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Fractal and Multifractal Time Series

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Encyclopedia of Complexity and Systems Science

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 bothequilibrium and non‐equilibrium situations, the natural fluctuations are often found to follow a scaling relation over several orders ofmagnitude. Such scaling laws allow for a characterization of the data and the generating complex system by fractal (or multifractal) scalingexponents, which can serve as characteristic fingerprints of the systems in comparisons with other systems and with models. Fractal scaling behavior hasbeen observed, e. g., in many data series from experimental physics, geophysics, medicine, physiology, and even social sciences. Although theunderlying causes of the observed fractal scaling are often not known in detail, the fractal or multifractal characterization can be used for generatingsurrogate (test) data, modeling the time series, and deriving predictions regarding extreme events or future...

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Abbreviations

Time series:

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

Complex system:

A system consisting of many non‐linearly interacting components. It cannot be split into simpler sub‐systems without tampering with the dynamical properties.

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 function of a scale parameter s (e. g., time scale, frequency) at least asymptotically: \( { F(s) \sim s^\alpha } \). The power law should be valid for a large range of s values, e. g., at least for one order of magnitude.

Fractal system:

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

Self‐affine system:

Generalization of a fractal system, where different magnifications s and \( { s^\prime=s^H } \) 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.

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.

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 \( { s_\times } \) in this article.

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.

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.

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.

Non‐stationarities:

If the mean or the standard deviation of the data values change 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.

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Acknowledgment

We thank Ronny Bartsch, Amir Bashan, Mikhail Bogachev, Armin Bunde, Jan Eichner, Shlomo Havlin, Diego Rybski, Aicko Schumann, and StephanZschiegner for helpful discussions and contribution. This work has been supported by the Deutsche Forschungsgemeinschaft (grant KA 1676/3) and the European Union (STREPproject DAPHNet, grant 018474-2).

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  1. Institute of Physics, Martin-Luther-University Halle-Wittenberg, Halle, Germany

    Jan W. Kantelhardt

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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Kantelhardt, J.W. (2009). Fractal and Multifractal Time Series. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_221

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