Skip to main content
SpringerLink
Log in
Book cover

Encyclopedia of Complexity and Systems Science pp 1–37Cite as

Fractal and Multifractal Time Series

  • Living reference work entry
  • First Online:

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...

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

Abbreviations

Complex system:

A system consisting of many nonlinearly interacting components. It cannot be split into simpler subsystems without tampering with the dynamical properties.

Crossover:

Change point in a scaling law, where one scaling exponent applies for small-scale parameters and another scaling exponent applies for large-scale parameters. The center of the crossover is denoted by its characteristic scale parameter s × in this entry.

Fractal system:

A system characterized by a scaling law with a fractal, i.e., a non-integer exponent. Fractal systems are self-similar, i.e., a magnification of a small part is statistically equivalent to the whole.

Long-term correlations:

Correlations that decay sufficiently slow that a characteristic correlation time scale cannot be defined; e.g., power-law correlations with an exponent between 0 and 1. Power-law scaling is observed on large time scales and asymptotically. The term long-range correlations should be used if the data is not a time series.

Multifractal system:

A system characterized by scaling laws with an infinite number of different fractal exponents. The scaling laws must be valid for the same range of the scale parameter.

Non-stationarities:

If the mean or the standard deviation of the data values changes with time, the weak definition of stationarity is violated. The strong definition of stationarity requires that all moments remain constant, i.e., the distribution density of the values does not change with time. Non-stationarities like monotonous, periodic, or step-like trends are often caused by external effects. In a more general sense, changes in the dynamics of the system also represent non-stationarities.

Persistence:

In a persistent time series, a large value is usually (i.e., with high statistical preference) followed by a large value and a small value is followed by a small value. A fractal scaling law holds at least for a limited range of scales.

Scaling law:

A power law with a scaling exponent (e.g., α) describing the behavior of a quantity F (e.g., fluctuation, spectral power) as a function of a scale parameter s (e.g., time scale, frequency) at least asymptotically: F(s) ∼ s α. The power law should be valid for a large range of s values, e.g., at least for one order of magnitude.

Self-affine system:

Generalization of a fractal system, where different magnifications s and s ′ = s H have to be used for different directions in order to obtain a statistically equivalent magnification. The exponent H is called Hurst exponent. Self-affine time series and time series becoming self-affine upon integration are commonly denoted as fractal using a less strict terminology.

Short-term correlations:

Correlations that decay sufficiently fast that they can be described by a characteristic correlation time scale; e.g., exponentially decaying correlations. A crossover to uncorrelated behavior is observed on larger scales.

Time series:

One-dimensional array of numbers (x i ), i = 1,…, N, representing values of an observable x usually measured equidistant (or nearly equidistant) in time.

Bibliography

  • Alessio E, Carbone A, Castelli G, Frappietro V (2002) Second-order moving average and scaling of stochastic time series. Eur Phys J B 27:197

    ADS  Google Scholar 

  • Altmann EG, Kantz H (2005) Recurrence time analysis, long-term correlations, and extreme events. Phys Rev E 71:056106

    Article  MathSciNet  ADS  Google Scholar 

  • Alvarez-Ramirez J, Rodriguez E, Echeverría JC (2005) Detrending fluctuation analysis based on moving average filtering. Phys A 354:199

    Article  Google Scholar 

  • Alvarez-Ramirez J, Rodriguez E, Echeverria JC (2009a) A DFA approach for assessing asymmetric correlations. Phys A 388:2263

    Article  Google Scholar 

  • Alvarez-Ramirez J, Rodriguez E, Echeverria JC (2009b) Using detrended fluctuation analysis for lagged correlation analysis of nonstationary signals. Phys Rev E 79:057202

    Article  ADS  Google Scholar 

  • 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 86:6026

    Article  ADS  Google Scholar 

  • Arneodo A, Bacry E, Graves PV, Muzy JF (1995) Characterizing long-range correlations in DNA sequences from wavelet analysis. Phys Rev Lett 74:3293

    Article  ADS  Google Scholar 

  • Arneodo A, Manneville S, Muzy JF (1998) Towards log-normal statistics in high Reynolds number turbulence. Eur Phys J B 1:129

    Article  ADS  Google Scholar 

  • 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, Berlin

    Google Scholar 

  • Ashkenazy Y, Ivanov PC, Havlin S, Peng CK, Goldberger AL, Stanley HE (2001) Magnitude and sign correlations in heartbeat fluctuations. Phys Rev Lett 86:1900

    Article  ADS  Google Scholar 

  • 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 323:19

    Article  MATH  Google Scholar 

  • Bacry E, Delour J, Muzy JF (2001) Multifractal random walk. Phys Rev E 64:026103

    Article  ADS  MATH  Google Scholar 

  • 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 56:454

    Article  ADS  Google Scholar 

  • Barabási AL, Vicsek T (1991) Multifractality of self-affine fractals. Phys Rev A 44:2730

    Article  ADS  Google Scholar 

  • Barnsley MF (1993) Fractals everywhere. Academic, San Diego

    MATH  Google Scholar 

  • Bartsch R, Henning T, Heinen A, Heinrichs S, Maass P (2005) Statistical analysis of fluctuations in the ECG morphology. Phys A 354:415

    Article  Google Scholar 

  • Bashan A, Bartsch R, Kantelhardt JW, Havlin S (2008) Comparison of detrending methods for fluctuation analysis. Phys A 387:5080

    Article  Google Scholar 

  • Bogachev MI, Eichner JF, Bunde A (2007) Effect of nonlinear correlations on the statistics of return intervals in multifractal data sets. Phys Rev Lett 99:240601

    Article  ADS  Google Scholar 

  • Bouchaud JP, Potters M (2003) Theory of financial risks: from statistical physics to risk management. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Box GEP, Jenkins GM, Reinsel GC (1994) Time-series analysis. Prentice Hall, Englewood Cliffs

    MATH  Google Scholar 

  • Bryce RM, Sprague KB (2012) Revisiting detrended fluctuation analysis. Sci Rep 2:315

    Article  ADS  Google Scholar 

  • Bunde A, Havlin S (1994) Fractals in science. Springer, Berlin

    Book  MATH  Google Scholar 

  • 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 85:3736

    Article  ADS  Google Scholar 

  • Bunde A, Kropp J, Schellnhuber HJ (2002) The science of disasters – climate disruptions, heart attacks, and market crashes. Springer, Berlin

    Google Scholar 

  • Bunde A, Eichner JF, Kantelhardt JW, Havlin S (2003) The effect of long-term correlations on the return periods of rare events. Phys A 330:1

    Article  MathSciNet  MATH  Google Scholar 

  • 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 94:048701

    Article  ADS  Google Scholar 

  • 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 69:026105

    Article  ADS  Google Scholar 

  • Carbone A, Castelli G, Stanley HE (2004b) Time-dependent Hurst exponent in financial time series. Phys A 344:267

    Article  MathSciNet  Google Scholar 

  • Chatfield C (2003) The analysis of time series. An introduction. Taylor & Francis, Boca Raton

    MATH  Google Scholar 

  • Chen Z, Ivanov PC, Hu K, Stanley HE (2002) Effect of non-stationarities on detrended fluctuation analysis. Phys Rev E 65:041107

    Article  ADS  Google Scholar 

  • 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 71:011104

    Article  ADS  Google Scholar 

  • Chianca CV, Ticona A, Penna TJP (2005) Fourier-detrended fluctuation analysis. Phys A 357:447

    Article  Google Scholar 

  • Daubechies I (1988) Orthogonal bases of compactly supported wavelets. Commun Pure Appl Math 41:909

    Article  MathSciNet  MATH  Google Scholar 

  • 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 50:525

    Article  MathSciNet  Google Scholar 

  • Eichner JF, Kantelhardt JW, Bunde A, Havlin S (2006) Extreme value statistics in records with long-term persistence. Phys Rev E 73:016130

    Article  ADS  Google Scholar 

  • Eichner JF, Kantelhardt JW, Bunde A, Havlin S (2007) Statistics of return intervals in long-term correlated records. Phys Rev E 75:011128

    Article  ADS  Google Scholar 

  • Feder J (1988) Fractals. Plenum Press, New York

    Book  MATH  Google Scholar 

  • Fisher RA, Tippett LHC (1928) Limiting forms of the frequency distribution of the largest or smallest member of a sample. Proc Camb Philol Soc 24:180

    Article  ADS  MATH  Google Scholar 

  • Galambos J (1978) The asymptotic theory of extreme order statistics. Wiley, New York

    MATH  Google Scholar 

  • Galambos J, Lechner J, Simin E (1994) Extreme value theory and applications. Kluwer, Dordrecht

    Book  Google Scholar 

  • 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 85:021915

    Article  ADS  Google Scholar 

  • Goupillaud P, Grossmann A, Morlet J (1984) Cycle-octave and related transforms in seismic signal analysis. Geoexploration 23:85

    Article  Google Scholar 

  • Grau-Carles P (2006) Bootstrap testing for detrended fluctuation analysis. Phys A 360:89

    Article  Google Scholar 

  • Grech D, Mazur Z (2005) Statistical properties of old and new techniques in detrended analysis of time series. Acta Phys Pol B 36:2403

    ADS  Google Scholar 

  • Grech D, Mazur Z (2013) On the scaling ranges of detrended fluctuation analysis for long-term memory correlated short series of data. Phys A 392:2384

    Article  Google Scholar 

  • Gu GF, Zhou WX (2006) Detrended fluctuation analysis for fractals and multifractals in higher dimensions. Phys Rev E 74:061104

    Article  ADS  Google Scholar 

  • Gu GF, Zhou WX (2010) Detrending moving average algorithm for multifractals. Phys Rev E 82:011136

    Article  ADS  Google Scholar 

  • Gulich D, Zunino L (2012) The effects of observational correlated noises on multifractal detrended fluctuation analysis. Phys A 391:4100

    Article  Google Scholar 

  • Gumbel EJ (1958) Statistics of extremes. Columbia University Press, New York

    MATH  Google Scholar 

  • 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 391:3770

    Article  Google Scholar 

  • Heneghan C, McDarby G (2000) Establishing the relation between detrended fluctuation analysis and power spectral density analysis for stochastic processes. Phys Rev E 62:6103

    Article  ADS  Google Scholar 

  • Hu K, Ivanov PC, Chen Z, Carpena P, Stanley HE (2001) Effect of trends on detrended fluctuation analysis. Phys Rev E 64:011114

    Article  ADS  Google Scholar 

  • Hunt GA (1951) Random Fourier transforms. Trans Am Math Soc 71:38

    Article  MathSciNet  MATH  Google Scholar 

  • Hurst HE (1951) Long-term storage capacity of reservoirs. Tran Am Soc Civ Eng 116:770

    Google Scholar 

  • Hurst HE, Black RP, Simaika YM (1965) Long-term storage: an experimental study. Constable, London

    Google Scholar 

  • Ivanov PC, Amaral LAN, Goldberger AL, Havlin S, Rosenblum MG, Struzik ZR, Stanley HE (1999) Multifractality in human heartbeat dynamics. Nature 399:461

    Article  ADS  Google Scholar 

  • Jánosi IM, Müller R (2005) Empirical mode decomposition and correlation properties of long daily ozone records. Phys Rev E 71:056126

    Article  ADS  Google Scholar 

  • Jorgenssen PET (2000) Analysis and probability: wavelets, signals, fractals. Springer, Berlin

    Google Scholar 

  • Kalisky T, Ashkenazy Y, Havlin S (2005) Volatility of linear and nonlinear time series. Phys Rev E 72:011913

    Article  MathSciNet  ADS  Google Scholar 

  • Kantelhardt JW, Roman HE, Greiner M (1995) Discrete wavelet approach to multifractality. Phys A 220:219

    Article  Google Scholar 

  • Kantelhardt JW, Koscielny-Bunde E, Rego HHA, Havlin S, Bunde A (2001) Detecting long-range correlations with detrended fluctuation analysis. Phys A 295:441

    Article  MATH  Google Scholar 

  • 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 316:87

    Article  MATH  Google Scholar 

  • 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 330:240

    Article  MATH  Google Scholar 

  • 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:D01106

    Article  ADS  MATH  Google Scholar 

  • Kantz H, Schreiber T (2003) Nonlinear time series analysis. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Kiyono K, Struzik ZR, Aoyagi N, Togo F, Yamamoto Y (2005) Phase transition in a healthy human heart rate. Phys Rev Lett 95:058101

    Article  ADS  Google Scholar 

  • 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 81:729

    Article  ADS  Google Scholar 

  • Koscielny-Bunde E, Kantelhardt JW, Braun P, Bunde A, Havlin S (2006) Long-term persistence and multifractality of river runoff records. J Hydrol 322:120

    Article  MATH  Google Scholar 

  • Leadbetter MR, Lindgren G, Rootzen H (1983) Extremes and related properties of random sequences and processes. Springer, New York

    Book  MATH  Google Scholar 

  • 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 390:2480

    Article  Google Scholar 

  • Makse HA, Havlin S, Schwartz M, Stanley HE (1996) Method for generating long-range correlations for large systems. Phys Rev E 53:5445

    Article  ADS  MATH  Google Scholar 

  • Mandelbrot BB (1971) A fast fractional Gaussian noise generator. Water Resour Res 7:543

    Article  ADS  Google Scholar 

  • Mandelbrot BB (1999) Multifractals and 1/f noise: wild self-affinity in physics. Springer, Berlin

    Book  MATH  Google Scholar 

  • Mandelbrot BB, van Ness JW (1968) Fractional Brownian motions, fractional noises and applications. SIAM Rev 10:422

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Mandelbrot BB, Wallis JR (1969) Some long-run properties of geophysical records. Water Resour Res 5:321–340

    Article  ADS  Google Scholar 

  • Mantegna RN, Stanley HE (2000) An introduction to econophysics – correlations and complexity in finance. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Mielniczuk J, Wojdyllo P (2007) Estimation of Hurst exponent revisited. Comput Stat Data Anal 51:4510

    Article  MathSciNet  MATH  Google Scholar 

  • Mirzayof D, Ashkenazy Y (2010) Preservation of long range temporal correlations under extreme random dilution. Phys A 389:5573

    Article  Google Scholar 

  • Muzy JF, Bacry E, Arneodo A (1991) Wavelets and multifractal formalism for singular signals: application to turbulence data. Phys Rev Lett 67:3515

    Article  ADS  Google Scholar 

  • Muzy JF, Bacry E, Arneodo A (1994) The multifractal formalism revisited with wavelets. Int J Bifurcation Chaos 4:245

    Article  MathSciNet  MATH  Google Scholar 

  • Nagarajan R (2006a) Effect of coarse-graining on detrended fluctuation analysis. Phys A 363:226

    Article  Google Scholar 

  • Nagarajan R (2006b) Reliable scaling exponent estimation of long-range correlated noise in the presence of random spikes. Phys A 366:1

    Article  Google Scholar 

  • Nagarajan R, Kavasseri RG (2005) Minimizing the effect of trends on detrended fluctuation analysis of long-range correlated noise. Phys A 354:182

    Article  MATH  Google Scholar 

  • Newell GF, Rosenblatt M (1962) Zero crossing probabilities for Gaussian stationary processes. Ann Math Stat 33:1306

    Article  MathSciNet  MATH  Google Scholar 

  • Oswiecimka P, Kwapien J, Drozdz S (2006) Wavelet versus detrended fluctuation analysis of multifractal structures. Phys Rev E 74:016103

    Article  ADS  Google Scholar 

  • Peitgen HO, Jürgens H, Saupe D (2004) Chaos and fractals. Springer, Berlin

    Book  MATH  Google Scholar 

  • Peng CK, Buldyrev SV, Goldberger AL, Havlin S, Sciortino F, Simons M, Stanley HE (1992) Long-range correlations in nucleotide sequences. Nature 356:168

    Article  ADS  Google Scholar 

  • 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 70:1343

    Article  ADS  Google Scholar 

  • Peng C-K, Buldyrev SV, Havlin S, Simons M, Stanley HE, Goldberger AL (1994) Mosaic organization of DNA nucleotides. Phys Rev E 49:1685

    Article  ADS  Google Scholar 

  • 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 390:4388

    Article  Google Scholar 

  • Rangarajan G, Ding M (2000) Integrated approach to the assessment of long range correlation in time series data. Phys Rev E 61:4991

    Article  MathSciNet  ADS  Google Scholar 

  • Rasmussen PF, Gautam N (2003) Alternative PWM-estimators of the Gumbel distribution. J Hydrol 280:265

    Article  Google Scholar 

  • Raudkivi AJ (1979) Hydrology. Pergamon Press, Oxford

    Google Scholar 

  • Rivera-Castro MA, Miranda JGV, Cajueiro DO, Andrade RFS (2012) Detecting switching points using asymmetric detrended fluctuation analysis. Phys A 391:170

    Article  Google Scholar 

  • Rodriguez E, Echeverria JC, Alvarez-Ramirez J (2007) Detrending fluctuation analysis based on high-pass filtering. Phys A 375:699

    Article  Google Scholar 

  • Rodriguez-Iturbe I, Rinaldo A (1997) Fractal river basins – change and self-organization. Cambridge University Press, Cambridge

    Google Scholar 

  • Santhanam MS, Bandyopadhyay JN, Angom D (2006) Quantum spectrum as a time series: fluctuation measures. Phys Rev E 73:015201

    Article  ADS  Google Scholar 

  • Schmitt DT, Schulz M (2006) Analyzing memory effects of complex systems from time series. Phys Rev E 73:056204

    Article  ADS  Google Scholar 

  • Schreiber T, Schmitz A (1996) Improved surrogate data for nonlinearity tests. Phys Rev Lett 77:635

    Article  ADS  Google Scholar 

  • Schreiber T, Schmitz A (2000) Surrogate time series. Physica D 142:346

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Schumann AY, Kantelhardt JW (2011) Multifractal moving average analysis and test of multifractal model with tuned correlations. Phys A 390:2637

    Article  Google Scholar 

  • 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 2:835

    Article  ADS  Google Scholar 

  • Sornette D (2004) Critical phenomena in natural sciences. Springer, Berlin

    MATH  Google Scholar 

  • Sornette D, Knopoff L (1997) The paradox of the expected time until the next earthquake. Bull Seismol Soc Am 87:789

    Google Scholar 

  • 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 349:582

    Article  Google Scholar 

  • Storch HV, Zwiers FW (2001) Statistical analysis in climate research. Cambridge University Press, Cambridge

    Google Scholar 

  • Sun J, Sheng H (2011) A hybrid detrending method for fractional Gaussian noise. Phys A 390:2995

    Article  Google Scholar 

  • Taqqu MS, Teverovsky V, Willinger W (1995) Estimators for long-range dependence: an empirical study. Fractals 3:785

    Article  MATH  Google Scholar 

  • te Chow V (1964) Handbook of applied hydrology. McGraw-Hill, New York

    Google Scholar 

  • 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 33:190

    Article  Google Scholar 

  • Voss RF (1985) Random fractal forgeries. In: Earnshaw RA (ed) Fundamental algorithms in computer graphics. Springer, Berlin, pp 805–835

    Google Scholar 

  • 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:L10206

    ADS  Google Scholar 

  • Weron R (2002) Estimating long-range dependence: finite sample properties and confidence intervals. Phys A 312:285

    Article  MathSciNet  MATH  Google Scholar 

  • 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 71:051101

    Article  ADS  Google Scholar 

  • 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 390:4057

    Article  Google Scholar 

Download references

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).

Author information

Authors and Affiliations

  1. Institute of Physics, Martin-Luther-University Halle-Wittenberg, von-Seckendorff-Platz 1, 06110, Halle (Saale), Germany

    Jan W. Kantelhardt

Authors
  1. Jan W. Kantelhardt
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jan W. Kantelhardt .

Editor information

Editors and Affiliations

  1. Ramtech Ltd., Larkspur, California, USA

    Robert A. Meyers

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer Science+Business Media New York

About this entry

Cite this entry

Kantelhardt, J.W. (2015). Fractal and Multifractal Time Series. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27737-5_221-3

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-27737-5_221-3

  • Received:

  • Accepted:

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Online ISBN: 978-3-642-27737-5

  • eBook Packages: Springer Reference Physics and AstronomyReference Module Physical and Materials ScienceReference Module Chemistry, Materials and Physics

Publish with us

Policies and ethics

Search

Navigation