An Improved Model of Wavelet Leader Covariance for Estimating Multifractal Properties

Abstract

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.