Encyclopedia of Complexity and Systems Science

Living Edition
| Editors: Robert A. Meyers

Fractal and Multifractal Time Series

  • Jan W. KantelhardtEmail author
Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27737-5_221-3

Definition of the Subject

Data series generated by complex systems exhibit fluctuations on a wide range of time scales and/or broad distributions of the values. In both equilibrium and nonequilibrium situations, the natural fluctuations are often found to follow a scaling relation over several orders of magnitude. Such scaling laws allow for a characterization of the data and the generating complex system by fractal (or multifractal) scaling exponents, which can serve as characteristic fingerprints of the systems in comparison with other systems and with models. Fractal scaling behavior has been observed, e.g., in many data series from experimental physics, geophysics, medicine, physiology, and even social sciences. Although the underlying causes of the observed fractal scaling are often not known in detail, the fractal or multifractal characterization can be used for generating surrogate (test) data, modeling the time series, and deriving predictions regarding extreme events or future...

Keywords

Empirical Mode Decomposition Hurst Exponent Scaling Behavior Detrended Fluctuation Analysis Return Interval 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Notes

Acknowledgment

We thank Ronny Bartsch, Amir Bashan, Mikhail Bogachev, Armin Bunde, Jan Eichner, Shlomo Havlin, Diego Rybski, Aicko Schumann, and Stephan Zschiegner for the helpful discussions and contributions. This work has been supported by the Deutsche Forschungsgemeinschaft (grants KA 1676/3 and KA 1676/4) and the European Union (FP6 project DAPHNet, grant 018474-2, and FP7 project SOCIONICAL, grant 231288).

Bibliography

  1. Alessio E, Carbone A, Castelli G, Frappietro V (2002) Second-order moving average and scaling of stochastic time series. Eur Phys J B 27:197ADSGoogle Scholar
  2. Altmann EG, Kantz H (2005) Recurrence time analysis, long-term correlations, and extreme events. Phys Rev E 71:056106MathSciNetADSCrossRefGoogle Scholar
  3. Alvarez-Ramirez J, Rodriguez E, Echeverría JC (2005) Detrending fluctuation analysis based on moving average filtering. Phys A 354:199CrossRefGoogle Scholar
  4. Alvarez-Ramirez J, Rodriguez E, Echeverria JC (2009a) A DFA approach for assessing asymmetric correlations. Phys A 388:2263CrossRefGoogle Scholar
  5. Alvarez-Ramirez J, Rodriguez E, Echeverria JC (2009b) Using detrended fluctuation analysis for lagged correlation analysis of nonstationary signals. Phys Rev E 79:057202ADSCrossRefGoogle Scholar
  6. Amaral LAN, Ivanov PC, Aoyagi N, Hidaka I, Tomono S, Goldberger AL, Stanley HE, Yamamoto Y (2001) Behavioral-independence features of complex heartbeat dynamics. Phys Rev Lett 86:6026ADSCrossRefGoogle Scholar
  7. Arneodo A, Bacry E, Graves PV, Muzy JF (1995) Characterizing long-range correlations in DNA sequences from wavelet analysis. Phys Rev Lett 74:3293ADSCrossRefGoogle Scholar
  8. Arneodo A, Manneville S, Muzy JF (1998) Towards log-normal statistics in high Reynolds number turbulence. Eur Phys J B 1:129ADSCrossRefGoogle Scholar
  9. Arneodo A, Audit B, Decoster N, Muzy JF, Vaillant C (2002) Wavelet based multifractal formalism: applications to DNA sequences, satellite images of the cloud structure, and stock market data. In: Bunde A, Kropp J, Schellnhuber HJ (eds) The science of disaster: climate disruptions, market crashes, and heart attacks. Springer, BerlinGoogle Scholar
  10. Ashkenazy Y, Ivanov PC, Havlin S, Peng CK, Goldberger AL, Stanley HE (2001) Magnitude and sign correlations in heartbeat fluctuations. Phys Rev Lett 86:1900ADSCrossRefGoogle Scholar
  11. Ashkenazy Y, Havlin S, Ivanov PC, Peng CK, Schulte-Frohlinde V, Stanley HE (2003) Magnitude and sign scaling in power-law correlated time series. Phys A 323:19CrossRefzbMATHGoogle Scholar
  12. Bacry E, Delour J, Muzy JF (2001) Multifractal random walk. Phys Rev E 64:026103ADSCrossRefzbMATHGoogle Scholar
  13. Bahar S, Kantelhardt JW, Neiman A, Rego HHA, Russell DF, Wilkens L, Bunde A, Moss F (2001) Long range temporal anti-correlations in paddlefish electro-receptors. Europhys Lett 56:454ADSCrossRefGoogle Scholar
  14. Barabási AL, Vicsek T (1991) Multifractality of self-affine fractals. Phys Rev A 44:2730ADSCrossRefGoogle Scholar
  15. Barnsley MF (1993) Fractals everywhere. Academic, San DiegozbMATHGoogle Scholar
  16. Bartsch R, Henning T, Heinen A, Heinrichs S, Maass P (2005) Statistical analysis of fluctuations in the ECG morphology. Phys A 354:415CrossRefGoogle Scholar
  17. Bashan A, Bartsch R, Kantelhardt JW, Havlin S (2008) Comparison of detrending methods for fluctuation analysis. Phys A 387:5080CrossRefGoogle Scholar
  18. Bogachev MI, Eichner JF, Bunde A (2007) Effect of nonlinear correlations on the statistics of return intervals in multifractal data sets. Phys Rev Lett 99:240601ADSCrossRefGoogle Scholar
  19. Bouchaud JP, Potters M (2003) Theory of financial risks: from statistical physics to risk management. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  20. Box GEP, Jenkins GM, Reinsel GC (1994) Time-series analysis. Prentice Hall, Englewood CliffszbMATHGoogle Scholar
  21. Bryce RM, Sprague KB (2012) Revisiting detrended fluctuation analysis. Sci Rep 2:315ADSCrossRefGoogle Scholar
  22. Bunde A, Havlin S (1994) Fractals in science. Springer, BerlinCrossRefzbMATHGoogle Scholar
  23. Bunde A, Havlin S, Kantelhardt JW, Penzel T, Peter JH, Voigt K (2000) Correlated and uncorrelated regions in heart-rate fluctuations during sleep. Phys Rev Lett 85:3736ADSCrossRefGoogle Scholar
  24. Bunde A, Kropp J, Schellnhuber HJ (2002) The science of disasters – climate disruptions, heart attacks, and market crashes. Springer, BerlinGoogle Scholar
  25. Bunde A, Eichner JF, Kantelhardt JW, Havlin S (2003) The effect of long-term correlations on the return periods of rare events. Phys A 330:1MathSciNetCrossRefzbMATHGoogle Scholar
  26. Bunde A, Eichner JF, Kantelhardt JW, Havlin S (2005) Long-term memory: a natural mechanism for the clustering of extreme events and anomalous residual times in climate records. Phys Rev Lett 94:048701ADSCrossRefGoogle Scholar
  27. Carbone A, Castelli G, Stanley HE (2004a) Analysis of clusters formed by the moving average of a long-range correlated time series. Phys Rev E 69:026105ADSCrossRefGoogle Scholar
  28. Carbone A, Castelli G, Stanley HE (2004b) Time-dependent Hurst exponent in financial time series. Phys A 344:267MathSciNetCrossRefGoogle Scholar
  29. Chatfield C (2003) The analysis of time series. An introduction. Taylor & Francis, Boca RatonzbMATHGoogle Scholar
  30. Chen Z, Ivanov PC, Hu K, Stanley HE (2002) Effect of non-stationarities on detrended fluctuation analysis. Phys Rev E 65:041107ADSCrossRefGoogle Scholar
  31. Chen Z, Hu K, Carpena P, Bernaola-Galvan P, Stanley HE, Ivanov PC (2005) Effect of nonlinear filters on detrended fluctuation analysis. Phys Rev E 71:011104ADSCrossRefGoogle Scholar
  32. Chianca CV, Ticona A, Penna TJP (2005) Fourier-detrended fluctuation analysis. Phys A 357:447CrossRefGoogle Scholar
  33. Daubechies I (1988) Orthogonal bases of compactly supported wavelets. Commun Pure Appl Math 41:909MathSciNetCrossRefzbMATHGoogle Scholar
  34. Delignieresa D, Ramdania S, Lemoinea L, Torrea K, Fortesb M, Ninot G (2006) Fractal analyses for ‘short’ time series: a re-assessment of classical methods. J Math Psychol 50:525CrossRefMathSciNetGoogle Scholar
  35. Eichner JF, Kantelhardt JW, Bunde A, Havlin S (2006) Extreme value statistics in records with long-term persistence. Phys Rev E 73:016130ADSCrossRefGoogle Scholar
  36. Eichner JF, Kantelhardt JW, Bunde A, Havlin S (2007) Statistics of return intervals in long-term correlated records. Phys Rev E 75:011128ADSCrossRefGoogle Scholar
  37. Feder J (1988) Fractals. Plenum Press, New YorkCrossRefzbMATHGoogle Scholar
  38. Fisher RA, Tippett LHC (1928) Limiting forms of the frequency distribution of the largest or smallest member of a sample. Proc Camb Philol Soc 24:180ADSCrossRefzbMATHGoogle Scholar
  39. Galambos J (1978) The asymptotic theory of extreme order statistics. Wiley, New YorkzbMATHGoogle Scholar
  40. Galambos J, Lechner J, Simin E (1994) Extreme value theory and applications. Kluwer, DordrechtCrossRefGoogle Scholar
  41. Gieraltowski J, Zebrowski JJ, Baranowski R (2012) Multiscale multifractal analysis of heart rate variability recordings with a large number of occurrences of arrhythmia. Phys Rev E 85:021915ADSCrossRefGoogle Scholar
  42. Goupillaud P, Grossmann A, Morlet J (1984) Cycle-octave and related transforms in seismic signal analysis. Geoexploration 23:85CrossRefGoogle Scholar
  43. Grau-Carles P (2006) Bootstrap testing for detrended fluctuation analysis. Phys A 360:89CrossRefGoogle Scholar
  44. Grech D, Mazur Z (2005) Statistical properties of old and new techniques in detrended analysis of time series. Acta Phys Pol B 36:2403ADSGoogle Scholar
  45. Grech D, Mazur Z (2013) On the scaling ranges of detrended fluctuation analysis for long-term memory correlated short series of data. Phys A 392:2384CrossRefGoogle Scholar
  46. Gu GF, Zhou WX (2006) Detrended fluctuation analysis for fractals and multifractals in higher dimensions. Phys Rev E 74:061104ADSCrossRefGoogle Scholar
  47. Gu GF, Zhou WX (2010) Detrending moving average algorithm for multifractals. Phys Rev E 82:011136ADSCrossRefGoogle Scholar
  48. Gulich D, Zunino L (2012) The effects of observational correlated noises on multifractal detrended fluctuation analysis. Phys A 391:4100CrossRefGoogle Scholar
  49. Gumbel EJ (1958) Statistics of extremes. Columbia University Press, New YorkzbMATHGoogle Scholar
  50. He LY, Qian WB (2012) A Monte Carlo simulation to the performance of the R/S and V/S methods-Statistical revisit and real world application. Phys A 391:3770CrossRefGoogle Scholar
  51. Heneghan C, McDarby G (2000) Establishing the relation between detrended fluctuation analysis and power spectral density analysis for stochastic processes. Phys Rev E 62:6103ADSCrossRefGoogle Scholar
  52. Hu K, Ivanov PC, Chen Z, Carpena P, Stanley HE (2001) Effect of trends on detrended fluctuation analysis. Phys Rev E 64:011114ADSCrossRefGoogle Scholar
  53. Hunt GA (1951) Random Fourier transforms. Trans Am Math Soc 71:38CrossRefMathSciNetzbMATHGoogle Scholar
  54. Hurst HE (1951) Long-term storage capacity of reservoirs. Tran Am Soc Civ Eng 116:770Google Scholar
  55. Hurst HE, Black RP, Simaika YM (1965) Long-term storage: an experimental study. Constable, LondonGoogle Scholar
  56. Ivanov PC, Amaral LAN, Goldberger AL, Havlin S, Rosenblum MG, Struzik ZR, Stanley HE (1999) Multifractality in human heartbeat dynamics. Nature 399:461ADSCrossRefGoogle Scholar
  57. Jánosi IM, Müller R (2005) Empirical mode decomposition and correlation properties of long daily ozone records. Phys Rev E 71:056126ADSCrossRefGoogle Scholar
  58. Jorgenssen PET (2000) Analysis and probability: wavelets, signals, fractals. Springer, BerlinGoogle Scholar
  59. Kalisky T, Ashkenazy Y, Havlin S (2005) Volatility of linear and nonlinear time series. Phys Rev E 72:011913MathSciNetADSCrossRefGoogle Scholar
  60. Kantelhardt JW, Roman HE, Greiner M (1995) Discrete wavelet approach to multifractality. Phys A 220:219CrossRefGoogle Scholar
  61. Kantelhardt JW, Koscielny-Bunde E, Rego HHA, Havlin S, Bunde A (2001) Detecting long-range correlations with detrended fluctuation analysis. Phys A 295:441CrossRefzbMATHGoogle Scholar
  62. Kantelhardt JW, Zschiegner SA, Bunde A, Havlin S, Koscielny-Bunde E, Stanley HE (2002) Multifractal detrended fluctuation analysis of non-stationary time series. Phys A 316:87CrossRefzbMATHGoogle Scholar
  63. Kantelhardt JW, Rybski D, Zschiegner SA, Braun P, Koscielny-Bunde E, Livina V, Havlin S, Bunde A (2003) Multifractality of river runoff and precipitation: comparison of fluctuation analysis and wavelet methods. Phys A 330:240CrossRefzbMATHGoogle Scholar
  64. Kantelhardt JW, Koscielny-Bunde E, Rybski D, Braun P, Bunde A, Havlin S (2006) Long-term persistence and multifractality of precipitation and river runoff records. J Geophys Res Atmos 111:D01106ADSCrossRefzbMATHGoogle Scholar
  65. Kantz H, Schreiber T (2003) Nonlinear time series analysis. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  66. Kiyono K, Struzik ZR, Aoyagi N, Togo F, Yamamoto Y (2005) Phase transition in a healthy human heart rate. Phys Rev Lett 95:058101ADSCrossRefGoogle Scholar
  67. Koscielny-Bunde E, Bunde A, Havlin S, Roman HE, Goldreich Y, Schellnhuber HJ (1998) Indication of a universal persistence law governing atmospheric variability. Phys Rev Lett 81:729ADSCrossRefGoogle Scholar
  68. Koscielny-Bunde E, Kantelhardt JW, Braun P, Bunde A, Havlin S (2006) Long-term persistence and multifractality of river runoff records. J Hydrol 322:120CrossRefzbMATHGoogle Scholar
  69. Leadbetter MR, Lindgren G, Rootzen H (1983) Extremes and related properties of random sequences and processes. Springer, New YorkCrossRefzbMATHGoogle Scholar
  70. Ludescher J, Bogachev MI, Kantelhardt JW, Schumann AY, Bunde A (2011) On spurious and corrupted multifractality: the effects of additive noise, short-term memory and periodic trends. Phys A 390:2480CrossRefGoogle Scholar
  71. Makse HA, Havlin S, Schwartz M, Stanley HE (1996) Method for generating long-range correlations for large systems. Phys Rev E 53:5445ADSCrossRefzbMATHGoogle Scholar
  72. Mandelbrot BB (1971) A fast fractional Gaussian noise generator. Water Resour Res 7:543ADSCrossRefGoogle Scholar
  73. Mandelbrot BB (1999) Multifractals and 1/f noise: wild self-affinity in physics. Springer, BerlinCrossRefzbMATHGoogle Scholar
  74. Mandelbrot BB, van Ness JW (1968) Fractional Brownian motions, fractional noises and applications. SIAM Rev 10:422MathSciNetADSCrossRefzbMATHGoogle Scholar
  75. Mandelbrot BB, Wallis JR (1969) Some long-run properties of geophysical records. Water Resour Res 5:321–340ADSCrossRefGoogle Scholar
  76. Mantegna RN, Stanley HE (2000) An introduction to econophysics – correlations and complexity in finance. Cambridge University Press, CambridgezbMATHGoogle Scholar
  77. Mielniczuk J, Wojdyllo P (2007) Estimation of Hurst exponent revisited. Comput Stat Data Anal 51:4510MathSciNetCrossRefzbMATHGoogle Scholar
  78. Mirzayof D, Ashkenazy Y (2010) Preservation of long range temporal correlations under extreme random dilution. Phys A 389:5573CrossRefGoogle Scholar
  79. Muzy JF, Bacry E, Arneodo A (1991) Wavelets and multifractal formalism for singular signals: application to turbulence data. Phys Rev Lett 67:3515ADSCrossRefGoogle Scholar
  80. Muzy JF, Bacry E, Arneodo A (1994) The multifractal formalism revisited with wavelets. Int J Bifurcation Chaos 4:245MathSciNetCrossRefzbMATHGoogle Scholar
  81. Nagarajan R (2006a) Effect of coarse-graining on detrended fluctuation analysis. Phys A 363:226CrossRefGoogle Scholar
  82. Nagarajan R (2006b) Reliable scaling exponent estimation of long-range correlated noise in the presence of random spikes. Phys A 366:1CrossRefGoogle Scholar
  83. Nagarajan R, Kavasseri RG (2005) Minimizing the effect of trends on detrended fluctuation analysis of long-range correlated noise. Phys A 354:182CrossRefzbMATHGoogle Scholar
  84. Newell GF, Rosenblatt M (1962) Zero crossing probabilities for Gaussian stationary processes. Ann Math Stat 33:1306MathSciNetCrossRefzbMATHGoogle Scholar
  85. Oswiecimka P, Kwapien J, Drozdz S (2006) Wavelet versus detrended fluctuation analysis of multifractal structures. Phys Rev E 74:016103ADSCrossRefGoogle Scholar
  86. Peitgen HO, Jürgens H, Saupe D (2004) Chaos and fractals. Springer, BerlinCrossRefzbMATHGoogle Scholar
  87. Peng CK, Buldyrev SV, Goldberger AL, Havlin S, Sciortino F, Simons M, Stanley HE (1992) Long-range correlations in nucleotide sequences. Nature 356:168ADSCrossRefGoogle Scholar
  88. Peng CK, Mietus J, Hausdorff JM, Havlin S, Stanley HE, Goldberger AL (1993) Long-range anti-correlations and non-Gaussian behaviour of the heartbeat. Phys Rev Lett 70:1343ADSCrossRefGoogle Scholar
  89. Peng C-K, Buldyrev SV, Havlin S, Simons M, Stanley HE, Goldberger AL (1994) Mosaic organization of DNA nucleotides. Phys Rev E 49:1685ADSCrossRefGoogle Scholar
  90. Qian XY, Gu GF, Zhou WX (2011) Modified detrended fluctuation analysis based on empirical mode decomposition for the characterization of anti-persistent processes. Phys A 390:4388CrossRefGoogle Scholar
  91. Rangarajan G, Ding M (2000) Integrated approach to the assessment of long range correlation in time series data. Phys Rev E 61:4991MathSciNetADSCrossRefGoogle Scholar
  92. Rasmussen PF, Gautam N (2003) Alternative PWM-estimators of the Gumbel distribution. J Hydrol 280:265CrossRefGoogle Scholar
  93. Raudkivi AJ (1979) Hydrology. Pergamon Press, OxfordGoogle Scholar
  94. Rivera-Castro MA, Miranda JGV, Cajueiro DO, Andrade RFS (2012) Detecting switching points using asymmetric detrended fluctuation analysis. Phys A 391:170CrossRefGoogle Scholar
  95. Rodriguez E, Echeverria JC, Alvarez-Ramirez J (2007) Detrending fluctuation analysis based on high-pass filtering. Phys A 375:699CrossRefGoogle Scholar
  96. Rodriguez-Iturbe I, Rinaldo A (1997) Fractal river basins – change and self-organization. Cambridge University Press, CambridgeGoogle Scholar
  97. Santhanam MS, Bandyopadhyay JN, Angom D (2006) Quantum spectrum as a time series: fluctuation measures. Phys Rev E 73:015201ADSCrossRefGoogle Scholar
  98. Schmitt DT, Schulz M (2006) Analyzing memory effects of complex systems from time series. Phys Rev E 73:056204ADSCrossRefGoogle Scholar
  99. Schreiber T, Schmitz A (1996) Improved surrogate data for nonlinearity tests. Phys Rev Lett 77:635ADSCrossRefGoogle Scholar
  100. Schreiber T, Schmitz A (2000) Surrogate time series. Physica D 142:346MathSciNetADSCrossRefzbMATHGoogle Scholar
  101. Schumann AY, Kantelhardt JW (2011) Multifractal moving average analysis and test of multifractal model with tuned correlations. Phys A 390:2637CrossRefGoogle Scholar
  102. Shao YH, Gu GF, Jiang ZQ, Zhou WX, Sornette D (2012) Comparing the performance of FA, DFA and DMA using different synthetic long-range correlated time series. Sci Rep 2:835ADSCrossRefGoogle Scholar
  103. Sornette D (2004) Critical phenomena in natural sciences. Springer, BerlinzbMATHGoogle Scholar
  104. Sornette D, Knopoff L (1997) The paradox of the expected time until the next earthquake. Bull Seismol Soc Am 87:789Google Scholar
  105. Staudacher M, Telser S, Amann A, Hinterhuber H, Ritsch-Marte M (2005) A new method for change-point detection developed for on-line analysis of the heart beat variability during sleep. Phys A 349:582CrossRefGoogle Scholar
  106. Storch HV, Zwiers FW (2001) Statistical analysis in climate research. Cambridge University Press, CambridgeGoogle Scholar
  107. Sun J, Sheng H (2011) A hybrid detrending method for fractional Gaussian noise. Phys A 390:2995CrossRefGoogle Scholar
  108. Taqqu MS, Teverovsky V, Willinger W (1995) Estimators for long-range dependence: an empirical study. Fractals 3:785CrossRefzbMATHGoogle Scholar
  109. te Chow V (1964) Handbook of applied hydrology. McGraw-Hill, New YorkGoogle Scholar
  110. Telser S, Staudacher M, Hennig B, Ploner Y, Amann A, Hinterhuber H, Ritsch-Marte M (2007) Temporally resolved fluctuation analysis of sleep-ECG. J Biol Phys 33:190CrossRefGoogle Scholar
  111. Voss RF (1985) Random fractal forgeries. In: Earnshaw RA (ed) Fundamental algorithms in computer graphics. Springer, Berlin, pp 805–835Google Scholar
  112. Vyushin D, Zhidkov I, Havlin S, Bunde A, Brenner S (2004) Volcanic forcing improves atmosphere–ocean coupled general circulation model scaling performance. Geophys Res Lett 31:L10206ADSGoogle Scholar
  113. Weron R (2002) Estimating long-range dependence: finite sample properties and confidence intervals. Phys A 312:285MathSciNetCrossRefzbMATHGoogle Scholar
  114. Xu L, Ivanov PC, Hu K, Chen Z, Carbone A, Stanley HE (2005) Quantifying signals with power-law correlations: a comparative study of detrended fluctuation analysis and detrended moving average techniques. Phys Rev E 71:051101ADSCrossRefGoogle Scholar
  115. Xu Y, Ma QDY, Schmitt DT, Bernaola-Galvan P, Ivanov PC (2011) Effects of coarse-graining on the scaling behavior of long-range correlated and anti-correlated signals. Phys A 390:4057CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Institute of PhysicsMartin-Luther-University Halle-WittenbergHalle (Saale)Germany