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Modeling the relationship between Higuchi’s fractal dimension and Fourier spectra of physiological signals

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Abstract

The exact mathematical relationship between FFT spectrum and fractal dimension (FD) of an experimentally recorded signal is not known. In this work, we tried to calculate signal FD directly from its Fourier amplitudes. First, dependence of Higuchi’s FD of mathematical sinusoids on their individual frequencies was modeled with a two-parameter exponential function. Next, FD of a finite sum of sinusoids was found to be a weighted average of their FDs, weighting factors being their Fourier amplitudes raised to a fractal degree. Exponent dependence on frequency was modeled with exponential, power and logarithmic functions. A set of 280 EEG signals and Weierstrass functions were analyzed. Cross-validation was done within EEG signals and between them and Weierstrass functions. Exponential dependence of fractal exponents on frequency was found to be the most accurate. In this work, signal FD was for the first time expressed as a fractal weighted average of FD values of its Fourier components, also allowing researchers to perform direct estimation of signal fractal dimension from its FFT spectrum.

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References

  1. Acharya RU, Faust O, Kannathal N, Chua T, Laxminarayan S (2005) Non-linear analysis of EEG signals at various sleep stages. Comput Method Program Biomed 80:37–45

    Article  Google Scholar 

  2. Bhattacharya J (2000) Complexity analysis of spontaneous EEG. Acta Neurobiol Exp 60:495–501

    CAS  Google Scholar 

  3. Bojić T, Vuckovic A, Kalauzi A (2010) Modeling EEG fractal dimension changes in wake and drowsy states in humans—a preliminary study. J Theor Biol 262(2):214–222

    Article  PubMed  Google Scholar 

  4. Eke A, Hermann P, Bassingthwaighte JB, Raymond GM, Percival DB, Cannon M, Balla I, Ikrenyi C (2000) Physiological time series: distinguishing fractal noises from motions. Pflug Arch 439:403–415

    Article  CAS  Google Scholar 

  5. Esteller R, Vachtsevanos G, Echauz J, Litt B (2001) A comparison of waveform fractal dimension algorithms. IEEE Trans Circ Syst I Fundam Theory Appl 48:177–183

    Article  Google Scholar 

  6. Fox CG (1989) Empirically derived relationships between fractal dimension and power law from frequency spectra. Pure Appl Geophys 131(1–2):211–239

    Article  Google Scholar 

  7. Higuchi T (1988) Approach to an irregular time series on the basis of the fractal theory. Phys D 31:277–283

    Article  Google Scholar 

  8. Higuchi T (1990) Relationship between the fractal dimension and the power law index for a time series: a numerical investigation. Physica D 46:254–264

    Article  Google Scholar 

  9. Kalauzi A, Bojić T, Rakić LJ (2008) Estimation of Higuchi fractal dimension for short signal epochs (<10 samples). In: The 10th experimental chaos conference. Catania, Italy, 3–6 June, pp 89

  10. Kalauzi A, Bojic T, Rakic Lj (2009) Extracting complexity waveforms from one-dimensional signals. Nonlin Biomed Phys 3:8

    Article  Google Scholar 

  11. Kalauzi A, Spasić S, Ćulić M, Grbić G, Martać LJ (2004) Correlation between fractal dimension and power spectra after unilateral cerebral injury in rat. In: FENS 2004, Abstracts, vol 2. Lisbon, A199.9

  12. Kalauzi A, Spasic S, Culic M, Grbic G, Martac Lj (2005) Consecutive differences as a method of signal fractal analysis. Fractals 13(4):283–292

    Article  Google Scholar 

  13. Katz M (1988) Fractals and the analysis of waveforms. Comput Biol Med 18(3):145–156

    Article  PubMed  CAS  Google Scholar 

  14. Kunhimangalam R, Joseph PK, Sujith OK (2008) Nonlinear analysis of EEG signals: surrogate data analysis. IRBM 29:239–244

    Article  Google Scholar 

  15. Lutzenberger W, Elbert T, Birbaumer N, Ray WJ, Schupp H (1992) The scalp distribution of the fractal dimension of the EEG and its validation with mental tasks. Brain Topogr 5(1):27–34

    Article  PubMed  CAS  Google Scholar 

  16. Navascués MA, Sebastián MV (2006) Spectral and affine fractal methods in signal processing. Int Math Forum 29:1405–1422

    Google Scholar 

  17. Petrosian A (1995) Kolmogorov complexity of finite sequences and recognition of different preictal EEG patterns. In: Proceedings of IEEE symposium computer-based med systems, pp 212–217

  18. Sebastián MV, Navascués MA (2008) A relation between fractal dimension and Fourier transform—electroencephalographic study using spectral and fractal parameters. Int J Comp Math 85(3–4):657–665

    Article  Google Scholar 

  19. Spasić S, Kalauzi A, Ćulić M, Grbić G, Martać Lj (2005) Estimation of parameter kmax in fractal analysis of rat brain activity. Ann NY Acad Sci 1048:427–429

    Article  PubMed  Google Scholar 

  20. Spasic S, Kalauzi A, Grbic G, Martac L, Culic M (2005) Fractal analysis of rat brain activity after injury. Med Biol Eng Comput 43:345–348

    Article  PubMed  CAS  Google Scholar 

  21. Spasic S, Kalauzi A, Kesic S, Obradovic M, Saponjic J (2011) Surrogate data modeling the relationship between high frequency amplitudes and Higuchi fractal dimension of EEG signals in anesthetized rats. J Theor Biol 289:160–166

    Article  PubMed  Google Scholar 

  22. Stam CJ (2005) Nonlinear dynamical analysis of EEG and EMG: review of an emerging field. Clin Neurophysiol 116:2266–2301

    Article  PubMed  CAS  Google Scholar 

  23. Theil H (1961) Economic forecasts and policy. North Holland, Amsterdam

    Google Scholar 

  24. Vuckovic A, Radivojevic V, Chen ACN, Popovic D (2002) Automatic recognition of alertness and drowsiness from EEG by an artificial neural network. Med Eng Phys 24:349–360

    Article  PubMed  Google Scholar 

  25. Weiss B, Clemens Z, Bόdisz R, Halász P (2011) Comparison of fractal and power EEG features: effects of topography and sleep stages. Brain Res Bull 84(6):359–375

    Article  PubMed  Google Scholar 

  26. Ziller M, Frick K, Herrmann WM, Kubicki S, Spieweg I, Winterer G (1995) Bivariate global frequency analysis versus chaos theory. A comparison for sleep EEG data. Neuropsychobiology 32(1):45–51

    Article  PubMed  CAS  Google Scholar 

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Acknowledgments

This work was financed by the Ministry of Education and Science of the Republic of Serbia (Projects OI 173022 and III 41028). We thank Dr. Vlada Radivojević and Mr. Predrag Šuković for their help and cooperation in obtaining and analyzing the data and Dr. Žarko Martinović for second opinions in visual analysis and scoring of EEG records.

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Correspondence to Aleksandar Kalauzi.

Appendix

Appendix

We shall derive an exact expression linking the frequency of a mathematical sinusoid with its fractal dimension, according to the procedure described in [7]. In order to apply trigonometric summation rules, instead of the absolute difference of signal samples, as in (1), we shall observe the sum of squares

$$ L_{m}^{2} (k) = \frac{1}{k}\left( {\sum\limits_{i = 1}^{{\left[ {\frac{N - m}{k}} \right]}} {\left( {x(m + ik) - x(m + (i - 1)k)} \right)^{2} } } \right)\frac{N - 1}{{\left[ {\frac{N - m}{k}} \right]k}}. $$
(18)

Consequently, FD should be calculated according to a slightly modified formula:

$$ {\text{FD}} = \frac{1}{2}({\text{slope}}(\ln (1/k),\ln (L^{2} (k)) + 1) = k = \, 1, \ldots ,k_{\max } , $$

where

$$ L^{2} (k) = \frac{1}{k}\sum\limits_{m = 1}^{k} {L_{m}^{2} (k)} . $$
(19)

For a given sinusoid \( x(i) = a\sin\, (\omega i\Updelta t + \varphi ) \), where i = 1, …, N; a and φ representing its amplitude and initial phase and Δt = 1/f samp, formula (18) becomes

$$ L_{m}^{2} (k) = \frac{1}{k}\left( {\sum\limits_{i = 1}^{{\left[ {\frac{N - m}{k}} \right]}} {\left( {a\sin \left( {(m + ik)\omega \Updelta t + \varphi } \right) - a\sin \left( {(m + (i - 1)k)\omega \Updelta t + \varphi } \right)} \right)^{2} } } \right)\frac{N - 1}{{\left[ {\frac{N - m}{k}} \right]k}}. $$

After transforming the difference of sinuses into product, and application of summation rule of squared cosines over the index i, we can write

$$ L_{m}^{2} (k) = A\left( {B + C\cos \left( {2((m - \frac{k}{2})\omega \Updelta t + \phi ) + \left[ {\frac{N - m}{k}} \right]k\omega \Updelta t} \right) - \cos^{2} \left( {(m - \frac{k}{2})\omega \Updelta t + \phi } \right)} \right) $$

where

$$ A = \frac{{4(N - 1)a^{2} }}{{\left[ {\frac{N - m}{k}} \right]k^{2} }}\sin^{2} \left( {\frac{k}{2}\omega \Updelta t} \right);\quad B = \frac{{\left[ {\frac{N - m}{k}} \right] + 1}}{2};\quad C = \frac{{\sin \left( {\left( {\left[ {\frac{N - m}{k}} \right] + 1} \right)k\omega \Updelta t} \right)}}{2\sin (k\omega \Updelta t)}. $$

As m appears within the square brackets as well, cosine summation rule over index m cannot be performed directly. Respecting that for most signals N ≫ m, therefore\( \left[ {\frac{N - m}{k}} \right] \approx \left[ \frac{N}{k} \right] \), after some elementary operations equation (19) could be written as

$$ L^{2} (k) = \frac{1}{k}\sum\limits_{m = 1}^{k} {\left( {AB + AC\cos\left( {m2\omega \Updelta t + \left( {\left[ \frac{N}{k} \right] - 1} \right)k\omega \Updelta t + \varphi } \right) - A\cos^{2}\left( {m\omega \Updelta t - \frac{k}{2}\omega \Updelta t + \varphi } \right)} \right)}. $$

Finally, according to the summation rules,

$$ L^{2} (k) = \frac{AB}{k} + \frac{AC}{k}L_{1} - \frac{A}{k}L_{2} $$
(20)

where

$$ L_{1} = \frac{{\cos\left( {\left[ \frac{N}{k} \right]k\omega \Updelta t + \varphi } \right)\sin \left( {(k + 1)\omega \Updelta t} \right)}}{{\sin \left( {\omega \Updelta t} \right)}} - \cos\left( {\left( {\left[ \frac{N}{k} \right] - 1} \right)k\omega \Updelta t + \varphi } \right); $$
$$ L_{2} = \left( {\frac{k + 1}{2} + \frac{{\sin \left( {(k + 1)\omega \Updelta t} \right)\cos (2\varphi )}}{2\sin (\omega \Updelta t)} - \cos^{2} \left( { - \frac{k}{2}\omega \Updelta t + \varphi } \right)} \right). $$

Expression (20) was verified by calculating FD of sinusoids for \( f = \frac{\omega }{2\pi } = 2, 4,\ldots, 30\,{\hbox{Hz}},\) for f samp = 256 Hz, and \( f = \frac{\omega }{2\pi } \)  = 1, 2,…, 14 Hz, for f samp = 128 Hz and the results presented as symbols in Fig. 1.

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Kalauzi, A., Bojić, T. & Vuckovic, A. Modeling the relationship between Higuchi’s fractal dimension and Fourier spectra of physiological signals. Med Biol Eng Comput 50, 689–699 (2012). https://doi.org/10.1007/s11517-012-0913-9

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