标题： 多分形涌现过程：乘法相互作用超越非线性组件特性 显示英文标题

标题： Multifractal emergent processes: Multiplicative interactions override nonlinear component properties

摘要： 在用于近似生物和心理过程中非线性相互作用的统计模型中，一种突出的框架是级联。 尽管几十年来一直使用多分形形式进行实证研究，但一个基本问题一直存在：跨尺度观察到的非线性相互作用的根源是乘法相互作用，还是它们固有地存在于组成部分过程中？ 本研究展示了严格的数值模拟结果，这些结果证明了乘法相互作用优于组件过程的内在非线性特性。 为了阐明这一点，我们进行了级联时间序列的模拟，其中组件过程在不同的时间尺度上运行，每个组件过程具有四种特性之一：多分形非线性、多分形线性（通过多分形非线性迭代振幅调整小波变换获得）、相位随机化线性（通过多分形非线性迭代振幅调整傅里叶变换获得）以及相位和振幅随机化（通过洗牌获得）。 我们的研究结果明确表明，组件之间的乘法相互作用，而不是组件过程本身的固有特性，决定了多分形的涌现特性。 值得注意的是，即使组件过程表现出纯粹的线性特征，当这些相互作用具有乘法性质时，它们也可以在不同尺度上产生非线性相互作用。 相反，组件过程之间的可加性必然导致线性结果。 这些结果为目前对多分形非线性的解释提供了坚实的理论基础，牢固地将其根源确立在生物和心理过程中跨尺度的乘法相互作用领域。 显示英文摘要

摘要： 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.