We introduce the wavelet scattering spectra which provide 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 correlation across time and scales of wavelet coefficients and their modulus. This correlation matrix is nearly diagonalized by a second wavelet transform, which defines the scattering spectra. We show that this vector of moments characterizes a wide range of non-Gaussian properties of multi-scale processes. We prove that self-similar processes have scattering spectra which are scale invariant. This property can be tested statistically on a single realization and defines a class of wide-sense self-similar processes. We build maximum entropy models conditioned by scattering spectra coefficients, and generate new time-series with a microcanonical sampling algorithm. Applications are shown for highly non-Gaussian financial and turbulence time-series.

翻译：我们引入小波散射谱，为具有平稳增量的时间序列提供非高斯模型。复小波变换可计算每个尺度上的信号变化。通过小波系数及其模量在时间和尺度上的联合相关性，捕获跨尺度的依赖关系。这种相关性矩阵通过二次小波变换实现近似对角化，由此定义散射谱。我们证明，该矩向量能够刻画多尺度过程广泛分布的非高斯特性。我们证明自相似过程的散射谱具有尺度不变性，该性质可通过单次实现进行统计检验，并定义了一类广义自相似过程。我们构建了以散射谱系数为约束条件的最大熵模型，并采用微正则采样算法生成新的时间序列。文中展示了针对高度非高斯金融与湍流时间序列的应用实例。