ABSTRACT

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.

摘要

本文讨论了仍然流行但不正确的想法，即单形（有时也称为“分形”）结构可能是确定的非线性的特征，以及作为必然结果，单分形分析是非线性分析。对于许多读者来说，这一点（即“分形=非线性”）是不正确的。我们怀疑对自相关的不熟悉已经使分形结构的线性从更受欢迎的欣赏中消失了。在本文中，为了解释单分数线的线性性质及其与多重分数线的区别，我们介绍了自相关函数，并回顾了短时滞记忆，非平稳运动以及常规分形分析量化的分数积分过程的中间集。 。从我们自己的经验中了解分形的线性是多么令人惊讶，我们试图用与数学一样多的图形描述来阐明我们的观点。