  Among the statistical models employed to approximate nonlinear interactions
in biological and psychological processes, one prominent framework is that of
cascades. Despite decades of empirical work using multifractal formalisms, a
fundamental question has persisted: Do the observed nonlinear interactions
across scales owe their origin to multiplicative interactions, or do they
inherently reside within the constituent processes? This study presents the
results of rigorous numerical simulations that demonstrate the supremacy of
multiplicative interactions over the intrinsic nonlinear properties of
component processes. To elucidate this point, we conducted simulations of
cascade time series featuring component processes operating at distinct
timescales, each characterized by one of four properties: multifractal
nonlinearity, multifractal linearity (obtained via the Iterative Amplitude
Adjusted Wavelet Transform of multifractal nonlinearity), phase-randomized
linearity (obtained via the Iterative Amplitude Adjustment Fourier Transform),
and phase- and amplitude-randomized (obtained via shuffling). Our findings
unequivocally establish that the multiplicative interactions among components,
rather than the inherent properties of the component processes themselves,
decisively dictate the multifractal emergent properties. Remarkably, even
component processes exhibiting purely linear traits can generate nonlinear
interactions across scales when these interactions assume a multiplicative
nature. In stark contrast, additivity among component processes inevitably
leads to a linear outcome. These outcomes provide a robust theoretical
underpinning for current interpretations of multifractal nonlinearity, firmly
anchoring its roots in the domain of multiplicative interactions across scales
within biological and psychological processes.
