Complex systems often produce multifractal signals defined by stationary increments that exhibit power-law scaling properties. The Legendre transform of the domain-dependent scaling function that defines the power law is known as the multifractal spectrum. The multifractal spectrum can also be defined by a power-series expansion of the scaling function and in practice the first two leading coefficients of that series are estimated from the discrete wavelet transform of the signal. To quantify, validate, and compare simulations of complex systems with data collected empirically from the actual system, practitioners require methods for approximating the variance associated with estimates of these coefficients. In this work, we generalize a previously developed semi-parametric statistical model for the values extracted from a discrete multi-scale wavelet transform to include both within-scale and between-scale covariance dependencies. We employ multiplicative cascades to simulate multifractals with known parameters to illustrate the necessity for this generalization and to test the precision of our improved model. The combined within- and between-scale model of covariance results in a more accurate estimate of the expected variance of the coefficients extracted from an empirical data set.

复杂系统通常会产生由具有幂律缩放特性的平稳增量定义的多重分形信号。定义幂律的域相关标度函数的勒让德变换称为多重分形谱。多重分形谱也可以通过缩放函数的幂级数展开来定义，并且实际上该级数的前两个主要系数是根据信号的离散小波变换来估计的。为了量化、验证复杂系统的模拟并将其与从实际系统中凭经验收集的数据进行比较，从业者需要近似与这些系数估计相关的方差的方法。在这项工作中，我们概括了先前开发的半参数统计模型，用于从离散多尺度小波变换中提取的值，以包括尺度内和尺度间协方差依赖性。我们采用乘法级联来模拟具有已知参数的多重分形，以说明这种推广的必要性并测试我们改进模型的精度。组合的尺度内和尺度间协方差模型可以更准确地估计从经验数据集中提取的系数的预期方差。