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On the multifractal effects generated by monofractal signals

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  • Dariusz Grech
  • Grzegorz Pamu{l}a

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.

Suggested Citation

  • Dariusz Grech & Grzegorz Pamu{l}a, 2013. "On the multifractal effects generated by monofractal signals," Papers 1307.2014, arXiv.org, revised Aug 2013.
    • Handle: RePEc:arx:papers:1307.2014
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    References listed on IDEAS

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    1. Ivanova, K. & Shirer, H.N. & Clothiaux, E.E. & Kitova, N. & Mikhalev, M.A. & Ackerman, T.P. & Ausloos, M., 2002. "A case study of stratus cloud base height multifractal fluctuations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 308(1), pages 518-532.
    2. Ausloos, M., 2012. "Measuring complexity with multifractals in texts. Translation effects," Chaos, Solitons & Fractals, Elsevier, vol. 45(11), pages 1349-1357.
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    Cited by:

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    2. Wu, Liang & Chen, Lei & Ding, Yiming & Zhao, Tongzhou, 2018. "Testing for the source of multifractality in water level records," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 508(C), pages 824-839.
    3. Chen, Feier & Tian, Kang & Ding, Xiaoxu & Miao, Yuqi & Lu, Chunxia, 2016. "Finite-size effect and the components of multifractality in transport economics volatility based on multifractal detrending moving average method," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 462(C), pages 1058-1066.
    4. Sierra-Porta, D., 2024. "A multifractal approach to understanding Forbush Decrease events: Correlations with geomagnetic storms and space weather phenomena," Chaos, Solitons & Fractals, Elsevier, vol. 185(C).
    5. Marcin Wk{a}torek & Marcin Kr'olczyk & Jaros{l}aw Kwapie'n & Tomasz Stanisz & Stanis{l}aw Dro.zd.z, 2024. "Approaching multifractal complexity in decentralized cryptocurrency trading," Papers 2411.05951, arXiv.org.
    6. Marcin Wk{a}torek & Stanis{l}aw Dro.zd.z & Jaros{l}aw Kwapie'n & Ludovico Minati & Pawe{l} O'swik{e}cimka & Marek Stanuszek, 2020. "Multiscale characteristics of the emerging global cryptocurrency market," Papers 2010.15403, arXiv.org, revised Mar 2021.
    7. Zhuang, Xiaoyang & Wei, Dan, 2022. "Asymmetric multifractality, comparative efficiency analysis of green finance markets: A dynamic study by index-based model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 604(C).
    8. Li, Tingyi & Xue, Leyang & Chen, Yu & Chen, Feier & Miao, Yuqi & Shao, Xinzeng & Zhang, Chenyi, 2018. "Insights from multifractality analysis of tanker freight market volatility with common external factor of crude oil price," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 505(C), pages 374-384.
    9. Kukacka, Jiri & Kristoufek, Ladislav, 2020. "Do ‘complex’ financial models really lead to complex dynamics? Agent-based models and multifractality," Journal of Economic Dynamics and Control, Elsevier, vol. 113(C).
    10. Olivares, Felipe & Zanin, Massimiliano, 2022. "Corrupted bifractal features in finite uncorrelated power-law distributed data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 603(C).
    11. Schadner, Wolfgang, 2022. "U.S. Politics from a multifractal perspective," Chaos, Solitons & Fractals, Elsevier, vol. 155(C).
    12. Schadner, Wolfgang, 2021. "On the persistence of market sentiment: A multifractal fluctuation analysis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 581(C).
    13. Hongtao Chang & Rui Ren & Yaqiong Wang & Jiaqi Li, 2022. "Evaluation of Air Pollutants in Extra-Long Road Tunnel with the Combination of Pollutants Nonlinear Evolution and Machine Learning Method," Sustainability, MDPI, vol. 14(17), pages 1-20, August.
    14. Olivares, Felipe & Sun, Xiaoqian & Wandelt, Sebastian & Zanin, Massimiliano, 2023. "Measuring landing independence and interactions using statistical physics," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 170(C).

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