IDEAS home Printed from https://ideas.repec.org/a/spr/eurphb/v94y2021i11d10.1140_epjb_s10051-021-00231-7.html
   My bibliography  Save this article

Definition of the fluctuation function in the detrended fluctuation analysis and its variants

Author

Listed:
  • Bastien Berthelot

    (THALES AVS France)

  • Eric Grivel

    (Bordeaux University-INP Bordeaux-IMS-UMR CNRS 5218)

  • Pierrick Legrand

    (Bordeaux University-IMB UMR CNRS 5251-ASTRAL team, INRIA)

  • Audrey Giremus

    (Bordeaux University-IMS-UMR CNRS 5218)

Abstract

The detrended fluctuation analysis (DFA) and its variants are popular methods to analyze the self-similarity of a signal. Two steps characterize them: firstly, the trend of the centered integrated signal is estimated and removed. Secondly, the properties of the so-called fluctuation function which is an approximation of the standard deviation of the resulting process is analyzed. However, it appears that the statistical mean was assumed to be equal to zero to obtain it. As there is no guarantee that this assumption is true a priori, this hypothesis is debatable. The purpose of this paper is to propose two alternative definitions of the fluctuation function. Then, we compare all of them based on a matrix formulation and the filter-based interpretation we recently proposed. This analysis will be useful to show that the approach proposed in the original paper remains a good compromise in terms of accuracy and computational cost. Graphic abstract

Suggested Citation

  • Bastien Berthelot & Eric Grivel & Pierrick Legrand & Audrey Giremus, 2021. "Definition of the fluctuation function in the detrended fluctuation analysis and its variants," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 94(11), pages 1-20, November.
    • Handle: RePEc:spr:eurphb:v:94:y:2021:i:11:d:10.1140_epjb_s10051-021-00231-7
    DOI: 10.1140/epjb/s10051-021-00231-7
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1140/epjb/s10051-021-00231-7
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1140/epjb/s10051-021-00231-7?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Tiwari, Aviral Kumar & Albulescu, Claudiu Tiberiu & Yoon, Seong-Min, 2017. "A multifractal detrended fluctuation analysis of financial market efficiency: Comparison using Dow Jones sector ETF indices," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 483(C), pages 182-192.
    2. Amir Bashan & Ronny Bartsch & Jan W. Kantelhardt & Shlomo Havlin, 2008. "Comparison of detrending methods for fluctuation analysis," Papers 0804.4081, arXiv.org.
    3. Salat, Hadrien & Murcio, Roberto & Arcaute, Elsa, 2017. "Multifractal methodology," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 473(C), pages 467-487.
    4. Kantelhardt, Jan W. & Zschiegner, Stephan A. & Koscielny-Bunde, Eva & Havlin, Shlomo & Bunde, Armin & Stanley, H.Eugene, 2002. "Multifractal detrended fluctuation analysis of nonstationary time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 316(1), pages 87-114.
    5. Marc Höll & Holger Kantz, 2015. "The relationship between the detrendend fluctuation analysis and the autocorrelation function of a signal," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 88(12), pages 1-7, December.
    6. Ayache, Antoine & Lévy Véhel, Jacques, 2004. "On the identification of the pointwise Hölder exponent of the generalized multifractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 111(1), pages 119-156, May.
    7. Ying-Hui Shao & Gao Feng Gu & Zhi-Qiang Jiang & Wei-Xing Zhou & Didier Sornette, 2012. "Comparing the performance of FA, DFA and DMA using different synthetic long-range correlated time series," Papers 1208.4158, arXiv.org.
    8. Osborn, Denise R, 1995. "Moving Average Detrending and the Analysis of Business Cycles," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 57(4), pages 547-558, November.
    9. Bashan, Amir & Bartsch, Ronny & Kantelhardt, Jan W. & Havlin, Shlomo, 2008. "Comparison of detrending methods for fluctuation analysis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(21), pages 5080-5090.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Kiyono, Ken & Tsujimoto, Yutaka, 2016. "Nonlinear filtering properties of detrended fluctuation analysis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 462(C), pages 807-815.
    2. Gu, Gao-Feng & Xiong, Xiong & Zhang, Yong-Jie & Chen, Wei & Zhang, Wei & Zhou, Wei-Xing, 2016. "Stylized facts of price gaps in limit order books," Chaos, Solitons & Fractals, Elsevier, vol. 88(C), pages 48-58.
    3. Jiang, Zhi-Qiang & Xie, Wen-Jie & Zhou, Wei-Xing, 2014. "Testing the weak-form efficiency of the WTI crude oil futures market," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 405(C), pages 235-244.
    4. Xiong, Gang & Yu, Wenxian & Zhang, Shuning, 2015. "Singularity power spectrum distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 431(C), pages 63-73.
    5. Gao-Feng Gu & Xiong Xiong & Yong-Jie Zhang & Wei Chen & Wei Zhang & Wei-Xing Zhou, 2014. "Stylized facts of price gaps in limit order books: Evidence from Chinese stocks," Papers 1405.1247, arXiv.org.
    6. Hongli Niu & Jun Wang, 2014. "Phase and multifractality analyses of random price time series by finite-range interacting biased voter system," Computational Statistics, Springer, vol. 29(5), pages 1045-1063, October.
    7. Gulich, Damián & Zunino, Luciano, 2014. "A criterion for the determination of optimal scaling ranges in DFA and MF-DFA," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 397(C), pages 17-30.
    8. Zhao, Xiaojun & Shang, Pengjian & Zhao, Chuang & Wang, Jing & Tao, Rui, 2012. "Minimizing the trend effect on detrended cross-correlation analysis with empirical mode decomposition," Chaos, Solitons & Fractals, Elsevier, vol. 45(2), pages 166-173.
    9. Xue Pan & Lei Hou & Mutua Stephen & Huijie Yang & Chenping Zhu, 2014. "Evaluation of Scaling Invariance Embedded in Short Time Series," PLOS ONE, Public Library of Science, vol. 9(12), pages 1-27, December.
    10. Xie, Wen-Jie & Jiang, Zhi-Qiang & Zhou, Wei-Xing, 2014. "Extreme value statistics and recurrence intervals of NYMEX energy futures volatility," Economic Modelling, Elsevier, vol. 36(C), pages 8-17.
    11. Yang, Yan-Hong & Shao, Ying-Hui & Shao, Hao-Lin & Stanley, H. Eugene, 2019. "Revisiting the weak-form efficiency of the EUR/CHF exchange rate market: Evidence from episodes of different Swiss franc regimes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 523(C), pages 734-746.
    12. Nurbanu Bursa & Hüseyin Tatlýdil, 2015. "Investigation of Credit Default Swaps using Detrended Fluctuation Analysis which is an Econophysical Technique," Eurasian Eononometrics, Statistics and Emprical Economics Journal, Eurasian Academy Of Sciences, vol. 2(2), pages 25-33, October.
    13. Ni, Xiao-Hui & Jiang, Zhi-Qiang & Gu, Gao-Feng & Ren, Fei & Chen, Wei & Zhou, Wei-Xing, 2010. "Scaling and memory in the non-Poisson process of limit order cancelation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(14), pages 2751-2761.
    14. Zhou, Yu & Leung, Yee & Chan, Lung Sang, 2017. "Oscillatory tendency of interevent direction in earthquake sequences," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 478(C), pages 120-130.
    15. Neelakshi, J. & Rosa, Reinaldo R. & Savio, Siomel & Stephany, Stephan & de Meneses, Francisco C. & Kherani, Esfhan Alam & Muralikrishna, P., 2022. "Multifractal characteristics of the low latitude equatorial ionospheric E–F valley region irregularities," Chaos, Solitons & Fractals, Elsevier, vol. 156(C).
    16. Wang, Lei & Liu, Lutao, 2020. "Long-range correlation and predictability of Chinese stock prices," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 549(C).
    17. Xiong, Gang & Zhang, Shuning & Liu, Qiang, 2012. "The time-singularity multifractal spectrum distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(20), pages 4727-4739.
    18. Akash P. POOJARI & Siva Kiran GUPTHA & G Raghavender RAJU, 2022. "Multifractal analysis of equities. Evidence from the emerging and frontier banking sectors," Theoretical and Applied Economics, Asociatia Generala a Economistilor din Romania - AGER, vol. 0(3(632), A), pages 61-80, Autumn.
    19. Rybski, Diego & Bunde, Armin, 2009. "On the detection of trends in long-term correlated records," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(8), pages 1687-1695.
    20. Lulu Qi & Zhilong Guo & Zhongxiang Qi & Jijun Guo, 2022. "Prospects of Precipitation Based on Reconstruction over the Last 2000 Years in the Qilian Mountains," Sustainability, MDPI, vol. 14(17), pages 1-15, August.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:eurphb:v:94:y:2021:i:11:d:10.1140_epjb_s10051-021-00231-7. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.