The Annals of Statistics

Intermittent process analysis with scattering moments

Joan Bruna, Stéphane Mallat, Emmanuel Bacry, and Jean-François Muzy

Full-text: Open access

Abstract

Scattering moments provide nonparametric models of random processes with stationary increments. They are expected values of random variables computed with a nonexpansive operator, obtained by iteratively applying wavelet transforms and modulus nonlinearities, which preserves the variance. First- and second-order scattering moments are shown to characterize intermittency and self-similarity properties of multiscale processes. Scattering moments of Poisson processes, fractional Brownian motions, Lévy processes and multifractal random walks are shown to have characteristic decay. The Generalized Method of Simulated Moments is applied to scattering moments to estimate data generating models. Numerical applications are shown on financial time-series and on energy dissipation of turbulent flows.

Article information

Source
Ann. Statist., Volume 43, Number 1 (2015), 323-351.

Dates
First available in Project Euclid: 6 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.aos/1423230082

Digital Object Identifier
doi:10.1214/14-AOS1276

Mathematical Reviews number (MathSciNet)
MR3311862

Zentralblatt MATH identifier
1308.62168

Subjects
Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 62M15: Spectral analysis 62M40: Random fields; image analysis

Keywords
Multifractal intermittency wavelet analysis spectral analysis Generalized method of moments

Citation

Bruna, Joan; Mallat, Stéphane; Bacry, Emmanuel; Muzy, Jean-François. Intermittent process analysis with scattering moments. Ann. Statist. 43 (2015), no. 1, 323--351. doi:10.1214/14-AOS1276. https://projecteuclid.org/euclid.aos/1423230082


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References

  • [1] Abry, P., Flandrin, P., Taqqu, M. S. and Veitch, D. (2000). Wavelets for the analysis, estimation and synthesis of scaling data. In Self-Similar Network Traffic and Performance Evaluation (K. Park and W. Willinger, eds.) 39–88. Wiley, New York.
  • [2] Anden, J. and Mallat, S. (2014). Deep scattering spectrum. IEEE Trans. Signal Process. 62 4114–4128.
  • [3] Arneodo, A. and Jaffard, S. (2004). L’analyse multi-fractale des signaux. Images des Mathématiques 2004 7–14.
  • [4] Bacry, E., Kozhemyak, A. and Muzy, J.-F. (2008). Continuous cascade models for asset returns. J. Econom. Dynam. Control 32 156–199.
  • [5] Bacry, E., Kozhemyak, A. and Muzy, J. F. (2013). Log-normal continuous cascade model of asset returns: Aggregation properties and estimation. Quant. Finance 13 795–818.
  • [6] Bacry, E. and Muzy, J. F. (2003). Log-infinitely divisible multifractal processes. Comm. Math. Phys. 236 449–475.
  • [7] Bacry, E., Muzy, J. F. and Arneodo, A. (1993). Singularity spectrum of fractal signals from wavelet analysis: Exact results. J. Stat. Phys. 70 635–674.
  • [8] Belomestny, D. (2010). Spectral estimation of the fractional order of a Lévy process. Ann. Statist. 38 317–351.
  • [9] Bruna, J. and Mallat, S. (2013). Invariant scattering convolution networks. IEEE Transactions of PAMI 35 1872–1886.
  • [10] Bruna, J., Mallat, S., Bacry, E. and Muzy, J.-F. (2014). Supplement to “Intermittent process analysis with scattering moments.” DOI:10.1214/14-AOS1276SUPP.
  • [11] Castaing, B. (2002). Lagrangian and Eulerian velocity intermittency. Eur. Phys. J. B 29 357–358.
  • [12] Chanal, O., Chabaud, B., Castaing, B. and Hebral, B. (2000). Intermittency in a turbulent low temperature gaseous helium jet. Eur. Phys. J. B 17 309–317.
  • [13] Chudacek, V., Anden, J., Mallat, S., Abry, P. and Doret, M. (2014). Scattering transform for intrapartum fetal heart rate variability fractal analysis: A case-control study. IEEE Trans. Biomed. Eng. 61 1100–1108.
  • [14] Delour, J., Muzy, J. F. and Arneodo, A. (2001). Intermittency of 1d velocity spatial profiles in turbulence: A magnitude cumulant analysis. Eur. Phys. J. B 23 243.
  • [15] Frisch, U. (1995). Turbulence: The Legacy of A. N. Kolmogorov. Cambridge Univ. Press, Cambridge.
  • [16] Hansen, L. P. (1982). Large sample properties of generalized method of moments estimators. Econometrica 50 1029–1054.
  • [17] Jaffard, S. (1997). Multifractal formalism for functions part I: Results valid for all functions. SIAM J. Math. Anal. 28 944–970.
  • [18] Jaffard, S. (1997). Multifractal formalism for functions part II: Self-similar functions. SIAM J. Math. Anal. 28 971–998.
  • [19] Jaffard, S. (2001). Wavelet expansions, function spaces and multifractal analysis. In Twentieth Century Harmonic Analysis—a Celebration (Il Ciocco, 2000). NATO Sci. Ser. II Math. Phys. Chem. 33 127–144. Kluwer Academic, Dordrecht.
  • [20] Jaffard, S., Lashermes, B. and Abry, P. (2007). Wavelet leaders in multifractal analysis. In Wavelet Analysis and Applications 201–246. Birkhäuser, Basel.
  • [21] Jaffard, S. and Meyer, Y. (1996). Wavelet methods for pointwise regularity and local oscillations of functions. Mem. Amer. Math. Soc. 123 x+110.
  • [22] Kahane, J. P. and Peyriere, J. (1975). Sur Certaines Martingales de B. Mandelbrot. Université Paris XI, Departement de mathematique.
  • [23] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.
  • [24] LeCun, Y., Kavukcuoglu, K. and Farabet, C. (2010). Convolutional networks and applications in vision. In Circuits and Systems (ISCAS), Proceedings of 2010 IEEE International Symposium on 253–256. IEEE.
  • [25] Mallat, S. (2012). Group invariant scattering. Comm. Pure Appl. Math. 65 1331–1398.
  • [26] Mandelbrot, B. B. (1969). On intermittent free turbulence. In Turbulence of Fluids and Plasmas. Interscience, New York.
  • [27] Mandelbrot, B. B. (1999). Intermittent turbulence in self-similar cascades: Divergence of high moments and dimension of the carrier. In Multifractals and $1/f$ Noise 317–357. Springer, New York.
  • [28] McFadden, D. (1989). A method of simulated moments for estimation of discrete response models without numerical integration. Econometrica 57 995–1026.
  • [29] Muzy, J. F. and Bacry, E. (2002). Multifractal stationary random measures and multifractal random walks with log infinitely divisible scaling laws. Phys. Rev. E 66 056121.
  • [30] Muzy, J.-F., Bacry, E. and Arneodo, A. (1993). Multifractal formalism for fractal signals: The structure-function approach versus the wavelet-transform modulus-maxima method. Phys. Rev. E 47 875.
  • [31] Muzy, J. F., Delour, J. and Bacry, E. (2000). Modeling fluctuations of financial time series: From cascade process to stochastic volatility model. Eur. Phys. J. B 17 537–548.
  • [32] Oppenheim, G., Doukhan, P. and Taqqu, M. S. (2003). Theory and Applications of Long-Range Dependence. Birkhäuser, Boston, MA.
  • [33] Robert, C. and Rosenbaum, M. (2011). A new approach for the dynamics of ultra high frequency data: The model with uncertainty zones. J. Financ. Econom. 9 344–366.
  • [34] Selesnick, I. W., Baraniuk, R. G. and Kingsbury, N. C. (2005). The dual-tree complex wavelet transform. IEEE Signal Process. Mag. 22 123–151.
  • [35] Sifre, L. and Mallat, S. (2012). Combined scattering for rotation invariant texture analysis. In European Symposium on Artificial Neural Networks.
  • [36] Wendt, H., Abry, P. and Jaffard, S. (2007). Bootstrap for empirical multifractal analysis. IEEE Signal Process. Mag. 24 38–48.
  • [37] Wendt, H., Roux, S. G., Abry, P. and Jaffard, S. (2009). Wavelet leaders and bootstrap for multifractal analysis of images. Signal Proces. 89 1100–1114.
  • [38] Yaglom, A. M. (1966). The influence of fluctuations in energy dissipation on the shape of turbulence characteristics in the inertial interval. Soviet Physics Dockl. 11 26.
  • [39] Yoshimatsu, K., Schneider, K., Okamoto, N., Kawahara, Y. and Farge, M. (2011). Intermittency and geometrical statistics of three-dimensional homogeneous magnetohydrodynamic turbulence: A wavelet viewpoint. Phys. Plasmas 18 092304.

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