The Annals of Statistics

Intermittent process analysis with scattering moments

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

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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

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

First available in Project Euclid: 6 February 2015

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Zentralblatt MATH identifier

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

Multifractal intermittency wavelet analysis spectral analysis Generalized method of moments


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.

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