A Brief Overview of Multifractal Time Series

• Part 1: Fractal behavior in time series
• Part 2: Using wavelets to detect singular behavior
• Part 3: The fractal dimension of the singular behavior
• Part 4: The singularity spectra of multifractal signals
• Part 5: What one learns from the singularity spectra of multifractal signals
• Part 6: Multifractality of healthy human heart rate
• Bibliography
  

1. Daubechies, I., Ten Lectures on Wavelets (S.I.A.M., Philadelphia, 1992).

2. Feder, J., Fractals (Plenum Press, 1988).

3. Muzy, J. F., Bacry, E. & Arneodo, A. (1994) The multifractal formalism revisited with wavelets. Int. J. Bifurc. Chaos. 4, 245-302.

4. Vicsek, T., Fractal Growth Phenomena, 2nd ed. (World Scientific, Singapore, 1993).

5. Struzik. Z. R. (2000) Determining local singularity strengths and their spectra with the wavelet transform. Fractals 8, 163-179.

6. Hurst, H. E. (1951) Long-term storage capacity of reservoirs. Trans. Am. Soc. Civ. Eng. 116, 770-808.

7. Bunde, A. & Havlin, S., Fractals and Disordered Systems, 2nd ed. (Springer-Verlag, Berlin, 1996).

8.Mandelbrot, B. B., The Fractal Geometry of Nature (W. H. Freeman, 1983).

9. Stanley, H.E. & Meakin, P. (1988) Multifractal phenomena in physics and chemistry. Nature, 335, 405-409.

10. Meneveau, C. & Sreenivasan, K.R. (1987) Simple multifractal cascade model for fully developed turbulence. Phys. Rev. Lett. 59, 1424-1427.

11. Ivanov, P. Ch., Amaral, L. A. N., Goldberger, A. L., Havlin, S., Rosenblum, M. B., Struzik, Z. & Stanley, H. E. (1999) Multifractality in healthy heartbeat dynamics. Nature 399, 461-465.

12. Amaral, L.A.N., Goldberger, A.L., Ivanov, P.Ch. & Stanley H.E. (1998) Scale-independent measures and pathologic cardiac dynamics. Phys. Rev. Lett. 81, 2388-2391.

13. Lin, D.C., & Hughson, R.L. (2001) Modeling heart rate variability in healthy humans: a turbulence analogy. Phys. Rev. Lett. 86, 1650-1653.

14. Ivanov, P.Ch., Amaral, L.A.N., Goldberger, A.L., Havlin, S., Rosenblum, M.G., Stanley, H.E., & Struzik, Z. (2001) From 1/f noise to multifractal cascades in heartbeat dynamics. Chaos 11, 641-652.

15. Goldberger, A.L., Amaral, L.A.N., Hausdorff, J.M., Ivanov, P.Ch., Peng, C.K., & Stanley, H.E. (2002) Fractal dynamics in physiology: alterations with disease and aging. Proc. Natl. Acad. Sci. (U.S.A.) 99[suppl 1], 2466-2472.

16. Kantelhardt, J.W., Zschiegner, S.A., Koscielny-Bunde, E., Havlin, S., Bunde, A., & Stanley, H.E. (2002) Multifractal detrended fluctuation analysis of nonstationary times series. Physica A 316, 87-114.

17. Lin, D.C., & Hughson, R.L. (2002) A phenomenology model of normal sinus rhythm in healthy humans. IEEE Trans. Biomed. Eng. 49, 97-109.

18. Ashkenazy, Y., Havlin, S., Ivanov, P.Ch., Peng, C.K., Schulte-Frohlinde, V., & Stanley, H.E. (2003) Magnitude and sign scaling in power-law correlated time series. Physica A 323, 19.

19. Fischer, R., Akay, M., Castiglioni, P., & Di Rienzo, M. (2003) Multi- and monofractal indices of short-term heart rate variability. Med. Biol. Eng. Comput. 41, 543-549.

20. West, B.J., & Scafetta, N. (2003) Nonlinear dynamical model of human gait. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 67, 051917.

21. Ching, E.S., Lin, D.C., & Zhang, C. (2004) Hierarchical structure in healthy and diseased human heart rate variability. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69, 051919.

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