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Fractal time series – A tutorial review. (English) Zbl 1191.37002
Summary: Fractal time series substantially differs from conventional one in its statistic properties. For instance, it may have a heavy-tailed probability distribution function (PDF), a slowly decayed autocorrelation function (ACF), and a power spectrum function (PSD) of $1/f$ type. It may have the statistical dependence, either long-range dependence (LRD) or short-range dependence (SRD), and global or local self-similarity. This article will give a tutorial review about those concepts. Note that a conventional time series can be regarded as the solution to a differential equation of integer order with the excitation of white noise in mathematics. In engineering, such as mechanical engineering or electronics engineering, engineers may usually consider it as the output or response of a differential system or filter of integer order under the excitation of white noise. In this paper, a fractal time series is taken as the solution to a differential equation of fractional order or a response of a fractional system or a fractional filter driven with a white noise in the domain of stochastic processes.
##### MSC:
 37-01 Instructional exposition (Dynamical systems and ergodic theory) 37M10 Time series analysis (dynamical systems) 37Jxx Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems 28Axx Classical measure theory