Multifractal nonlinearity as a robust estimator of multiplicative cascade dynamics

Area of Science:

• Natural and behavioral sciences
• Complex systems analysis
• Time series modeling

Background:

• Measurement time series are increasingly modeled as random multiplicative cascades.
• Multifractal formalisms study these cascades, but multifractality can be ambiguous without surrogate comparison.
• Existing methods are sensitive to series length, linear correlations, or additive dynamics.

Purpose of the Study:

• To develop a robust method for identifying random multiplicative cascades.
• To quantify cross-scale interactivity and ergodicity breaking in these processes.
• To address the ambiguity of multifractality in time series analysis.

Main Methods:

• Constructed random cascades with variations in length, noise type (additive white Gaussian noise, fractional Gaussian noise), and noise operations (addition vs. multiplication).
• Developed and applied a t-statistic, tMF, comparing original multifractal spectrum width to surrogate spectra width.
• Analyzed the sensitivity of tMF to series length, noise type, and number of generations.

Main Results:

• The tMF statistic reliably identifies random multiplicative cascades irrespective of series length or noise type.
• tMF is more sensitive to interactivity and the number of generations than to series length.
• Multiplicative cascades exhibit stronger ergodicity breaking than additive cascades, increasing with correlated noise and generations.

Conclusions:

• tMF is a powerful metric for identifying multiplicative cascade processes and quantifying ergodicity breaking.
• This metric advances understanding of cascading mechanisms in natural and behavioral sciences.
• tMF shows potential for extending group-level findings to individual-level analyses.