, pp. 28-46

[11] R. Fontugne; P. Abry; K. Fukuda; D. Veitch; K. Cho; P. Borgnat; H. Wendt Scaling in Internet traffic: a 14 year and 3 day longitudinal study, with multiscale analyses and random projections, IEEE/ACM Trans. Netw., Volume 25 (2017) no. 4, pp. 2152-2165

[12] B. Mandelbrot Information theory and psycholinguistics (B.B. Wolman; E. Nagel, eds.), Scientific Psychology: Principles and Approaches, Basic Books, New York, 1965

[13] L. Calvet; A. Fisher; B. Mandelbrot The multifractal model of asset returns, Cowles Foundation Discussion Papers, vol. 1164, 1997

[14] L. Calvet; A. Fisher Multifractality in assets returns: theory and evidence, Rev. Econ. Stat., Volume LXXXIV (2002) no. 84, pp. 381-406

[15] P. Frankhauser L'approche fractale: un nouvel outil dans l'analyse spatiale des agglomerations urbaines, Population, Volume 4 (1997), pp. 1005-1040

[16] P. Abry; S. Jaffard; H. Wendt When Van Gogh meets Mandelbrot: multifractal classification of painting's texture, Signal Process., Volume 93 (2013) no. 3, pp. 554-572

[17] R. Leonarduzzi, P. Abry, S. Jaffard, H. Wendt, L. Gournay, T. Kyriacopoulou, C. Martineau, C. Martinez, p-Leader multifractal analysis for text type identification, in: Proc. 42nd IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP2017, New Orleans, LA, USA, 5–9 March 2017.

[18] L.S. Liebovitch; A.T. Todorov Invited editorial on “Fractal dynamics of human gait: stability of long-range correlations in stride interval fluctuations”, J. Appl. Physiol. (1996), pp. 1446-1447

[19] P.C. Ivanov Scale-invariant aspects of cardiac dynamics, IEEE Eng. Med. Biol. Mag., Volume 26 (2007) no. 6, pp. 33-37

[20] M. Doret; H. Helgason; P. Abry; P. Gonçalvès; Cl. Gharib; P. Gaucherand Multifractal analysis of fetal heart rate variability in fetuses with and without severe acidosis during labor, Am. J. Perinatol., Volume 28 (2011) no. 4, pp. 259-266

[21] T. Nakamura; K. Kiyono; H. Wendt; P. Abry; Y. Yamamoto Multiscale analysis of intensive longitudinal biomedical signals and its clinical applications, Proc. IEEE, Volume 104 (2016) no. 2, SI, pp. 242-261

[22] H. Wendt; P. Abry; K. Kiyono; J. Hayano; E. Watanabe; Y. Yamamoto Wavelet p-leader non Gaussian multiscale expansions for heart rate variability analysis in congestive heart failure patients, IEEE Trans. Biomed. Eng., Volume 66 (2019) no. 1, pp. 80-88

[23] G. Werner Fractals in the nervous system: conceptual implications for theoretical neuroscience, Front. Physiol., Volume 1 (2010)

[24] B.J. He Scale-free brain activity: past, present, and future, Trends Cogn. Sci., Volume 18 (2014) no. 9, pp. 480-487

[25] B. Maniscalco; J.L. Lee; P. Abry; A. Lin; T. Holroyd; B.J. He Neural integration of stimulus history underlies prediction for naturalistically evolving sequences, J. Neurosci., Volume 38 (2018) no. 6, pp. 1541-1557

[26] D. La Rocca; N. Zilber; P. Abry; V. van Wassenhove; P. Ciuciu Self-similarity and multifractality in human brain activity: a wavelet-based analysis of scale-free brain dynamics, J. Neurosci. Methods, Volume 309:175–187 (2018)

[27] U. Frisch Turbulence, the Legacy of A.N. Kolmogorov, Addison-Wesley, 1993

[28] D. Schertzer; S. Lovejoy Physically based rain and cloud modeling by anisotropic, multiplicative turbulent cascades, J. Geophys. Res., Volume 92 (1987), pp. 9693-9714

[29] B. Lashermes; S.G. Roux; P. Abry; S. Jaffard Comprehensive multifractal analysis of turbulent velocity using the wavelet leaders, Eur. Phys. J. B, Volume 61 (2008) no. 2, pp. 201-215

[30] Wavelets in Geophysics (E. Foufoula-Georgiou; P. Kumar, eds.), Academic Press, San Diego, CA, USA, 1994

[31] S. Lovejoy; D. Schertzer Scaling and multifractal fields in the solid Earth and topography, Nonlinear Process. Geophys., Volume 14 (2007) no. 4, pp. 465-502

[32] B. Mandelbrot; W. Wallis Noah, Joseph, and operational hydrology, Water Resour. Res., Volume 4 (1968) no. 5, pp. 909-918

[33] B. Mandelbrot The Fractal Geometry of Nature, 1982 (New York)

[34] P. Abry; S. Jaffard; H. Wendt Irregularities and scaling in signal and image processing: multifractal analysis (M. Frame; N. Cohen, eds.), Benoît Mandelbrot: a Life in Many Dimensions, World Scientific Publishing, 2015, pp. 31-116

[35] P. Ciuciu; P. Abry; B.J. He Interplay between functional connectivity and scale-free dynamics in intrinsic fMRI networks, NeuroImage, Volume 95 (2014), pp. 248-263

[36] I. Daubechies Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1992

[37] S. Mallat A Wavelet Tour of Signal Processing, Academic Press, San Diego, CA, USA, 1998

[38] P. Flandrin On the spectrum of fractional Brownian motions, IEEE Trans. Inf. Theory, Volume IT-35 (1989) no. 1, pp. 197-199

[39] P. Flandrin Wavelet analysis and synthesis of fractional Brownian motions, IEEE Trans. Inf. Theory, Volume 38 (1992), pp. 910-917

[40] P. Abry; P. Gonçalvès; P. Flandrin Wavelets, spectrum estimation and 1/f processes, Wavelets and Statistics, Springer-Verlag, New York, 1995 (chapter 103, Lecture Notes in Statistics)

[41] P. Abry; D. Veitch Wavelet analysis of long-range dependent traffic, IEEE Trans. Inf. Theory, Volume 44 (1998) no. 1, pp. 2-15

[42] G. Didier; V. Pipiras Integral representations and properties of operator fractional Brownian motions, Bernoulli, Volume 17 (2011) no. 1, pp. 1-33

[43] P. Abry; G. Didier Wavelet estimation for operator fractional Brownian motion, Bernoulli, Volume 24 ( May 2018 ) no. 2, pp. 895-928

[44] P. Abry; G. Didier Wavelet eigenvalue regression for n-variate operator fractional Brownian motion, J. Multivar. Anal., Volume 168 (2018), pp. 75-104

[45] C. Meneveau; K.R. Sreenivasan; P. Kailasnath; M.S. Fan Joint multifractal measures – theory and applications to turbulence, Phys. Rev. A, Volume 41 (1990) no. 2, pp. 894-913

[46] S. Jaffard; S. Seuret; H. Wendt; R. Leonarduzzi; S. Roux; P. Abry Multivariate multifractal analysis, Appl. Comput. Harmon. Anal., Volume 46 (2019) no. 3, pp. 653-663

[47] P. Flandrin Time-Frequency/Time-Scale Analysis, vol. 10, Academic Press, 1998

[48] B. Whitcher; P. Guttorp; D.B. Percival Wavelet analysis of covariance with application to atmospheric time series, J. Geophys. Res., Atmos., Volume 105 (2000) no. D11, pp. 14941-14962

[49] H. Wendt; G. Didier; S. Combrexelle; P. Abry Multivariate Hadamard self-similarity: testing fractal connectivity, Physica D, Volume 356 (2017), pp. 1-36

[50] N. Zilber; P. Ciuciu; A. Gramfort; V. van Wassenhove Supramodal processing optimizes visual perceptual learning and plasticity, Neuroimage, Volume 93 (2014) no. Pt 1, pp. 32-46

[51] B. Mandelbrot; J.W. van Ness Fractional Brownian motion, fractional noises and applications, SIAM Rev., Volume 10 (1968), pp. 422-437

[52] G. Samorodnitsky; M. Taqqu Stable Non-Gaussian Random Processes, Chapman and Hall, New York, 1994

[53] M. Maejima; J.D. Mason Operator-self-similar stable processes, Stoch. Process. Appl., Volume 54 (1994) no. 1, pp. 139-163

[54] J.D. Mason; Y. Xiao Sample path properties of operator-self-similar Gaussian random fields, Theory Probab. Appl., Volume 46 (2002) no. 1, pp. 58-78

[55] G. Didier; V. Pipiras Exponents, symmetry groups and classification of operator fractional Brownian motions, J. Theor. Probab., Volume 25 (2012) no. 2, pp. 353-395

[56] C.-F. Chung Sample means, sample autocovariances, and linear regression of stationary multivariate long memory processes, Econom. Theory, Volume 18 (2002), pp. 51-78

[57] H. Dai Convergence in law to operator fractional Brownian motions, J. Theor. Probab., Volume 26 (2013) no. 3, pp. 676-696

[58] P. Abry; G. Didier; H. Li Two-step wavelet-based estimation for Gaussian mixed fractional processes, Stat. Inference Stoch. Process., Volume 22 (2019) no. 2, pp. 157-185

[59] P.M. Robinson Multiple local Whittle estimation in stationary systems, Ann. Stat., Volume 36 (2008) no. 5, pp. 2508-2530

[60] S. Achard; D.S. Bassett; A. Meyer-Lindenberg; E. Bullmore Fractal connectivity of long-memory networks, Phys. Rev. E, Volume 77 (2008) no. 3

[61] D. La Rocca, P. Ciuciu, V. van Wassenhove, H. Wendt, P. Abry, R. Leonarduzzi, Scale-free functional connectivity analysis from source reconstructed MEG data, in: Proc. European Signal Processing Conference (EUSIPCO 2018), Rome, Italy, 3–7 September 2018.

[62] J. Frecon; G. Didier; N. Pustelnik; P. Abry Non-linear wavelet regression and branch & bound optimization for the full identification of bivariate operator fractional Brownian motion, IEEE Trans. Signal Process., Volume 64 (2016) no. 15, pp. 4040-4049

[63] R.H. Riedi Multifractal processes (P. Doukhan; G. Oppenheim; M.S. Taqqu, eds.), Theory and Applications of Long Range Dependence, Birkhäuser, 2003, pp. 625-717

[64] S. Jaffard Wavelet techniques in multifractal analysis (M. Lapidus; M. van Frankenhuijsen, eds.), Fractal Geometry and Applications: a Jubilee of Benoît Mandelbrot, Proc. Symp. Pure Math., vol. 72(2), American Mathematical Society, Providence, RI, USA, 2004, pp. 91-152

[65] H. Wendt; P. Abry; S. Jaffard Bootstrap for empirical multifractal analysis, IEEE Signal Process. Mag., Volume 24 (2007) no. 4, pp. 38-48

[66] S. Jaffard; C. Melot; R. Leonarduzzi; H. Wendt; P. Abry; S.G. Roux; M.E. Torres p-exponent and p-leaders, part I: negative pointwise regularity, Physica A, Volume 448 (2016), pp. 300-318

[67] R. Leonarduzzi; H. Wendt; P. Abry; S. Jaffard; C. Melot; S.G. Roux; M.E. Torres p-exponent and p-leaders, part II: multifractal analysis. Relations to detrended fluctuation analysis, Physica A, Volume 448 (2016), pp. 319-339

[68] D. Schertzer; S. Lovejoy Physically based rain and cloud modeling by anisotropic, multiplicative turbulent cascades, J. Geophys. Res., Volume 92.D8 (1987), pp. 9693-9714

[69] M. Ben Slimane Baire typical results for mixed Hölder spectra on product of continuous Besov or oscillation spaces, Mediterr. J. Math., Volume 13 (2016), pp. 1513-1533

[70] B. Castaing; Y. Gagne; M. Marchand Log-similarity for turbulent flows, Physica D, Volume 68 (1993) no. 3–4, pp. 387-400

[71] A. Arneodo; E. Bacry; J.F. Muzy The thermodynamics of fractals revisited with wavelets, Physica A, Volume 213 (1995) no. 1–2, pp. 232-275

[72] H. Wendt, R. Leonarduzzi, P. Abry, S. Roux, S. Jaffard, S. Seuret, Assessing cross-dependencies using bivariate multifractal analysis, in: 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP 2018), Calgary, Alberta, Canada, 15–20 April 2018.

[73] R. Leonarduzzi, P. Abry, S.G. Roux, H. Wendt, S. Jaffard, S. Seuret, Multifractal characterization for bivariate data, in: Proc. European Signal Processing Conference (EUSIPCO 2018), Rome, Italy, 3–7 September 2018.

[74] E. Bacry; J. Delour; J.F. Muzy Multifractal random walk, Phys. Rev. E, Volume 64 (2001) no. 2

[75] H. Helgason; V. Pipiras; P. Abry Synthesis of multivariate stationary series with prescribed marginal distributions and covariance using circulant matrix embedding, Signal Process., Volume 91 (2011), pp. 1741-1758

[76] H. Wendt, P. Abry, G. Didier, Wavelet domain bootstrap for testing the equality of bivariate self-similarity exponents, in: Proc. IEEE Workshop Statistical Signal Proces. (SSP), Freiburg, Germany, 10–13 June 2018.

[77] P. Abry, H. Wendt, G. Didier, Detecting and estimating multivariate self-similar sources in high-dimensional noisy mixtures, in: Proc. IEEE Workshop Statistical Signal Proces. (SSP), Freiburg, Germany, 10–13 June 2018.

Cité par Sources :

Commentaires - Politique