On the multifractal effects generated by monofractal signals

TL;DR

This paper investigates how finite data length and long-term memory can produce false multifractal signals in time series analysis, providing formulas to distinguish genuine multifractality from artifacts.

Contribution

It offers a quantitative framework to identify and separate true multifractal effects from apparent ones caused by data limitations and memory effects.

Findings

False multifractal signals depend linearly on autocorrelation exponent.

The strength of apparent multifractality decays as a power law with series length.

Formulas enable differentiation between real and artificial multifractal properties.

Abstract

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 $\Delta h$ or a threshold in the width of multifractal spectrum $\Delta \alpha$ below which multifractal properties of the system are only apparent, i.e. do not exist, despite $\Delta\alpha\neq0$ or $\Delta h\neq 0$. We find this effect quite important for shorter or persistent series and we argue it is linear with respect to autocorrelation exponent $\gamma$. 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.