Intermittent process analysis with scattering moments. (English) Zbl 1308.62168

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

Ann. Stat. 43, No. 1, 323-351 (2015).

Summary: 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.

MSC:

62M10	Time series, auto-correlation, regression, etc. in statistics (GARCH)
62M15	Inference from stochastic processes and spectral analysis
62M40	Random fields; image analysis
62G05	Nonparametric estimation
62P05	Applications of statistics to actuarial sciences and financial mathematics

Keywords:

multifractal; intermittency; wavelet analysis; spectral analysis; generalized method of moments

Software:

Multifractal; ScatNet

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