Multifractality Versus (Mono-) Fractality as Evidence of Nonlinear Interactions Across Timescales: Disentangling the Belief in Nonlinearity From the Diagnosis of Nonlinearity in Empirical Data

This article addresses the still popular but incorrect idea that monofractal (sometimes called “fractal” for short) structure might be a definitive signature of nonlinearity and, as a corollary, that monofractal analyses are nonlinear analyses. That this point (i.e., “fractal = nonlinear”) is incorrect remains novel to many readers. We suspect that unfamiliarity with autocorrelation has helped eclipse the linearity of fractal structure from more popular appreciation. In this article, in order to explain the linear nature of monofractality and its difference from multifractality, we present an introduction to the autocorrelation function and review short-lag memory, nonstationary motions, and the intermediary set of fractionally integrated processes that conventional fractal analyses quantify. Understanding from our own experiences how surprising the linearity of fractals is to accept, we attempt to make our points clear with as much graphic depiction as math. We hope to share our own experiences in struggling with this potentially strange-sounding idea that, really, monofractals are linear while at the same time contrasting them to multifractals that can indicate nonlinearity.