单分形信号的多重分形效应
On the multifractal effects generated by monofractal signals
摘要：
定量研究了多重分形消除趋势波动分析（MF-DFA）中时间序列多重分形现象分析时可能遇到的虚假多重分形信号的水平。所研究的效果出现作为有限长度的使用的数据系列的结果，并通过数据最终可能包含的长期记忆被放大。我们提供了详细的定量描述，这种明显的多重分形背景信号作为一个阈值的广义赫斯特指数值$Δh$或阈值的宽度的多重分形谱$Δα$低于该系统的多重分形属性是唯一明显的，即不存在，尽管$Δα\neq0$或$Δh\neq 0$。我们发现这种效应对于较短或持续的序列非常重要，我们认为它与自相关指数$γ$是线性的。其强度随时间序列长度的增加按幂律衰减。本文还研究了初始数据在不同长记忆水平下的基本线性和非线性变换的影响。这提供了额外的一组半分析结果。所得到的公式是重要的多重分形的任何跨学科的应用，包括物理学，金融数据分析或生理学，因为它们允许分离的“真正的”多重分形现象从明显的（人工）多重分形效应。他们应该是一个有用的工具的第一选择，以决定我们是否在特定情况下与真正的多尺度特性的信号或没有。
英文摘要：
We study quantitatively the level of false multifractal signal one may encounter while analyzing multifractal phenomena in time series within multifractal detrended fluctuation analysis (MF-DFA). The investigated effect appears as a result of finite length of used data series and is additionally amplified by the long-term memory the data eventually may contain. We provide the detailed quantitative description of such apparent multifractal background signal as a threshold in spread of generalized Hurst exponent values $Δh$ or a threshold in the width of multifractal spectrum $Δα$ below which multifractal properties of the system are only apparent, i.e. do not exist, despite $Δα\neq0$ or $Δh\neq 0$. We find this effect quite important for shorter or persistent series and we argue it is linear with respect to autocorrelation exponent $γ$. Its strength decays according to power law with respect to the length of time series. The influence of basic linear and nonlinear transformations applied to initial data in finite time series with various level of long memory is also investigated. This provides additional set of semi-analytical results. The obtained formulas are significant in any interdisciplinary application of multifractality, including physics, financial data analysis or physiology, because they allow to separate the 'true' multifractal phenomena from the apparent (artificial) multifractal effects. They should be a helpful tool of the first choice to decide whether we do in particular case with the signal with real multiscaling properties or not.
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