Scale Dependencies and Self-Similarity Through Wavelet Scattering Covariance

We introduce a scattering covariance matrix which provides non-Gaussian models of time-series having stationary increments. A complex wavelet transform computes signal variations at each scale. Dependencies across scales are captured by the joint covariance across time and scales of complex wavelet coefficients and their modulus. This covariance is nearly diagonalized by a second wavelet transform, which defines the scattering covariance. We show that this set of moments characterizes a wide range of non-Gaussian properties of multi-scale processes. This is analyzed for a variety of processes, including fractional Brownian motions, Poisson, multifractal random walks and Hawkes processes. We prove that self-similar processes have a scattering covariance matrix which is scale invariant. This property can be estimated numerically and defines a class of wide-sense self-similar processes. We build maximum entropy models conditioned by scattering covariance coefficients, and generate new time-series with a microcanonical sampling algorithm. Applications are shown for highly non-Gaussian financial and turbulence time-series.