Multifractal nonlinearity as a robust estimator of multiplicative cascade dynamics

Abstract

Multifractal formalisms provide an apt framework to study random cascades in which multifractal spectrum width $\Delta\alpha$ fluctuates depending on the number of estimable power-law relationships. Then again, multifractality without surrogate comparison can be ambiguous: the original measurement series' multifractal spectrum width $\Delta\alpha_\mathrm{Orig}$ can be sensitive to the series length, ergodicity-breaking linear temporal correlations (e.g., fractional Gaussian noise, $fGn$), or additive cascade dynamics. To test these threats, we built a suite of random cascades that differ by the length, type of noise (i.e., additive white Gaussian noise, $awGn$, or $fGn$), and mixtures of $awGn$ or $fGn$ across generations (progressively more $awGn$, progressively more $fGn$, and a random sampling by generation), and operations applying noise (i.e., addition vs. multiplication). The so-called ``multifractal nonlinearity'' $t_\mathrm{MF}$ (i.e., a $t$-statistic comparing $\Delta\alpha_\mathrm{Orig}$ and multifractal spectra width for phase-randomized linear surrogates $\Delta\alpha_\mathrm{Surr}$) is a robust indicator of random multiplicative rather than random additive cascade processes irrespective of the series length or type of noise. $t_\mathrm{MF}$ is more sensitive to the number of generations than the series length. Furthermore, the random additive cascades exhibited much stronger ergodicity breaking than all multiplicative analogs. Instead, ergodicity breaking in random multiplicative cascades more closely followed the ergodicity-breaking of the constituent noise types -- breaking ergodicity much less when arising from ergodic $awGn$ and more so for noise incorporating relatively more correlated $fGn$. Hence, $t_\mathrm{MF}$ is a robust multifractal indicator of multiplicative cascade processes and not spuriously sensitive to ergodicity breaking.