Skip to main content
SpringerLink
Log in

Fractal Time Series: Background, Estimation Methods, and Performances

  • Chapter
  • First Online:
The Fractal Geometry of the Brain

Part of the book series: Advances in Neurobiology ((NEUROBIOL,volume 36))

  • 293 Accesses

Abstract

Over the past 40 years, from its classical application in the characterization of geometrical objects, fractal analysis has been progressively applied to study time series in several different disciplines. In neuroscience, starting from identifying the fractal properties of neuronal and brain architecture, attention has shifted to evaluating brain signals in the time domain. Classical linear methods applied to analyzing neurophysiological signals can lead to classifying irregular components as noise, with a potential loss of information. Thus, characterizing fractal properties, namely, self-similarity, scale invariance, and fractal dimension (FD), can provide relevant information on these signals in physiological and pathological conditions. Several methods have been proposed to estimate the fractal properties of these neurophysiological signals. However, the effects of signal characteristics (e.g., its stationarity) and other signal parameters, such as sampling frequency, amplitude, and noise level, have partially been tested. In this chapter, we first outline the main properties of fractals in the domain of space (fractal geometry) and time (fractal time series). Then, after providing an overview of the available methods to estimate the FD, we test them on synthetic time series (STS) with different sampling frequencies, signal amplitudes, and noise levels. Finally, we describe and discuss the performances of each method and the effect of signal parameters on the accuracy of FD estimation.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 189.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 249.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Abbreviations

BC:

Box counting

DFA:

Detrended fluctuation analysis

fBm:

Fractional Brownian motion

FD:

Fractal dimension

fGn:

Fractional Gaussian noise

fs:

Sampling frequency

GHE:

Generalized Hurst exponent

H:

Hurst exponent

HFD:

Higuchi's fractal dimension

KFD:

Katz's fractal dimension

SNR:

Signal-to-noise ratio

sPSD:

Slope of power spectral density

STS:

Synthetic time series

TLF:

Takagi–Landsberg function

WCF:

Weierstrass cosine function

References

  1. Abry P, Sellan F. The wavelet-based synthesis for fractional Brownian motion proposed by F. Sellan and Y. Meyer: remarks and fast implementation. Appl Comput Harmon Anal. 1996;3:377–83. https://doi.org/10.1006/acha.1996.0030.

    Article  Google Scholar 

  2. Affinito M, Carrozzi M, Accardo A, Bouquet F. Use of the fractal dimension for the analysis of electroencephalographic time series. Biol Cybern. 1997;77:339–50. https://doi.org/10.1007/s004220050394.

    Article  PubMed  Google Scholar 

  3. Barabási A-L, Vicsek T. Multifractality of self-affine fractals. Phys Rev A. 1991;44:2730–3. https://doi.org/10.1103/PhysRevA.44.2730.

    Article  PubMed  Google Scholar 

  4. Borri A, Cerasa A, Tonin P, et al. Characterizing fractal genetic variation in the human genome from the Hapmap project. Int J Neural Syst. 2022;32:2250028. https://doi.org/10.1142/S0129065722500289.

    Article  PubMed  Google Scholar 

  5. Bryce RM, Sprague KB. Revisiting detrended fluctuation analysis. Sci Rep. 2012;2:315. https://doi.org/10.1038/srep00315.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  6. Buczkowski S, Hildgen P, Cartilier L. Measurements of fractal dimension by box-counting: a critical analysis of data scatter. Phys Stat Mech Its Appl. 1998;252:23–34. https://doi.org/10.1016/S0378-4371(97)00581-5.

    Article  Google Scholar 

  7. Churchill NW, Spring R, Grady C, et al. The suppression of scale-free fMRI brain dynamics across three different sources of effort: aging, task novelty and task difficulty. Sci Rep. 2016;6:30895. https://doi.org/10.1038/srep30895.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  8. Cottone C, Porcaro C, Cancelli A, et al. Neuronal electrical ongoing activity as a signature of cortical areas. Brain Struct Funct. 2017;222:2115–26. https://doi.org/10.1007/s00429-016-1328-4.

    Article  PubMed  Google Scholar 

  9. de Amo E, Díaz Carrillo M, Fernández-Sánchez J. Singular functions with applications to fractal dimensions and generalized Takagi functions. Acta Appl Math. 2012;119:129–48. https://doi.org/10.1007/s10440-011-9665-z.

    Article  Google Scholar 

  10. Delignieres D, Ramdani S, Lemoine L, et al. Fractal analyses for ‘short’ time series: a re-assessment of classical methods. J Math Psychol. 2006;50:525–44. https://doi.org/10.1016/j.jmp.2006.07.004.

    Article  Google Scholar 

  11. Delignières D, Torre K, Bernard P-L. Transition from persistent to anti-persistent correlations in postural sway indicates velocity-based control. PLoS Comput Biol. 2011;7:e1001089. https://doi.org/10.1371/journal.pcbi.1001089.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  12. Di Ieva A, editor. The fractal geometry of the brain. New York, NY: Springer; 2016.

    Google Scholar 

  13. Di Ieva A, Bruner E, Widhalm G, et al. Computer-assisted and fractal-based morphometric assessment of microvascularity in histological specimens of gliomas. Sci Rep. 2012;2:429. https://doi.org/10.1038/srep00429.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  14. Di Ieva A, Grizzi F, Jelinek H, et al. Fractals in the neurosciences, part I: general principles and basic neurosciences. Neuroscientist. 2014;20:403–17. https://doi.org/10.1177/1073858413513927.

    Article  CAS  PubMed  Google Scholar 

  15. Dick EO, Murav’eva SV, Lebedev VS, Shelepin Yu E. Fractal structure of brain electrical activity of patients with mental disorders. Front Physiol. 2022;13:905318. https://doi.org/10.3389/fphys.2022.905318.

    Article  Google Scholar 

  16. Dong J, Jing B, Ma X, et al. Hurst exponent analysis of resting-state fMRI signal complexity across the adult lifespan. Front Neurosci. 2018;12:34. https://doi.org/10.3389/fnins.2018.00034.

    Article  PubMed  PubMed Central  Google Scholar 

  17. Dubuc B, Dubuc S. Error bounds on the estimation of fractal dimension. SIAM J Numer Anal. 1996;33:602–26. https://doi.org/10.1137/0733032.

    Article  Google Scholar 

  18. Eke A, Herman P, Kocsis L, Kozak LR. Fractal characterization of complexity in temporal physiological signals. Physiol Meas. 2002;23:R1–R38. https://doi.org/10.1088/0967-3334/23/1/201.

    Article  CAS  PubMed  Google Scholar 

  19. Esteller R, Vachtsevanos G, Echauz J, Litt B. A comparison of waveform fractal dimension algorithms. IEEE Trans Circuits Syst Fundam Theory Appl. 2001;48:177–83. https://doi.org/10.1109/81.904882.

    Article  Google Scholar 

  20. Evertz R, Hicks DG, Liley DTJ. Alpha blocking and 1/fβ spectral scaling in resting EEG can be accounted for by a sum of damped alpha band oscillatory processes. PLoS Comput Biol. 2022;18:e1010012. https://doi.org/10.1371/journal.pcbi.1010012.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  21. Feder J. Fractals. Boston, MA: Springer; 1988.

    Book  Google Scholar 

  22. Gazit Y, Berk DA, Leunig M, et al. Scale-invariant behavior and vascular network formation in Normal and tumor tissue. Phys Rev Lett. 1995;75:2428–31. https://doi.org/10.1103/PhysRevLett.75.2428.

    Article  CAS  PubMed  Google Scholar 

  23. Gentili C, Vanello N, Cristea I, et al. Proneness to social anxiety modulates neural complexity in the absence of exposure: a resting state fMRI study using Hurst exponent. Psychiatry Res Neuroimaging. 2015;232:135–44. https://doi.org/10.1016/j.pscychresns.2015.03.005.

    Article  Google Scholar 

  24. Grizzi F, Chiriva-Internati M. The complexity of anatomical systems. Theor Biol Med Model. 2005;2:26. https://doi.org/10.1186/1742-4682-2-26.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  25. Grosu GF, Hopp AV, Moca VV, et al. The fractal brain: scale-invariance in structure and dynamics. Cereb Cortex bhac. 2022;363 https://doi.org/10.1093/cercor/bhac363.

  26. Hardstone R, Poil S-S, Schiavone G, et al. Detrended fluctuation analysis: a scale-free view on neuronal oscillations. Front Physiol. 2012:3. https://doi.org/10.3389/fphys.2012.00450.

  27. Higuchi T. Approach to an irregular time series on the basis of the fractal theory. Phys Nonlinear Phenom. 1988;31:277–83. https://doi.org/10.1016/0167-2789(88)90081-4.

    Article  Google Scholar 

  28. Higuchi T. Relationship between the fractal dimension and the power law index for a time series: a numerical investigation. Phys Nonlinear Phenom. 1990;46:254–64. https://doi.org/10.1016/0167-2789(90)90039-R.

    Article  Google Scholar 

  29. Hurst HE. Long-term storage capacity of reservoirs. Trans Am Soc Civ Eng. 1951;116:770–99. https://doi.org/10.1061/TACEAT.0006518.

    Article  Google Scholar 

  30. Jiang B, Brandt S. A fractal perspective on scale in geography. ISPRS Int J Geo Inf. 2016;5:95. https://doi.org/10.3390/ijgi5060095.

    Article  Google Scholar 

  31. Katz MJ. Fractals and the analysis of waveforms. Comput Biol Med. 1988;18:145–56. https://doi.org/10.1016/0010-4825(88)90041-8.

    Article  CAS  PubMed  Google Scholar 

  32. Kesić S, Spasić SZ. Application of Higuchi’s fractal dimension from basic to clinical neurophysiology: a review. Comput Methods Programs Biomed. 2016;133:55–70. https://doi.org/10.1016/j.cmpb.2016.05.014.

    Article  PubMed  Google Scholar 

  33. Lagarias JC (2011) The Takagi function and its properties. https://doi.org/10.48550/ARXIV.1112.4205.

  34. Lee C-Y. The fractal dimension as a measure for characterizing genetic variation of the human genome. Comput Biol Chem. 2020;87:107278. https://doi.org/10.1016/j.compbiolchem.2020.107278.

    Article  CAS  PubMed  Google Scholar 

  35. Lee J-S, Yang B-H, Lee J-H, et al. Detrended fluctuation analysis of resting EEG in depressed outpatients and healthy controls. Clin Neurophysiol. 2007;118:2489–96. https://doi.org/10.1016/j.clinph.2007.08.001.

    Article  PubMed  Google Scholar 

  36. Lloyd EH, Hurst HE, Black RP, Simaika YM. Long-term storage: an experimental study. J R Stat Soc Ser Gen. 1966;129:591. https://doi.org/10.2307/2982267.

    Article  Google Scholar 

  37. Losa GA. The fractal geometry of life. Riv Biol. 2009;102:29–59.

    PubMed  Google Scholar 

  38. Losa GA, Nonnenmacher TF. Self-similarity and fractal irregularity in pathologic tissues. Mod Pathol Off J U S Can Acad Pathol Inc. 1996;9:174–82.

    CAS  Google Scholar 

  39. Losa GA, Di Ieva A, Grizzi F, De Vico G. On the fractal nature of nervous cell system. Front Neuroanat. 2011;5 https://doi.org/10.3389/fnana.2011.00045.

  40. Malamud BD, Turcotte DL. Self-affine time series: I. generation and analyses. In: Advances in geophysics. Elsevier; 1999. p. 1–90.

    Google Scholar 

  41. Mandelbrot B. How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science. 1967;156:636–8. https://doi.org/10.1126/science.156.3775.636.

    Article  CAS  PubMed  Google Scholar 

  42. Mandelbrot BB, Van Ness JW. Fractional Brownian motions, fractional noises and applications. SIAM Rev. 1968;10:422–37. https://doi.org/10.1137/1010093.

    Article  Google Scholar 

  43. Mandelbrot BB, Wallis JR. Noah, Joseph, and operational hydrology. Water Resour Res. 1968;4:909–18. https://doi.org/10.1029/WR004i005p00909.

    Article  Google Scholar 

  44. Mandelbrot BB, Wallis JR. Computer experiments with fractional Gaussian noises: part 2, rescaled ranges and spectra. Water Resour Res. 1969;5:242–59. https://doi.org/10.1029/WR005i001p00242.

    Article  Google Scholar 

  45. Mandelbrot BB, Wheeler JA. The fractal geometry of nature. Am J Physiol. 1983;51:286–7. https://doi.org/10.1119/1.13295.

    Article  Google Scholar 

  46. Marino M, Liu Q, Samogin J, et al. Neuronal dynamics enable the functional differentiation of resting state networks in the human brain. Hum Brain Mapp. 2019;40:1445–57. https://doi.org/10.1002/hbm.24458.

    Article  PubMed  Google Scholar 

  47. Nolte G, Aburidi M, Engel AK. Robust calculation of slopes in detrended fluctuation analysis and its application to envelopes of human alpha rhythms. Sci Rep. 2019;9:6339. https://doi.org/10.1038/s41598-019-42732-7.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  48. Olejarczyk E, Sobieszek A, Rudner R, Marciniak R, Wartak M, Stasiowski M, Jalowiecki P. Evaluation of the EEG-signal during volatile anaesthesia: methodological approach. Biocybern Biomed Eng. 2009;29(1):3–28.

    Google Scholar 

  49. Olejarczyk E, Gotman J, Frauscher B. Region-specific complexity of the intracranial EEG in the sleeping human brain. Sci Rep. 2022;12(1):451.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  50. Ouyang G, Hildebrandt A, Schmitz F, Herrmann CS. Decomposing alpha and 1/f brain activities reveals their differential associations with cognitive processing speed. Neuroimage. 2020;205:116304. https://doi.org/10.1016/j.neuroimage.2019.116304.

    Article  PubMed  Google Scholar 

  51. Paramanathan P, Uthayakumar R. Application of fractal theory in analysis of human electroencephalographic signals. Comput Biol Med. 2008;38:372–8. https://doi.org/10.1016/j.compbiomed.2007.12.004.

    Article  CAS  PubMed  Google Scholar 

  52. Paumgartner D, Losa G, Weibel ER. Resolution effect on the stereological estimation of surface and volume and its interpretation in terms of fractal dimensions. J Microsc. 1981;121:51–63. https://doi.org/10.1111/j.1365-2818.1981.tb01198.x.

    Article  CAS  PubMed  Google Scholar 

  53. Peano G. Sur une courbe, qui remplit toute une aire plane. Math Ann. 1890;36:157–60. https://doi.org/10.1007/BF01199438.

    Article  Google Scholar 

  54. Peng C-K, Buldyrev SV, Havlin S, et al. Mosaic organization of DNA nucleotides. Phys Rev E. 1994;49:1685–9. https://doi.org/10.1103/PhysRevE.49.1685.

    Article  CAS  Google Scholar 

  55. Pieter CA, Kawamura K. The Takagi Function: a Survey. Real Anal Exch. 2012;37:1. https://doi.org/10.14321/realanalexch.37.1.0001.

    Article  Google Scholar 

  56. Porcaro C, Cottone C, Cancelli A, Rossini PM, Zito G, Tecchio F. Cortical neurodynamics changes mediate the efficacy of a personalized neuromodulation against multiple sclerosis fatigue. Sci Rep. 2019;9(1):18213. https://doi.org/10.1038/s41598-019-54595-.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  57. Porcaro C, Di Renzo A, Tinelli E, et al. Haemodynamic activity characterization of resting state networks by fractal analysis and thalamocortical morphofunctional integrity in chronic migraine. J Headache Pain. 2020a;21:112. https://doi.org/10.1186/s10194-020-01181-8.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  58. Porcaro C, Mayhew SD, Marino M, et al. Characterisation of Haemodynamic activity in resting state networks by fractal analysis. Int J Neural Syst. 2020b;30:2050061. https://doi.org/10.1142/S0129065720500616.

    Article  PubMed  Google Scholar 

  59. Porcaro C, Di Renzo A, Tinelli E, Di Lorenzo G, Seri S, Di Lorenzo C, Parisi V, Caramia F, Fiorelli M, Di Piero V, Pierelli F, Coppola G. Hypothalamic structural integrity and temporal complexity of cortical information processing at rest in migraine without aura patients between attacks. Sci Rep. 2021;11(1):18701. https://doi.org/10.1038/s41598-021-98213-3.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  60. Porcaro C, Di Renzo A, Tinelli E, et al. A hypothalamic mechanism regulates the duration of a migraine attack: insights from microstructural and temporal complexity of cortical functional networks analysis. Int J Mol Sci. 2022a;23:13238. https://doi.org/10.3390/ijms232113238.

    Article  PubMed  PubMed Central  Google Scholar 

  61. Porcaro C, Marino M, Carozzo S, et al. Fractal dimension feature as a signature of severity in disorders of consciousness: an EEG study. Int J Neural Syst. 2022b;32:2250031. https://doi.org/10.1142/S0129065722500319.

    Article  PubMed  Google Scholar 

  62. Raghavendra BS, Dutt DN. Computing fractal dimension of signals using multiresolution box-counting method. Int J Inf Math Sci. 2010;6(1):50–65. https://doi.org/10.5281/ZENODO.1057349.

    Article  Google Scholar 

  63. Raghavendra BS, Narayana Dutt D. A note on fractal dimensions of biomedical waveforms. Comput Biol Med. 2009;39:1006–12. https://doi.org/10.1016/j.compbiomed.2009.08.001.

    Article  CAS  PubMed  Google Scholar 

  64. Reishofer G, Studencnik F, Koschutnig K, et al. Age is reflected in the fractal dimensionality of MRI diffusion based tractography. Sci Rep. 2018;8:5431. https://doi.org/10.1038/s41598-018-23769-6.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  65. Richards GR. A fractal forecasting model for financial time series. J Forecast. 2004;23:586–601. https://doi.org/10.1002/for.927.

    Article  Google Scholar 

  66. Rigaut JP. An empirical formulation relating boundary lengths to resolution in specimens showing ‘non-ideally fractal’ dimensions. J Microsc. 1984;133:41–54. https://doi.org/10.1111/j.1365-2818.1984.tb00461.x.

    Article  Google Scholar 

  67. Rudner R, Jalowiecki P, Willand M, Klonowski W, Olejarczyk E, Stepien R, Hagihira S. Fractal dimension - a new EEG-based method of assessing depth of anaesthesia in comparison with BIS during induction and recovery from anaesthesia. Eur J Anaesthesiol. 2005;22(suppl. 34, A-118):32–3.

    Article  Google Scholar 

  68. Schmittbuhl J, Vilotte J-P, Roux S. Reliability of self-affine measurements. Phys Rev E. 1995;51:131–47. https://doi.org/10.1103/PhysRevE.51.131.

    Article  CAS  Google Scholar 

  69. Shao Z-G, Ditlevsen PD. Contrasting scaling properties of interglacial and glacial climates. Nat Commun. 2016;7:10951. https://doi.org/10.1038/ncomms10951.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  70. Shi C-T. Signal pattern recognition based on fractal features and machine learning. Appl Sci. 2018;8:1327. https://doi.org/10.3390/app8081327.

    Article  Google Scholar 

  71. Smith JH, Rowland C, Harland B, et al. How neurons exploit fractal geometry to optimize their network connectivity. Sci Rep. 2021;11:2332. https://doi.org/10.1038/s41598-021-81421-2.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  72. Smits FM, Porcaro C, Cottone C, et al. Electroencephalographic fractal dimension in healthy ageing and Alzheimer’s disease. PloS One. 2016;11:e0149587. https://doi.org/10.1371/journal.pone.0149587.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  73. Varley TF, Craig M, Adapa R, et al. Fractal dimension of cortical functional connectivity networks & severity of disorders of consciousness. PloS One. 2020;15:e0223812. https://doi.org/10.1371/journal.pone.0223812.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  74. Willand M, Rudner R, Olejarczyk E, Wartak M, Marciniak R, Stasiowski M, Byrczek T, Jalowiecki P. Fractal dimension – a new EEG-based method of assessing the depth of anaesthesia. Anaesthesiol Intens Ther. 2008;4:217–22.

    Google Scholar 

  75. Zappasodi F, Olejarczyk E, Marzetti L, Assenza G, Pizzella V, Tecchio F. Fractal dimension of EEG activity senses neuronal impairment in acute stroke. PloS One. 2014;9(6):e100199.

    Article  PubMed  PubMed Central  Google Scholar 

  76. Zappasodi F, Marzetti L, Olejarczyk E, Tecchio F, Pizzella V. Age-Related Changes in Electroencephalographic Signal Complexity. PloS One. 2015;10(11):e0141995.

    Article  PubMed  PubMed Central  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Neuroscience (DNS) and Padova Neuroscience Center (PNC), University of Padova, Padua, Italy

    Camillo Porcaro

  2. Institute of Cognitive Sciences and Technologies (ISTC) National Research Council (CNR), Rome, Italy

    Camillo Porcaro

  3. Centre for Human Brain Health, School of Psychology, University of Birmingham, Birmingham, UK

    Camillo Porcaro

  4. Department of Neuroscience (DNS), University of Padova, Padua, Italy

    Sadaf Moaveninejad

  5. Padova Neuroscience Center (PNC), University of Padova, Padua, Italy

    Valentina D’Onofrio

  6. Computational NeuroSurgery (CNS) Lab & Macquarie Neurosurgery, Macquarie Medical School, Faculty of Medicine, Human and Health Sciences, Macquarie University, Sydney, NSW, Australia

    Antonio DiIeva

Authors
  1. Camillo Porcaro
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sadaf Moaveninejad
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Valentina D’Onofrio
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Antonio DiIeva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Camillo Porcaro .

Editor information

Editors and Affiliations

  1. Computational NeuroSurgery (CNS) Lab & Macquarie Neurosurgery Macquarie Medical School, Faculty of Medicine, Human and Health Sciences, Macquarie University, Sydney, NSW, Australia

    Antonio Di Ieva

Rights and permissions

Reprints and permissions

Copyright information

© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Porcaro, C., Moaveninejad, S., D’Onofrio, V., DiIeva, A. (2024). Fractal Time Series: Background, Estimation Methods, and Performances. In: Di Ieva, A. (eds) The Fractal Geometry of the Brain. Advances in Neurobiology, vol 36. Springer, Cham. https://doi.org/10.1007/978-3-031-47606-8_5

Download citation

Publish with us

Policies and ethics

Search

Navigation