Fractal and Multifractal Time Series

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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., a) 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. 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 s0 = s have to be used for different directions in order to obtain a statistically equivalent magnification. The exponentH 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.